Vetor (matemática) – Wikipédia, a enciclopédia livre

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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Definição_formal" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definição_formal"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definição formal</span> </div> </a> <ul id="toc-Definição_formal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Importância_dos_vetores" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Importância_dos_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Importância dos vetores</span> </div> </a> <ul id="toc-Importância_dos_vetores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Módulo_ou_norma_do_vetor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Módulo_ou_norma_do_vetor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Módulo ou norma do vetor</span> </div> </a> <ul id="toc-Módulo_ou_norma_do_vetor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operações_com_vetores" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operações_com_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Operações com vetores</span> </div> </a> <button aria-controls="toc-Operações_com_vetores-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Operações com vetores</span> </button> <ul id="toc-Operações_com_vetores-sublist" class="vector-toc-list"> <li id="toc-Adição" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adição"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Adição</span> </div> </a> <ul id="toc-Adição-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subtração" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtração"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Subtração</span> </div> </a> <ul id="toc-Subtração-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicação_por_escalares[5]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplicação_por_escalares[5]"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Multiplicação por escalares<sup><span>[</span>5<span>]</span></sup></span> </div> </a> <ul id="toc-Multiplicação_por_escalares[5]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Produto_escalar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Produto_escalar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Produto escalar</span> </div> </a> <ul id="toc-Produto_escalar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriedades_do_produto_escalar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriedades_do_produto_escalar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Propriedades do produto escalar</span> </div> </a> <ul id="toc-Propriedades_do_produto_escalar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Produto_vetorial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Produto_vetorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Produto vetorial</span> </div> </a> <ul id="toc-Produto_vetorial-sublist" class="vector-toc-list"> <li id="toc-Propriedades_do_produto_vetorial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Propriedades_do_produto_vetorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6.1</span> <span>Propriedades do produto vetorial</span> </div> </a> <ul id="toc-Propriedades_do_produto_vetorial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ângulo_entre_dois_vetores" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ângulo_entre_dois_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Ângulo entre dois vetores</span> </div> </a> <ul id="toc-Ângulo_entre_dois_vetores-sublist" class="vector-toc-list"> <li id="toc-Relação_entre_o_ângulo_e_o_produto_escalar_de_dois_vetores" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relação_entre_o_ângulo_e_o_produto_escalar_de_dois_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7.1</span> <span>Relação entre o ângulo e o produto escalar de dois vetores</span> </div> </a> <ul id="toc-Relação_entre_o_ângulo_e_o_produto_escalar_de_dois_vetores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Demonstração" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Demonstração"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7.2</span> <span>Demonstração</span> </div> </a> <ul id="toc-Demonstração-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relação_entre_produto_vetorial_e_ângulo_entre_vetores" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relação_entre_produto_vetorial_e_ângulo_entre_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Relação entre produto vetorial e ângulo entre vetores</span> </div> </a> <ul id="toc-Relação_entre_produto_vetorial_e_ângulo_entre_vetores-sublist" class="vector-toc-list"> <li id="toc-Interpretação_do_produto_vetorial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Interpretação_do_produto_vetorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.1</span> <span>Interpretação do produto vetorial</span> </div> </a> <ul id="toc-Interpretação_do_produto_vetorial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ângulos_diretores" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ângulos_diretores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.2</span> <span>Ângulos diretores</span> </div> </a> <ul id="toc-Ângulos_diretores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projeções_sobre_vetores" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Projeções_sobre_vetores"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.3</span> <span>Projeções sobre vetores</span> </div> </a> <ul id="toc-Projeções_sobre_vetores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_desigualdade_de_Cauchy-Schwarz" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#A_desigualdade_de_Cauchy-Schwarz"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8.4</span> <span>A desigualdade de Cauchy-Schwarz</span> </div> </a> <ul id="toc-A_desigualdade_de_Cauchy-Schwarz-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Produto_misto" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Produto_misto"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Produto misto</span> </div> </a> <button aria-controls="toc-Produto_misto-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Produto misto</span> </button> <ul id="toc-Produto_misto-sublist" class="vector-toc-list"> <li id="toc-Propriedades_do_produto_misto" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriedades_do_produto_misto"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Propriedades do produto misto</span> </div> </a> <ul id="toc-Propriedades_do_produto_misto-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vetores_na_Física" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vetores_na_Física"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Vetores na Física</span> </div> </a> <button aria-controls="toc-Vetores_na_Física-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Vetores na Física</span> </button> <ul id="toc-Vetores_na_Física-sublist" class="vector-toc-list"> <li id="toc-Vetores_velocidade_e_aceleração" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vetores_velocidade_e_aceleração"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Vetores velocidade e aceleração</span> </div> </a> <ul id="toc-Vetores_velocidade_e_aceleração-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Velocidade_e_aceleração_relativas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocidade_e_aceleração_relativas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Velocidade e aceleração relativas</span> </div> </a> <ul id="toc-Velocidade_e_aceleração_relativas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Produto_escalar_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Produto_escalar_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Produto escalar</span> </div> </a> <ul id="toc-Produto_escalar_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vetores_deslizantes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vetores_deslizantes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Vetores deslizantes</span> </div> </a> <ul id="toc-Vetores_deslizantes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adição_de_forças" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adição_de_forças"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Adição de forças</span> </div> </a> <ul id="toc-Adição_de_forças-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Versor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Versor"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Versor</span> </div> </a> <ul id="toc-Versor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Alternar o índice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Vetor (matemática)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 95 línguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-95" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">95 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektor_(Wiskunde)" title="Vektor (Wiskunde) — africanês" lang="af" hreflang="af" data-title="Vektor (Wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Vektor" title="Vektor — alemão suíço" lang="gsw" hreflang="gsw" data-title="Vektor" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%A8%E1%88%A8%E1%88%AD" title="ጨረር — amárico" lang="am" hreflang="am" data-title="ጨረር" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%AC%D9%87" title="متجه — árabe" lang="ar" hreflang="ar" data-title="متجه" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Vector" title="Vector — asturiano" lang="ast" hreflang="ast" data-title="Vector" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Vektor_(h%C9%99nd%C9%99s%C9%99)" title="Vektor (həndəsə) — azerbaijano" lang="az" hreflang="az" data-title="Vektor (həndəsə)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DB%8C%D8%A4%D9%86%D8%A6%DB%8C_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="یؤنئی (هندسه) — South Azerbaijani" lang="azb" hreflang="azb" data-title="یؤنئی (هندسه)" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — bashkir" lang="ba" hreflang="ba" data-title="Вектор (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%B0%D1%80_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Вектар (матэматыка) — bielorrusso" lang="be" hreflang="be" data-title="Вектар (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%92%D1%8D%D0%BA%D1%82%D0%B0%D1%80" title="Вэктар — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Вэктар" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — búlgaro" lang="bg" hreflang="bg" data-title="Вектор" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%A6%E0%A6%BF%E0%A6%95_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF" title="সদিক রাশি — bengalês" lang="bn" hreflang="bn" data-title="সদিক রাশি" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Euklidski_vektor" title="Euklidski vektor — bósnio" lang="bs" hreflang="bs" data-title="Euklidski vektor" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Vector_(matem%C3%A0tiques)" title="Vector (matemàtiques) — catalão" lang="ca" hreflang="ca" data-title="Vector (matemàtiques)" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Hi%C3%B3ng-li%C3%B4ng" title="Hióng-liông — Mindong" lang="cdo" hreflang="cdo" data-title="Hióng-liông" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%D8%A7%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D8%A8%DA%95%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="ئاڕاستەبڕی ئیقلیدسی — curdo central" lang="ckb" hreflang="ckb" data-title="ئاڕاستەبڕی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektor" title="Vektor — checo" lang="cs" hreflang="cs" data-title="Vektor" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Вектор (геометри) — chuvash" lang="cv" hreflang="cv" data-title="Вектор (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Fector" title="Fector — galês" lang="cy" hreflang="cy" data-title="Fector" data-language-autonym="Cymraeg" data-language-local-name="galês" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Vektor_(geometri)" title="Vektor (geometri) — dinamarquês" lang="da" hreflang="da" data-title="Vektor (geometri)" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vektor" title="Vektor — alemão" lang="de" hreflang="de" data-title="Vektor" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%BF_%CE%B4%CE%B9%CE%AC%CE%BD%CF%85%CF%83%CE%BC%CE%B1" title="Ευκλείδειο διάνυσμα — grego" lang="el" hreflang="el" data-title="Ευκλείδειο διάνυσμα" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclidean_vector" title="Euclidean vector — inglês" lang="en" hreflang="en" data-title="Euclidean vector" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektoro" title="Vektoro — esperanto" lang="eo" hreflang="eo" data-title="Vektoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector" title="Vector — espanhol" lang="es" hreflang="es" data-title="Vector" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vektor" title="Vektor — estónio" lang="et" hreflang="et" data-title="Vektor" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore_(matematika)" title="Bektore (matematika) — basco" lang="eu" hreflang="eu" data-title="Bektore (matematika)" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="بردار اقلیدسی — persa" lang="fa" hreflang="fa" data-title="بردار اقلیدسی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektori" title="Vektori — finlandês" lang="fi" hreflang="fi" data-title="Vektori" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vecteur_euclidien" title="Vecteur euclidien — francês" lang="fr" hreflang="fr" data-title="Vecteur euclidien" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Vektor" title="Vektor — frísio setentrional" lang="frr" hreflang="frr" data-title="Vektor" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Veicteoir" title="Veicteoir — irlandês" lang="ga" hreflang="ga" data-title="Veicteoir" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Bheactor" title="Bheactor — gaélico escocês" lang="gd" hreflang="gd" data-title="Bheactor" data-language-autonym="Gàidhlig" data-language-local-name="gaélico escocês" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Vector" title="Vector — galego" lang="gl" hreflang="gl" data-title="Vector" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="וקטור אוקלידי — hebraico" lang="he" hreflang="he" data-title="וקטור אוקלידי" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%B0%E0%A4%BE%E0%A4%B6%E0%A4%BF" title="सदिश राशि — hindi" lang="hi" hreflang="hi" data-title="सदिश राशि" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vektor" title="Vektor — croata" lang="hr" hreflang="hr" data-title="Vektor" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Vekt%C3%A8" title="Vektè — haitiano" lang="ht" hreflang="ht" data-title="Vektè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektor" title="Vektor — húngaro" lang="hu" hreflang="hu" data-title="Vektor" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Vektor_Euklides" title="Vektor Euklides — indonésio" lang="id" hreflang="id" data-title="Vektor Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vektoro" title="Vektoro — ido" lang="io" hreflang="io" data-title="Vektoro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vigur_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Vigur (stærðfræði) — islandês" lang="is" hreflang="is" data-title="Vigur (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandês" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Vettore_(matematica)" title="Vettore (matematica) — italiano" lang="it" hreflang="it" data-title="Vettore (matematica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%BA%E9%96%93%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB" title="空間ベクトル — japonês" lang="ja" hreflang="ja" data-title="空間ベクトル" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%95%E1%83%94%E1%83%A5%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98" title="ვექტორი — georgiano" lang="ka" hreflang="ka" data-title="ვექტორი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — cazaque" lang="kk" hreflang="kk" data-title="Вектор" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EB%B2%A1%ED%84%B0" title="유클리드 벡터 — coreano" lang="ko" hreflang="ko" data-title="유클리드 벡터" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Vector_(mathematica)" title="Vector (mathematica) — latim" lang="la" hreflang="la" data-title="Vector (mathematica)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Vettor_(matematega)" title="Vettor (matematega) — lombardo" lang="lmo" hreflang="lmo" data-title="Vettor (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorius" title="Vektorius — lituano" lang="lt" hreflang="lt" data-title="Vektorius" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vektors" title="Vektors — letão" lang="lv" hreflang="lv" data-title="Vektors" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — Eastern Mari" lang="mhr" hreflang="mhr" data-title="Вектор" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="artigo destacado"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — macedónio" lang="mk" hreflang="mk" data-title="Вектор" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%82_(%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF)" title="സദിശം (ജ്യാമിതി) — malaiala" lang="ml" hreflang="ml" data-title="സദിശം (ജ്യാമിതി)" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Евклидийн вектор — mongol" lang="mn" hreflang="mn" data-title="Евклидийн вектор" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Vektor" title="Vektor — malaio" lang="ms" hreflang="ms" data-title="Vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Vettur_ewklidju" title="Vettur ewklidju — maltês" lang="mt" hreflang="mt" data-title="Vettur ewklidju" data-language-autonym="Malti" data-language-local-name="maltês" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — erzya" lang="myv" hreflang="myv" data-title="Вектор (геометрия)" data-language-autonym="Эрзянь" data-language-local-name="erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Vekter" title="Vekter — baixo-alemão" lang="nds" hreflang="nds" data-title="Vekter" data-language-autonym="Plattdüütsch" data-language-local-name="baixo-alemão" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vector_(wiskunde)" title="Vector (wiskunde) — neerlandês" lang="nl" hreflang="nl" data-title="Vector (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektor" title="Vektor — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Vektor" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektor_(matematikk)" title="Vektor (matematikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Vektor (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kalqabee" title="Kalqabee — oromo" lang="om" hreflang="om" data-title="Kalqabee" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektor" title="Wektor — polaco" lang="pl" hreflang="pl" data-title="Wektor" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Vetor" title="Vetor — Piedmontese" lang="pms" hreflang="pms" data-title="Vetor" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%A7%D9%82%D9%84%D9%8A%D8%AF%D8%B3_%D9%84%D9%88%D8%B1%DB%8C" title="د اقليدس لوری — pastó" lang="ps" hreflang="ps" data-title="د اقليدس لوری" data-language-autonym="پښتو" data-language-local-name="pastó" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vector_euclidian" title="Vector euclidian — romeno" lang="ro" hreflang="ro" data-title="Vector euclidian" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — russo" lang="ru" hreflang="ru" data-title="Вектор (геометрия)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вектор (геометрия) — sakha" lang="sah" hreflang="sah" data-title="Вектор (геометрия)" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Vettura_euclideu" title="Vettura euclideu — siciliano" lang="scn" hreflang="scn" data-title="Vettura euclideu" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vektor" title="Vektor — servo-croata" lang="sh" hreflang="sh" data-title="Vektor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BA%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%BD%E0%B7%92%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%AF%E0%B7%9B%E0%B7%81%E0%B7%92%E0%B6%9A%E0%B6%BA" title="යුක්ලිඩියානු දෛශිකය — cingalês" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දෛශිකය" data-language-autonym="සිංහල" data-language-local-name="cingalês" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vector" title="Vector — Simple English" lang="en-simple" hreflang="en-simple" data-title="Vector" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) — eslovaco" lang="sk" hreflang="sk" data-title="Vektor (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) — esloveno" lang="sl" hreflang="sl" data-title="Vektor (matematika)" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Vektor" title="Vektor — inari sami" lang="smn" hreflang="smn" data-title="Vektor" data-language-autonym="Anarâškielâ" data-language-local-name="inari sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vektori" title="Vektori — albanês" lang="sq" hreflang="sq" data-title="Vektori" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор — sérvio" lang="sr" hreflang="sr" data-title="Вектор" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/V%C3%A9ktor_(rohangan)" title="Véktor (rohangan) — sundanês" lang="su" hreflang="su" data-title="Véktor (rohangan)" data-language-autonym="Sunda" data-language-local-name="sundanês" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vektor" title="Vektor — sueco" lang="sv" hreflang="sv" data-title="Vektor" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Wekt%C5%AFr" title="Wektůr — Silesian" lang="szl" hreflang="szl" data-title="Wektůr" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%A9%E0%AF%8D" title="திசையன் — tâmil" lang="ta" hreflang="ta" data-title="திசையன்" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A7%E0%B8%81%E0%B9%80%E0%B8%95%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="เวกเตอร์ — tailandês" lang="th" hreflang="th" data-title="เวกเตอร์" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Wektor_ululyklar" title="Wektor ululyklar — turcomano" lang="tk" hreflang="tk" data-title="Wektor ululyklar" data-language-autonym="Türkmençe" data-language-local-name="turcomano" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Euclidyanong_bektor" title="Euclidyanong bektor — tagalo" lang="tl" hreflang="tl" data-title="Euclidyanong bektor" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Vekt%C3%B6r" title="Vektör — turco" lang="tr" hreflang="tr" data-title="Vektör" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D1%96%D0%B2_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Евклідів вектор — ucraniano" lang="uk" hreflang="uk" data-title="Евклідів вектор" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C_%D8%B3%D9%85%D8%AA%DB%8C%DB%81" title="اقلیدسی سمتیہ — urdu" lang="ur" hreflang="ur" data-title="اقلیدسی سمتیہ" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) — usbeque" lang="uz" hreflang="uz" data-title="Vektor (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Vect%C6%A1" title="Vectơ — vietnamita" lang="vi" hreflang="vi" data-title="Vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — wu" lang="wuu" hreflang="wuu" data-title="向量" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%A2%D7%A7%D7%98%D7%90%D7%A8" title="וועקטאר — iídiche" lang="yi" hreflang="yi" data-title="וועקטאר" data-language-autonym="ייִדיש" data-language-local-name="iídiche" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — chinês" lang="zh" hreflang="zh" data-title="向量" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng" title="Hiòng-liōng — min nan" lang="nan" hreflang="nan" data-title="Hiòng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 — cantonês" lang="yue" hreflang="yue" data-title="向量" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44528#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Vetor_(matem%C3%A1tica)" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon 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.tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Esta página <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">cita fontes</a>, mas que <b><a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">não cobrem</a> todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=Vetor+%28matem%C3%A1tica%29">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Vetor+%28matem%C3%A1tica%29&oq=Vetor+%28matem%C3%A1tica%29">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Vetor+%28matem%C3%A1tica%29">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Vetor+%28matem%C3%A1tica%29">A</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Agosto de 2013</span>)</i></small></div></td></tr></tbody></table> <div class="hatnote"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/20px-Disambig_grey.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/30px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/40px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> <b>Nota:</b> Para outros significados de vetor, veja <a href="/wiki/Vetor" class="mw-disambig" title="Vetor">Vetor</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:VectorAB.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/220px-VectorAB.svg.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/330px-VectorAB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/440px-VectorAB.svg.png 2x" data-file-width="169" data-file-height="81" /></a><figcaption>Representação gráfica de um vetor.</figcaption></figure> <p>Em <a href="/wiki/Geometria_anal%C3%ADtica" title="Geometria analítica">geometria analítica</a>, um <b>vetor</b> é uma <a href="/wiki/Classe_(teoria_dos_conjuntos)" title="Classe (teoria dos conjuntos)">classe</a> de <a href="/wiki/Equipol%C3%AAncia" title="Equipolência">equipolência</a> de <a href="/wiki/Segmento_de_recta" class="mw-redirect" title="Segmento de recta">segmentos de reta</a> orientados, que possuem todos a mesma intensidade (também designada por norma ou módulo), mesma direção e mesmo sentido.<sup id="cite_ref-santos_1-0" class="reference"><a href="#cite_note-santos-1"><span>[</span>1<span>]</span></a></sup> Em alguns dos casos, a expressão <b>vetor espacial</b> também é utilizada.<span style="color:gray"><sup>[</sup></span><sup><a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes"><span title="Esta afirmação precisa de uma referência para confirmá-la desde agosto de 2013." style="color:gray"><i>carece de fontes</i></span></a><span class="printfooter">?</span><span style="color:gray">]</span></sup> </p><p>Neste contexto, um vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> pode ser representado por qualquer segmento de <a href="/wiki/Reta" title="Reta">reta</a> orientado, que seja membro da classe deste vetor (isto é, por qualquer segmento de reta orientado que possua o mesmo módulo, mesma direção e mesmo sentido de qualquer outro segmento da referida classe). Se o segmento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f91c39c977790f3cc4e768d5aad89bb1696110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.622ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}}"></span> (segmento de reta orientado do ponto A para o ponto B) for um representante do vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d76b79e18147442c016d8d5f7e8117aa922f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.946ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} ,}"></span> então podemos dizer que o vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> é igual ao vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {AB}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AB}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6456460139935ea147f2f27e5f3c406584eab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.372ex; width:4.284ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {AB}}.}"></span> </p><p>Podemos ainda representar um vetor como um número complexo na forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+bi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef52538a0077e8dff45f937a232364d68ca6124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+bi}"></span>, onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> representa a abcissa e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> representa a ordenada desse vetor.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definição_formal"><span id="Defini.C3.A7.C3.A3o_formal"></span>Definição formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=1" title="Editar secção: Definição formal" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=1" title="Editar código-fonte da secção: Definição formal"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um vetor é definido como sendo uma classe de <a href="/wiki/Equipol%C3%AAncia" title="Equipolência">equipolência</a> de segmentos de reta orientados de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376a8d173c8a56b6c993d77885c535cf945d1f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {V} ^{n}}"></span>,<sup id="cite_ref-cruz_3-0" class="reference"><a href="#cite_note-cruz-3"><span>[</span>3<span>]</span></a></sup> em que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376a8d173c8a56b6c993d77885c535cf945d1f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {V} ^{n}}"></span> representa um <a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">espaço vetorial</a> de n dimensões. Assim sendo, em um espaço vetorial de 3 dimensões (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a458f6c9ca144bd96fc4b28948f0769bc9ca66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {V} ^{3}}"></span>), cada vetor será dotado de três coordenadas, comumente denominadas <i>x</i>, <i>y</i> e <i>z</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Importância_dos_vetores"><span id="Import.C3.A2ncia_dos_vetores"></span>Importância dos vetores</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=2" title="Editar secção: Importância dos vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=2" title="Editar código-fonte da secção: Importância dos vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os vetores desempenham um papel importante na <a href="/wiki/F%C3%ADsica" title="Física">física</a>: <a href="/wiki/Velocidade" title="Velocidade">velocidade</a> e <a href="/wiki/Acelera%C3%A7%C3%A3o" title="Aceleração">aceleração</a> de um objeto e as <a href="/wiki/For%C3%A7a" title="Força">forças</a> que agem sobre ele são descritas por vetores. Os componentes de um vetor dependem do <a href="/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas">sistema de coordenadas</a> usado para descrevê-lo. Outros objetos usados para descrever quantidades físicas são os pseudovetores e <a href="/wiki/Tensor" title="Tensor">tensores</a>. </p><p>Os vetores têm aplicação em várias áreas do conhecimento, tanto técnico quanto científico, como física, <a href="/wiki/Engenharia" title="Engenharia">engenharia</a> e <a href="/wiki/Economia" title="Economia">economia</a>, por exemplo, sendo os elementos a partir dos quais se constrói o <a href="/wiki/C%C3%A1lculo_Vetorial" class="mw-redirect" title="Cálculo Vetorial">Cálculo Vetorial</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Módulo_ou_norma_do_vetor"><span id="M.C3.B3dulo_ou_norma_do_vetor"></span>Módulo ou norma do vetor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=3" title="Editar secção: Módulo ou norma do vetor" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=3" title="Editar código-fonte da secção: Módulo ou norma do vetor"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O módulo, norma, magnitude ou intensidade, de um vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> representa-se por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||\mathbf {a} ||}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||\mathbf {a} ||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b7a9f10cc01eaa9ac6b300b97152fee12f4695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.887ex; height:2.843ex;" alt="{\displaystyle ||\mathbf {a} ||}"></span> e corresponde, do ponto de vista geométrico, ao seu <a href="/wiki/Comprimento" title="Comprimento">comprimento</a><sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span>[</span>4<span>]</span></a></sup> (que, na figura acima, seria a distância AB). </p><p>Fórmula de cálculo (para uma <a href="/wiki/Base_ortonormal" title="Base ortonormal">base ortonormal</a>): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||\mathbf {a} ||={\sqrt {x^{2}+y^{2}+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||\mathbf {a} ||={\sqrt {x^{2}+y^{2}+z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6305e6e3b13dcc4afe748664fc031c2cfbe36c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.733ex; height:4.843ex;" alt="{\displaystyle ||\mathbf {a} ||={\sqrt {x^{2}+y^{2}+z^{2}}}}"></span> (dedução a partir do <a href="/wiki/Pit%C3%A1goras#Teorema_de_Pitágoras" title="Pitágoras">Teorema de Pitágoras</a>) </p> <div class="mw-heading mw-heading2"><h2 id="Operações_com_vetores"><span id="Opera.C3.A7.C3.B5es_com_vetores"></span>Operações com vetores</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=4" title="Editar secção: Operações com vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=4" title="Editar código-fonte da secção: Operações com vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vector_Addition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/220px-Vector_Addition.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/330px-Vector_Addition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Vector_Addition.svg/440px-Vector_Addition.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Adição vetorial pela regra do paralelogramo.</figcaption></figure><p>Muitas <a href="/wiki/Opera%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Operação (matemática)">operações algébricas</a> nos <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a> possuem formas <a href="/wiki/Analogia" title="Analogia">análogas</a> para vetores. Os vetores podem ser <a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição">adicionados</a>, <a href="/wiki/Subtra%C3%A7%C3%A3o" title="Subtração">subtraídos</a>, <a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">multiplicados</a> por um <a href="/wiki/N%C3%BAmero" title="Número">número</a> e invertidos. Essas operações obedecem às conhecidas leis da <a href="/wiki/%C3%81lgebra" title="Álgebra">álgebra</a>: <a href="/wiki/Comutatividade" title="Comutatividade">comutatividade</a>, <a href="/wiki/Associatividade" title="Associatividade">associatividade</a> e <a href="/wiki/Distributividade" title="Distributividade">distributividade</a>. A soma de dois vetores com o mesmo ponto inicial pode ser encontrada geometricamente usando a <a href="/wiki/Regra_do_paralelogramo" title="Regra do paralelogramo">regra do paralelogramo</a>. A multiplicação por um número positivo (comumente chamado <i><a href="/wiki/Escalar" class="mw-redirect" title="Escalar">escalar</a></i>), nesse contexto, se resume a alterar a magnitude do vetor, isto é, alongando ou encurtando-o porém mantendo o seu sentido. A multiplicação por -1 preserva a magnitude mas inverte o sentido. As <a href="/wiki/Coordenadas_cartesianas" class="mw-redirect" title="Coordenadas cartesianas">coordenadas cartesianas</a> fornecem uma maneira sistemática de descrever e operar vetores. Como não há a multiplicação nem a divisão, também não há a potenciação nem a radiciação. </p><div class="mw-heading mw-heading3"><h3 id="Adição"><span id="Adi.C3.A7.C3.A3o"></span>Adição</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=5" title="Editar secção: Adição" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=5" title="Editar código-fonte da secção: Adição"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A decomposição dos vetores nos seus componentes horizontais e verticais, nos revela componentes de triângulos retângulos, nos quais podemos observar claramente a propriedade da adição<sup id="cite_ref-:1_5-0" class="reference"><a href="#cite_note-:1-5"><span>[</span>5<span>]</span></a></sup> dos vetores. </p><p>Observemos o gráfico: </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_soma.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Vetor_soma.png/250px-Vetor_soma.png" decoding="async" width="250" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3b/Vetor_soma.png 1.5x" data-file-width="319" data-file-height="361" /></a><figcaption>Adição de vetores</figcaption></figure> <p>Podemos verificar que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}={\vec {v}}+{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}={\vec {v}}+{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8717083a8090cc4eee1f98054ae62499e9645b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.108ex; height:2.509ex;" alt="{\displaystyle {\vec {w}}={\vec {v}}+{\vec {u}}}"></span> </p><p>e que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{x}=v_{x}+u_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{x}=v_{x}+u_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8941af1962ee9b1e64f52424d75a9888752a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.578ex; height:2.343ex;" alt="{\displaystyle w_{x}=v_{x}+u_{x}}"></span> </p><p>assim como: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{y}=v_{y}+u_{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{y}=v_{y}+u_{y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8730598c08c2bd6ab500caf9a4fed53c330a832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.855ex; height:2.676ex;" alt="{\displaystyle w_{y}=v_{y}+u_{y}.}"></span> </p><p>Logo temos que, dados dois vetores: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f021faec30bf5c1787f3688cd100626ae7ca03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.539ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}}}"></span> </p><p>a sua adição resulta em: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b894fb172fa7c23fde9833b463f4a4ef887600d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.645ex; height:3.009ex;" alt="{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y}\rangle }"></span> </p><p>Expandindo para a forma tridimensional temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y},v_{z}+u_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y},v_{z}+u_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8b77104fee892de4e9e4a917d8b00a0a02c618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.98ex; height:3.009ex;" alt="{\displaystyle {\vec {w}}=\langle v_{x}+u_{x},v_{y}+u_{y},v_{z}+u_{z}\rangle }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Subtração"><span id="Subtra.C3.A7.C3.A3o"></span>Subtração</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=6" title="Editar secção: Subtração" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=6" title="Editar código-fonte da secção: Subtração"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vector_subtraction.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Vector_subtraction.png/150px-Vector_subtraction.png" decoding="async" width="150" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Vector_subtraction.png/225px-Vector_subtraction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Vector_subtraction.png/300px-Vector_subtraction.png 2x" data-file-width="868" data-file-height="609" /></a><figcaption></figcaption></figure> <p>Da mesma forma que no caso anterior temos a subtração<sup id="cite_ref-:1_5-1" class="reference"><a href="#cite_note-:1-5"><span>[</span>5<span>]</span></a></sup> como já aprendemos, também podemos demonstrar esta propriedade usando a decomposição em triângulos retângulos: </p><p>Observemos o gráfico: </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_subtrac.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Vetor_subtrac.png" decoding="async" width="215" height="354" class="mw-file-element" data-file-width="215" data-file-height="354" /></a><figcaption>Subtração de vetores</figcaption></figure> <p>Podemos verificar que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}={\vec {v}}-{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}={\vec {v}}-{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cfb9a6078e0c0599dade0eee92a18179e34cf49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.108ex; height:2.509ex;" alt="{\displaystyle {\vec {w}}={\vec {v}}-{\vec {u}}}"></span> </p><p>e que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{x}=v_{x}-u_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{x}=v_{x}-u_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d0e413c539a3b7bcb2e5fdf80f7d85be45bfdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.578ex; height:2.343ex;" alt="{\displaystyle w_{x}=v_{x}-u_{x}}"></span> </p><p>assim como: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{y}=v_{y}-u_{y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{y}=v_{y}-u_{y}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8684f2c458efdbee6400a6f257817ac9c822f08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.855ex; height:2.676ex;" alt="{\displaystyle w_{y}=v_{y}-u_{y}.}"></span> </p><p>Logo temos que, dados dois vetores: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f021faec30bf5c1787f3688cd100626ae7ca03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.539ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}}}"></span> </p><p>a sua subtração resulta em: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af0d2bf5fcf76b14ff04ae9f8eca735bf8ac344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.645ex; height:3.009ex;" alt="{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y}\rangle }"></span> </p><p>Expandindo para a forma tridimensional temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y},v_{z}-u_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y},v_{z}-u_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ce4c3130d7a7bb73e1edacdb06e4d390c32e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.98ex; height:3.009ex;" alt="{\displaystyle {\vec {w}}=\langle v_{x}-u_{x},v_{y}-u_{y},v_{z}-u_{z}\rangle }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Multiplicação_por_escalares[5]"><span id="Multiplica.C3.A7.C3.A3o_por_escalares.5B5.5D"></span>Multiplicação por escalares<sup id="cite_ref-:1_5-2" class="reference"><a href="#cite_note-:1-5"><span>[</span>5<span>]</span></a></sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=7" title="Editar secção: Multiplicação por escalares[5]" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=7" title="Editar código-fonte da secção: Multiplicação por escalares[5]"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Scalar_multiplication_of_vectors.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Scalar_multiplication_of_vectors.png/150px-Scalar_multiplication_of_vectors.png" decoding="async" width="150" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Scalar_multiplication_of_vectors.png/225px-Scalar_multiplication_of_vectors.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Scalar_multiplication_of_vectors.png/300px-Scalar_multiplication_of_vectors.png 2x" data-file-width="930" data-file-height="1739" /></a><figcaption></figcaption></figure> <p>Definimos que se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ef490c028656282fd8b18c44c4939bbfff750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }"></span> expressando apenas valor numérico, então o denominamos <b>escalar</b>. </p><p>O produto de um escalar por um vetor é encontrado pela notação </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=c\ {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>c</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=c\ {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e424f792d69cb94d3a6f1d8238d6545467bc8043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.58ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}=c\ {\vec {a}}}"></span> </p><p>que operamos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=\langle c\ a_{x},c\ a_{y},c\ a_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>c</mi> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>c</mi> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>c</mi> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=\langle c\ a_{x},c\ a_{y},c\ a_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab1a15ce04979f206de25cba0b9465be382b00d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.315ex; height:3.009ex;" alt="{\displaystyle {\vec {w}}=\langle c\ a_{x},c\ a_{y},c\ a_{z}\rangle }"></span> </p><p>onde: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}"></span> é o vetor resultante;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> é o vetor parâmetro original;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> é o escalar.</li></ul> <p>Esta operação pode ser observada graficamente ao lado: </p><p>Note que todos os vetores gerados pela multiplicação por escalares são paralelos ao original, quando multiplicamos um vetor por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311a5a9dcc81d10b9313cd9e32e254200165f2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle (-1)}"></span> temos uma inversão de sentido e qualquer valor de escalar diferente de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> altera a magnitude do vetor. </p> <div class="mw-heading mw-heading3"><h3 id="Produto_escalar">Produto escalar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=8" title="Editar secção: Produto escalar" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=8" title="Editar código-fonte da secção: Produto escalar"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O <a href="/wiki/Produto_escalar" title="Produto escalar">produto escalar</a>, também denominado <a href="/wiki/Produto_interno" title="Produto interno">produto interno</a>, é o produto de dois vetores que resulta em um escalar, a operação que define o seu valor definimos abaixo. </p><p>Consideremos dois vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06480e308fad59f91bdee0eb3f2a3dc6d8290ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.186ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}},}"></span> cujos componentes são notados por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac4aa20f05fda74ee30f9f9842229d087f33133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.22ex; height:2.009ex;" alt="{\displaystyle v_{d}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ba98f8e7acb185246a1f1b02be733529eca07f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.009ex;" alt="{\displaystyle u_{d}}"></span> respectivamente, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> uma das dimensões: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y,z\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,y,z\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5738613a098266a2028c572b89d028f5f40a9736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.613ex; height:2.843ex;" alt="{\displaystyle \{x,y,z\},}"></span> então o produto escalar é definido como: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\ \cdot \ {\vec {u}}=v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>⋅</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\ \cdot \ {\vec {u}}=v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65728a732611e5492838a5892acaecc214661f3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.944ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}\ \cdot \ {\vec {u}}=v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z}}"></span><sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span>[</span>4<span>]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Propriedades_do_produto_escalar">Propriedades do produto escalar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=9" title="Editar secção: Propriedades do produto escalar" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=9" title="Editar código-fonte da secção: Propriedades do produto escalar"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As propriedades<sup id="cite_ref-:0_4-2" class="reference"><a href="#cite_note-:0-4"><span>[</span>4<span>]</span></a></sup> do produto escalar são facilmente demonstráveis e estão na tabela abaixo, na mesma convencionamos que: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc19e1f57588a10e647a1851a53fd02c1a10e3b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"></span> são vetores em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae613b45e932009c0ba932442199b5e65b9bffc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3};}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> é um escalar.</li></ul> <table class="wikitable"> <tbody><tr> <th>Propriedade </th> <th>Operação </th></tr> <tr> <td>Produto nulo </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\cdot {\vec {v}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\cdot {\vec {v}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97cf302b5cbae3ae4140ef99f0aabb4bdf0ead68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.278ex; height:2.343ex;" alt="{\displaystyle 0\cdot {\vec {v}}=0}"></span> </td></tr> <tr> <td>Comutativa do produto escalar </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {u}}={\vec {u}}\cdot {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {u}}={\vec {u}}\cdot {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc3f55419a9f73c7915e21d77b97e3c0701b87f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.467ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {u}}={\vec {u}}\cdot {\vec {v}}}"></span> </td></tr> <tr> <td>Associativa entre produto escalar e produto por escalares </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c({\vec {v}}\cdot {\vec {u}})=(c{\vec {v}})\cdot {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c({\vec {v}}\cdot {\vec {u}})=(c{\vec {v}})\cdot {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d864f01493da5bf6fb04d07fb14e8b30454d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.099ex; height:2.843ex;" alt="{\displaystyle c({\vec {v}}\cdot {\vec {u}})=(c{\vec {v}})\cdot {\vec {u}}}"></span> </td></tr> <tr> <td>Distributiva </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}+{\vec {u}})\cdot {\vec {w}}={\vec {v}}\cdot {\vec {w}}+{\vec {u}}\cdot {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}+{\vec {u}})\cdot {\vec {w}}={\vec {v}}\cdot {\vec {w}}+{\vec {u}}\cdot {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567f948a95ad5805ce222a99550e3d521ac3ea31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.628ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}+{\vec {u}})\cdot {\vec {w}}={\vec {v}}\cdot {\vec {w}}+{\vec {u}}\cdot {\vec {w}}}"></span> </td></tr> <tr> <td>Escalar quadrado </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {v}}=|{\vec {v}}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {v}}=|{\vec {v}}|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f123fd628ca1537c04950bfea4777206eab9613a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.652ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {v}}=|{\vec {v}}|^{2}}"></span> </td></tr></tbody></table> <p>A demonstração das propriedades não é difícil, uma vez que todas são intuitivas. Para isto, basta efetuar a operação do produto escalar e utilizar regras básicas de operações algébricas. </p> <div class="mw-heading mw-heading3"><h3 id="Produto_vetorial">Produto vetorial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=10" title="Editar secção: Produto vetorial" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=10" title="Editar código-fonte da secção: Produto vetorial"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As operações com vetores podem, muitas vezes, parecer estranhas a princípio, porém depois que entendemos a sua finalidade e o conceito do fundamento que está por trás de seu comportamento nos habituamos, podendo aproveitar dos recursos que estas operações podem nos oferecer. Um dos cálculos mais intrigantes dentro do universo dos vetores é o chamado <b><a href="/wiki/Produto_vetorial" title="Produto vetorial">produto vetorial</a></b>, que é definido pela seguinte operação: </p><p>Sejam os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70aee709a1214828b1d9876870a5a308f6f6dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.956ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"></span> o produto vetorial dos mesmos é: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},u_{x}v_{z}-v_{x}u_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},u_{x}v_{z}-v_{x}u_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf08c22c66810e8c9fa91454cdb9af520881e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.48ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},u_{x}v_{z}-v_{x}u_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }"></span> </p><p>A razão desta definição está na operação geométrica entre dois vetores, que neste caso, objetiva-se em encontrar um vetor que seja, ao mesmo tempo, perpendicular aos dois vetores operados. Como a operação resulta em um novo vetor, ela é denominada de <b>produto vetorial</b>. </p><p>A operação resume-se em encontrar coordenadas em cada eixo que sejam perpendiculares entre elas e de módulo igual a <a href="/wiki/%C3%81rea" title="Área">área</a> formada pelo <a href="/wiki/Paralelogramo" title="Paralelogramo">paralelogramo</a> criado pela imagem dos dois vetores em cada um dos planos primários. </p><p>Podemos observar que cada componente é igual à resultante de um <a href="/wiki/Determinante" title="Determinante">determinante</a>, o que nos habilita representá-los da seguinte forma: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}=\left\langle {\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}},{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}},{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}=\left\langle {\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}},{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}},{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5276c84e99969b14fd11a9e14cff5d8456b545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.526ex; height:6.509ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}=\left\langle {\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}},{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}},{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}\right\rangle }"></span> </p><p>Como temos um vetor tridimensional, podemos adotar a notação: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}}{\vec {i}}+{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}}{\vec {j}}+{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}{\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}}{\vec {i}}+{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}}{\vec {j}}+{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}{\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74e0c8be9ec4570111e99702772e6278207c44d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.052ex; height:6.509ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}v_{y}&v_{z}\\u_{y}&u_{z}\end{vmatrix}}{\vec {i}}+{\begin{vmatrix}v_{z}&v_{x}\\u_{z}&u_{x}\end{vmatrix}}{\vec {j}}+{\begin{vmatrix}v_{x}&v_{y}\\u_{x}&u_{y}\end{vmatrix}}{\vec {k}}}"></span> </p><p>Que pode ser simplificada ainda mais se adotarmos a notação de determinante com três variáveis: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8753b56127ff8a868a1e473370f57c667c228279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.347ex; height:10.176ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"></span> </p><p>Em decorrência disto, temos um produto que se comporta de maneira idêntica a um determinante, onde todas as propriedades são iguais às apresentadas pelos mesmos. </p> <div class="mw-heading mw-heading4"><h4 id="Propriedades_do_produto_vetorial">Propriedades do produto vetorial</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=11" title="Editar secção: Propriedades do produto vetorial" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=11" title="Editar código-fonte da secção: Propriedades do produto vetorial"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O produto vetorial é basicamente uma operação em forma de determinante. A tabela abaixo introduz as propriedades do mesmo: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc19e1f57588a10e647a1851a53fd02c1a10e3b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"></span> são vetores em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{3};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{3};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e6d2dae53622538d995d41d847f16fc8bac32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.618ex; height:3.009ex;" alt="{\displaystyle V^{3};}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> é um escalar.</li></ul> <table class="wikitable"> <tbody><tr> <th>Propriedade </th> <th>Operação </th></tr> <tr> <td>Produto vetorial inverso </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}=-{\vec {u}}\times {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}=-{\vec {u}}\times {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bd64a9a1585641395527097c5da7938d6803c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.597ex; height:2.509ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}=-{\vec {u}}\times {\vec {v}}}"></span> </td></tr> <tr> <td>múltiplo de escalar por produto vetorial </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c({\vec {v}}\times {\vec {u}})=(c{\vec {v}})\times {\vec {u}}={\vec {v}}\times (c{\vec {u}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c({\vec {v}}\times {\vec {u}})=(c{\vec {v}})\times {\vec {u}}={\vec {v}}\times (c{\vec {u}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19df82cdc7abadda3c03b122b79a5f603d36da4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.682ex; height:2.843ex;" alt="{\displaystyle c({\vec {v}}\times {\vec {u}})=(c{\vec {v}})\times {\vec {u}}={\vec {v}}\times (c{\vec {u}})}"></span> </td></tr> <tr> <td>Distributiva a direita </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}\times ({\vec {v}}+{\vec {u}})={\vec {w}}\times {\vec {v}}+{\vec {w}}\times {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}\times ({\vec {v}}+{\vec {u}})={\vec {w}}\times {\vec {v}}+{\vec {w}}\times {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab3f03a562db18c1aef03076537d8921e66529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.112ex; height:2.843ex;" alt="{\displaystyle {\vec {w}}\times ({\vec {v}}+{\vec {u}})={\vec {w}}\times {\vec {v}}+{\vec {w}}\times {\vec {u}}}"></span> </td></tr> <tr> <td>Distributiva a esquerda </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}+{\vec {u}})\times {\vec {w}}={\vec {v}}\times {\vec {w}}+{\vec {u}}\times {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}+{\vec {u}})\times {\vec {w}}={\vec {v}}\times {\vec {w}}+{\vec {u}}\times {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2b591117d36dadf1602c9d99e840c60a90bc2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.112ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}+{\vec {u}})\times {\vec {w}}={\vec {v}}\times {\vec {w}}+{\vec {u}}\times {\vec {w}}}"></span> </td></tr> <tr> <td>Conversão em vetores com produtos escalares </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {u}}\times {\vec {v}})\times {\vec {w}}=({\vec {u}}\cdot {\vec {w}}){\vec {v}}-({\vec {v}}\cdot {\vec {w}}){\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {u}}\times {\vec {v}})\times {\vec {w}}=({\vec {u}}\cdot {\vec {w}}){\vec {v}}-({\vec {v}}\cdot {\vec {w}}){\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6556a8d5d34b4f5c280a6560b905a64fe6ffa5b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.913ex; height:2.843ex;" alt="{\displaystyle ({\vec {u}}\times {\vec {v}})\times {\vec {w}}=({\vec {u}}\cdot {\vec {w}}){\vec {v}}-({\vec {v}}\cdot {\vec {w}}){\vec {u}}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ângulo_entre_dois_vetores"><span id=".C3.82ngulo_entre_dois_vetores"></span>Ângulo entre dois vetores</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=12" title="Editar secção: Ângulo entre dois vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=12" title="Editar código-fonte da secção: Ângulo entre dois vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Observemos o gráfico: </p><p><span typeof="mw:File"><a href="/wiki/Ficheiro:Angle_between_two_vectors.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Angle_between_two_vectors.svg/300px-Angle_between_two_vectors.svg.png" decoding="async" width="300" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Angle_between_two_vectors.svg/450px-Angle_between_two_vectors.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Angle_between_two_vectors.svg/600px-Angle_between_two_vectors.svg.png 2x" data-file-width="470" data-file-height="400" /></a></span> </p> <div class="mw-heading mw-heading4"><h4 id="Relação_entre_o_ângulo_e_o_produto_escalar_de_dois_vetores"><span id="Rela.C3.A7.C3.A3o_entre_o_.C3.A2ngulo_e_o_produto_escalar_de_dois_vetores"></span>Relação entre o <a href="/wiki/%C3%82ngulo" title="Ângulo">ângulo</a> e o produto escalar de dois vetores</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=13" title="Editar secção: Relação entre o ângulo e o produto escalar de dois vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=13" title="Editar código-fonte da secção: Relação entre o ângulo e o produto escalar de dois vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dados dois vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}},{\vec {B}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}},{\vec {B}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c0cad0c87b1132685c72c2e03ec309a356260b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.188ex; height:3.343ex;" alt="{\displaystyle {\vec {A}},{\vec {B}},}"></span> é possível demonstrar que o cosseno do ângulo entre os dois vetores é proporcional ao produto interno (escalar), relacionando-se da seguinte forma: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}}\cdot {\vec {B}}=|{\vec {A}}||{\vec {B}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}\cdot {\vec {B}}=|{\vec {A}}||{\vec {B}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d57ec06f7e81cd36f63301cd735f44ac49eb0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.777ex; height:3.509ex;" alt="{\displaystyle {\vec {A}}\cdot {\vec {B}}=|{\vec {A}}||{\vec {B}}|\cos(\theta )}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Demonstração"><span id="Demonstra.C3.A7.C3.A3o"></span>Demonstração</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=14" title="Editar secção: Demonstração" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=14" title="Editar código-fonte da secção: Demonstração"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Observemos o gráfico abaixo: </p><p><span typeof="mw:File"><a href="/wiki/Ficheiro:Angulo_vetores.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Angulo_vetores.png/300px-Angulo_vetores.png" decoding="async" width="300" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/88/Angulo_vetores.png 1.5x" data-file-width="449" data-file-height="307" /></a></span> </p><p>O gráfico mostra dois vetores relacionados pelo ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082e8402f1cbddec479b88e2ff0d1be5e9b95bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.737ex; height:2.176ex;" alt="{\displaystyle \theta .}"></span> É importante notar que temos um triângulo, que pode ser generalizado pela lei dos cossenos, o que permite fazer a relação entre o ângulo e os dois vetores. </p><p>Considerando os vetores na ilustração acima, é possível fazer o cálculo do módulo de sua diferença utilizando a conhecida relação da lei dos cossenos para o triângulo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}-{\vec {u}}|^{2}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}-{\vec {u}}|^{2}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdebe549b2a309761b2ef7b88dee81dbebfe8c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.326ex; height:3.343ex;" alt="{\displaystyle |{\vec {v}}-{\vec {u}}|^{2}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}-{\vec {u}})\cdot ({\vec {v}}-{\vec {u}})=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}-{\vec {u}})\cdot ({\vec {v}}-{\vec {u}})=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896eb17f179ccc7e3d46efa74a755465c56677c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.621ex; height:3.343ex;" alt="{\displaystyle ({\vec {v}}-{\vec {u}})\cdot ({\vec {v}}-{\vec {u}})=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {v}}-2{\vec {v}}\cdot {\vec {u}}+{\vec {u}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {v}}-2{\vec {v}}\cdot {\vec {u}}+{\vec {u}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e51c3e12803aa62b323650004653698019c856f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.029ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {v}}-2{\vec {v}}\cdot {\vec {u}}+{\vec {u}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2{\vec {v}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2{\vec {v}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c2ee88f3eae7f3ff2f3ceed9fe0320da6df469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.861ex; height:3.343ex;" alt="{\displaystyle |{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2{\vec {v}}\cdot {\vec {u}}=|{\vec {v}}|^{2}+|{\vec {u}}|^{2}-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2{\vec {v}}\cdot {\vec {u}}=-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2{\vec {v}}\cdot {\vec {u}}=-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a678b0b4d16cbd536345cdcae12ebe90b0630c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.714ex; height:2.843ex;" alt="{\displaystyle -2{\vec {v}}\cdot {\vec {u}}=-2|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p>Portanto: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363c3ff8432cbe3e665ab8bb5d74fbb0690f4df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.773ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relação_entre_produto_vetorial_e_ângulo_entre_vetores"><span id="Rela.C3.A7.C3.A3o_entre_produto_vetorial_e_.C3.A2ngulo_entre_vetores"></span>Relação entre produto vetorial e ângulo entre vetores</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=15" title="Editar secção: Relação entre produto vetorial e ângulo entre vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=15" title="Editar código-fonte da secção: Relação entre produto vetorial e ângulo entre vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seja os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> vetores em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5506da51d4507becec0424c53968ff69daed4763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.618ex; height:3.009ex;" alt="{\displaystyle V^{3},}"></span> é possível provar que o módulo do vetor resultante do produto vetorial destes, é proporcional ao seno do ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> entre os dois vetores, relacionando-se da seguinte forma: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8e37432b883c73b644d263481cf03fb3608c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"></span> <b>Comprovação</b>: Façamos a operação conforme indicado na definição para verificar a evolução: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},v_{x}u_{z}-u_{x}v_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},v_{x}u_{z}-u_{x}v_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0875546e5c26ebfe11f43ddc10900d2892f70df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.48ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}=\langle v_{y}u_{z}-u_{y}v_{z},v_{x}u_{z}-u_{x}v_{z},v_{x}u_{y}-u_{x}v_{y}\rangle }"></span> </p><p>De onde calculamos o seu módulo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{y}u_{z}-u_{y}v_{z})^{2}+(v_{x}u_{z}-u_{x}v_{z})^{2}+(v_{x}u_{y}-u_{x}v_{y})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{y}u_{z}-u_{y}v_{z})^{2}+(v_{x}u_{z}-u_{x}v_{z})^{2}+(v_{x}u_{y}-u_{x}v_{y})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d085e9dbfc3168c7b0883e1a6eba73c5eec983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.222ex; height:3.509ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{y}u_{z}-u_{y}v_{z})^{2}+(v_{x}u_{z}-u_{x}v_{z})^{2}+(v_{x}u_{y}-u_{x}v_{y})^{2}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=v_{y}^{2}u_{z}^{2}-2v_{y}u_{z}u_{y}v_{z}+u_{y}^{2}v_{z}^{2}+v_{x}^{2}u_{z}^{2}-2v_{x}u_{z}u_{x}v_{z}+u_{x}^{2}v_{z}^{2}+v_{x}^{2}u_{y}^{2}-2v_{x}u_{y}u_{x}v_{y}+u_{x}^{2}v_{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=v_{y}^{2}u_{z}^{2}-2v_{y}u_{z}u_{y}v_{z}+u_{y}^{2}v_{z}^{2}+v_{x}^{2}u_{z}^{2}-2v_{x}u_{z}u_{x}v_{z}+u_{x}^{2}v_{z}^{2}+v_{x}^{2}u_{y}^{2}-2v_{x}u_{y}u_{x}v_{y}+u_{x}^{2}v_{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df87a15abdc7d8643290d59f6f6646084ed5800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:92.508ex; height:3.509ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=v_{y}^{2}u_{z}^{2}-2v_{y}u_{z}u_{y}v_{z}+u_{y}^{2}v_{z}^{2}+v_{x}^{2}u_{z}^{2}-2v_{x}u_{z}u_{x}v_{z}+u_{x}^{2}v_{z}^{2}+v_{x}^{2}u_{y}^{2}-2v_{x}u_{y}u_{x}v_{y}+u_{x}^{2}v_{y}^{2}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})-(v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})-(v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1893a60c427b5ba0754f98e5899495a443d5fad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.909ex; height:3.509ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})-(v_{x}u_{x}+v_{y}u_{y}+v_{z}u_{z})^{2}}"></span> </p><p>Como já sabemos, podemos aplicar as propriedades já estudadas para os vetores: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-({\vec {v}}\cdot {\vec {u}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-({\vec {v}}\cdot {\vec {u}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac089e566a10c1a049733f8f93812d8b5876f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.881ex; height:3.343ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-({\vec {v}}\cdot {\vec {u}})^{2}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-|{\vec {v}}|^{2}|{\vec {u}}|^{2}\cos ^{2}(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-|{\vec {v}}|^{2}|{\vec {u}}|^{2}\cos ^{2}(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f036ccfd3e9de7c59a10b55b9f8b1d27b50d02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.486ex; height:3.343ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}-|{\vec {v}}|^{2}|{\vec {u}}|^{2}\cos ^{2}(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\left[1-\cos ^{2}(\theta )\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\left[1-\cos ^{2}(\theta )\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c872e8a4b9da09a1c02f277e5adcb717ef58804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.387ex; height:3.509ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\left[1-\cos ^{2}(\theta )\right]}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\operatorname {sen} ^{2}(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\operatorname {sen} ^{2}(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e10b97476a90595befbeed0f2f47e0824c92ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.575ex; height:3.343ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|^{2}=|{\vec {v}}|^{2}|{\vec {u}}|^{2}\operatorname {sen} ^{2}(\theta )}"></span> </p><p>Lembremos que, se: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}=\operatorname {sen} ^{2}(\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}=\operatorname {sen} ^{2}(\theta ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71709666fc5b97d178d2b7a88ae1aeb5bb54970c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.225ex; height:3.176ex;" alt="{\displaystyle a^{2}=\operatorname {sen} ^{2}(\theta ),}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\operatorname {sen} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\operatorname {sen} (\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3addfa214cf9cf35225ec22d8eebb5afc481d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.469ex; height:2.843ex;" alt="{\displaystyle a=\operatorname {sen} (\theta )}"></span> </p><p>Quando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d6a6521db7f3738e48ea472c8d88f0c4e0c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.782ex; height:2.343ex;" alt="{\displaystyle 0\leq \theta \leq \pi }"></span> </p><p>logo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8e37432b883c73b644d263481cf03fb3608c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Interpretação_do_produto_vetorial"><span id="Interpreta.C3.A7.C3.A3o_do_produto_vetorial"></span>Interpretação do produto vetorial</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=16" title="Editar secção: Interpretação do produto vetorial" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=16" title="Editar código-fonte da secção: Interpretação do produto vetorial"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O vetor resultante do produto vetorial apresenta módulo igual à área do paralelogramo delimitado pelos dois vetores que lhe deram origem, observemos o gráfico abaixo: </p> <figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Ficheiro:Prod_vetorial.png" class="mw-file-description" title="Produto vetorial"><img alt="Produto vetorial" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Prod_vetorial.png/300px-Prod_vetorial.png" decoding="async" width="300" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Prod_vetorial.png/450px-Prod_vetorial.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Prod_vetorial.png/600px-Prod_vetorial.png 2x" data-file-width="648" data-file-height="392" /></a><figcaption>Produto vetorial</figcaption></figure> <p>Como já sabemos, os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f021faec30bf5c1787f3688cd100626ae7ca03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.539ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}}}"></span> que mantêm um ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> entre eles, quando multiplicados, podem ser expressos desta forma: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8e37432b883c73b644d263481cf03fb3608c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}||{\vec {u}}|\operatorname {sen} (\theta )}"></span> </p><p>Considere a seguinte separação: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\ \underbrace {|{\vec {u}}|\operatorname {sen} (\theta )} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\ \underbrace {|{\vec {u}}|\operatorname {sen} (\theta )} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b92175795c635640693b2eeeb4dc32f1bc39b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.028ex; width:22.354ex; height:4.509ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\ \underbrace {|{\vec {u}}|\operatorname {sen} (\theta )} }"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\cdot \quad h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mspace width="1em" /> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\cdot \quad h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33515e768a77193e8180b9d225d8fd779c520a77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.547ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\times {\vec {u}}|=|{\vec {v}}|\cdot \quad h}"></span> </p><p>Muito convenientemente, podemos verificar que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> é a altura do paralelogramo, que multiplicado pela norma do vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> nos dá a área do paralelogramo. Da mesma forma verifiquemos que o <b>produto vetorial</b> nos fornece um vetor perpendicular aos dois que lhe deram origem... Façamos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}\cdot \langle v_{x},v_{y},v_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>⋅</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}\cdot \langle v_{x},v_{y},v_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250059811dde37dbb61363e807c15ad879790634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:39.327ex; height:10.176ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}\cdot \langle v_{x},v_{y},v_{z}\rangle }"></span> </p><p>Pelas propriedades dos determinantes e do produto escalar temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}v_{x}&{\vec {j}}v_{y}&{\vec {k}}v_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}v_{x}&{\vec {j}}v_{y}&{\vec {k}}v_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccabad3d7560866f098a190a1ca8d418ddc7a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:29.958ex; height:10.509ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}={\begin{vmatrix}{\vec {i}}v_{x}&{\vec {j}}v_{y}&{\vec {k}}v_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"></span> </p><p>Sendo o determinante acima nulo, uma vez que uma das suas linhas é múltipla de outra: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a12ba8da2e4d7c9a9037a2de53545032015bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.424ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {u}}=0}"></span> </p><p>Também teremos o mesmo resultado para o segundo vetor, visto que o mesmo também é uma das linhas do determinante, portanto o produto vetorial fornece um vetor perpendicular aos dois que lhe deram origem, visto que o cosseno do ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> é nulo, ou seja, o ângulo é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe4f48f7e0ba038a0dce9c299f02cbb63e0f500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.815ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Ângulos_diretores"><span id=".C3.82ngulos_diretores"></span>Ângulos diretores</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=17" title="Editar secção: Ângulos diretores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=17" title="Editar código-fonte da secção: Ângulos diretores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A relação entre produto escalar e cosseno do ângulo entre os vetores nos fornece uma outra possibilidade de referenciar os vetores aos eixos do sistema cartesiano, uma vez que temos versores primários para os eixos, podemos verificar qual o resultado do produto escalar entre estes e um vetor qualquer no espaço. Seja o vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19c08dec703a7fa59149aff3fd1ebb67d6cde7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.404ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,}"></span> calculemos o produto escalar entre ele e os vetores relacionados aos eixos: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j,k:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j,k:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33df30f6e5c1e59cd11e3842a446becc0a4a722e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.332ex; height:2.509ex;" alt="{\displaystyle i,j,k:}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot i=|{\vec {v}}||i|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot i=|{\vec {v}}||i|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb90dd61e81509c4feccb8c10b984b2c17d8040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.718ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\cdot i=|{\vec {v}}||i|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x}=|{\vec {v}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x}=|{\vec {v}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58cecba3ce9e4a36c541fb3fcb6606748b9dc33c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.266ex; height:2.843ex;" alt="{\displaystyle v_{x}=|{\vec {v}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1488f2c7f9c71c8929a95dd4ce651db6f7b5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.414ex; height:5.509ex;" alt="{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\theta )}"></span> </p><p>Para este caso inicialmente façamos uma breve reflexão: </p><p>Se para cada vetor de módulo unitário for feita a mesma operação acima, teremos três ângulos para <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a45ee89f965ec5498b347cb54f45243ef88b218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.509ex;" alt="{\displaystyle \theta ,}"></span> os quais são os ângulos do vetor no espaço em relação aos eixos, por esta razão convencionou-se chamá-los de nomes especiais, que são <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }"></span> respectivamente, para os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j,k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j,k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2efb4cff831463db35e5d914f938d78053c2eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.686ex; height:2.509ex;" alt="{\displaystyle i,j,k.}"></span> Observemos, também, que a operação será sempre a mesma para cada eixo e o resultado será o valor da componente do vetor para o eixo dividido pela norma do mesmo, o que nos fornece três cossenos: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b91f4a81845d6da985e8cb6e2a177c7c782b7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.812ex; height:5.509ex;" alt="{\displaystyle {\frac {v_{x}}{|{\vec {v}}|}}=\cos(\alpha )}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{y}}{|{\vec {v}}|}}=\cos(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{y}}{|{\vec {v}}|}}=\cos(\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb4f6395452ba4da03a076a4328c681105088a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.656ex; height:5.843ex;" alt="{\displaystyle {\frac {v_{y}}{|{\vec {v}}|}}=\cos(\beta )}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{z}}{|{\vec {v}}|}}=\cos(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{z}}{|{\vec {v}}|}}=\cos(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2b23587ffcb6e5080693e0f0b0029b32ea8c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.586ex; height:5.509ex;" alt="{\displaystyle {\frac {v_{z}}{|{\vec {v}}|}}=\cos(\gamma )}"></span></li></ul> <p>Os quais chamamos de cossenos diretores, pois direcionam o vetor no espaço sob a referência dos eixos. Em consequência disto, também chamamos os ângulos de <i>ângulos diretores</i>. </p><p>Em consequência do que já vimos nas seções anteriores, temos a seguinte <a href="/wiki/Equa%C3%A7%C3%A3o" title="Equação">equação</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}(\alpha )+\cos ^{2}(\beta )+\cos ^{2}(\gamma )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}(\alpha )+\cos ^{2}(\beta )+\cos ^{2}(\gamma )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a746fc9e4abf7ca5c8c1253da8a18bff240fa6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.948ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}(\alpha )+\cos ^{2}(\beta )+\cos ^{2}(\gamma )=1}"></span> </p><p>Também temos outra forma de referenciar o vetor, usando os cossenos diretores: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\langle |{\vec {v}}|\cos(\alpha ),|{\vec {v}}|\cos(\beta ),|{\vec {v}}|\cos(\gamma )\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\langle |{\vec {v}}|\cos(\alpha ),|{\vec {v}}|\cos(\beta ),|{\vec {v}}|\cos(\gamma )\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf2fcd1c8d7cb172212fd29b7c280cb7078eae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.563ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=\langle |{\vec {v}}|\cos(\alpha ),|{\vec {v}}|\cos(\beta ),|{\vec {v}}|\cos(\gamma )\rangle }"></span> </p><p>Que será útil em determinadas análises, inclusive quando parâmetros polares forem considerados. </p> <div class="mw-heading mw-heading4"><h4 id="Projeções_sobre_vetores"><span id="Proje.C3.A7.C3.B5es_sobre_vetores"></span>Projeções sobre vetores</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=18" title="Editar secção: Projeções sobre vetores" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=18" title="Editar código-fonte da secção: Projeções sobre vetores"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O produto escalar nos dá a possibilidade de encontrar a projeção de um vetor sobre o outro sem a necessidade de sabermos qual o ângulo entre os dois, isto é possível devido à equivalência algébrica do produto escalar com o cosseno do ângulo entre os dois vetores como vimos anteriormente. </p><p>Sejam os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70aee709a1214828b1d9876870a5a308f6f6dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.956ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"></span> o produto escalar dos mesmos é: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363c3ff8432cbe3e665ab8bb5d74fbb0690f4df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.773ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p>de onde concluímos que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{|{\vec {v}}|}}{\vec {v}}\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{|{\vec {v}}|}}{\vec {v}}\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca71c875a897e2d4442589eee2cc2882929956ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.609ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{|{\vec {v}}|}}{\vec {v}}\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a05b4b8cda5fe7021f8bbb735680272d0e72b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.855ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}=|{\vec {u}}|\cos(\theta )}"></span> </p><p>Observando o lado direito da equação observamos que a expressão corresponde à projeção do vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> sobre o vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daed28269ba38a9fccf0f713362611a0a302b649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.822ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},}"></span> porém sob a forma escalar, ou seja, o valor corresponde ao comprimento (norma) da projeção, não contendo a informação acerca da direção e do sentido. Definimos, então, a projeção escalar do vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> sobre o vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daed28269ba38a9fccf0f713362611a0a302b649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.822ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},}"></span> como: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Prj</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae849b3769cd46b2bfbb66464a7f97bec56f82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.42ex; height:6.343ex;" alt="{\displaystyle \operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {\vec {v}}{|{\vec {v}}|}}\right)\cdot {\vec {u}}}"></span> </p><p>Observemos ainda outro fato esclarecedor: a projeção escalar de um vetor sobre o outro é o produto escalar do versor do vetor sobre o qual será projetado e o outro vetor. Intuitivamente, percebemos que o versor que contém a informação sobre a direção e sentido do vetor onde será projetado o valor e portanto, determina-o, visto que é no mesmo onde temos a informação sobre a inclinação. </p><p>Seguindo este mesmo raciocínio, se multiplicarmos esta projeção, que é um valor escalar, pelo vetor unitário (versor), que usamos no cálculo anterior, teremos um <i>vetor projeção</i> criado como "imagem" do outro. </p><p>Fazendo isto teremos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {{\vec {v}}\cdot {\vec {u}}}{|{\vec {v}}|^{2}}}\right){\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msub> <mi>Prj</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {{\vec {v}}\cdot {\vec {u}}}{|{\vec {v}}|^{2}}}\right){\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b10f10fafbddaed8dc9ff3ac4a4c39a6f0aea3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.123ex; height:7.509ex;" alt="{\displaystyle V\operatorname {Prj} _{\vec {v}}\ {\vec {u}}=\left({\frac {{\vec {v}}\cdot {\vec {u}}}{|{\vec {v}}|^{2}}}\right){\vec {v}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="A_desigualdade_de_Cauchy-Schwarz">A desigualdade de Cauchy-Schwarz</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=19" title="Editar secção: A desigualdade de Cauchy-Schwarz" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=19" title="Editar código-fonte da secção: A desigualdade de Cauchy-Schwarz"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sejam os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70aee709a1214828b1d9876870a5a308f6f6dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.956ex; height:3.009ex;" alt="{\displaystyle {\vec {v}}=\langle v_{x},v_{y},v_{z}\rangle ,{\vec {u}}=\langle u_{x},u_{y},u_{z}\rangle ,}"></span> é possível provar que o produto escalar relaciona-se com os módulos dos vetores de forma a satisfazer a seguinte desigualdade </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6657c838807d12a82b58b59278b5823be758a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"></span> </p><p>Comprovação: </p><p>Analisemos o produto escalar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d97a5604d6052149c59585bbfb4e32d309ffa70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.184ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {u}}}"></span> separadamente: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363c3ff8432cbe3e665ab8bb5d74fbb0690f4df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.773ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\cdot {\vec {u}}=|{\vec {v}}||{\vec {u}}|\cos(\theta )}"></span> </p><p>Se quisermos o módulo do produto escalar teremos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|=|{\vec {v}}||{\vec {u}}||\cos(\theta )|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\cdot {\vec {u}}|=|{\vec {v}}||{\vec {u}}||\cos(\theta )|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbfce4f3d77789dc8ccb8faf9e2028b1c06aaa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.36ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|=|{\vec {v}}||{\vec {u}}||\cos(\theta )|}"></span> </p><p>porém, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq |\cos(\theta )|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq |\cos(\theta )|\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7749f710eec7f0b72802db02149e56c4170a8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.213ex; height:2.843ex;" alt="{\displaystyle 0\leq |\cos(\theta )|\leq 1}"></span> </p><p>então: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq {\frac {|{\vec {v}}\cdot {\vec {u}}|}{|{\vec {v}}||{\vec {u}}|}}\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq {\frac {|{\vec {v}}\cdot {\vec {u}}|}{|{\vec {v}}||{\vec {u}}|}}\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10b1dce756ae1a35c3e374187dafef4cea00812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.836ex; height:6.509ex;" alt="{\displaystyle 0\leq {\frac {|{\vec {v}}\cdot {\vec {u}}|}{|{\vec {v}}||{\vec {u}}|}}\leq 1}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7334402fbc9e1f0a7a7494fb27b6b3b406d777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.93ex; height:2.843ex;" alt="{\displaystyle 0\leq |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"></span> </p><p>Como o lado esquerdo da inequação faz parte do módulo, podemos simplificar para: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6657c838807d12a82b58b59278b5823be758a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.669ex; height:2.843ex;" alt="{\displaystyle |{\vec {v}}\cdot {\vec {u}}|\leq |{\vec {v}}||{\vec {u}}|}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Produto_misto">Produto misto</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=20" title="Editar secção: Produto misto" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=20" title="Editar código-fonte da secção: Produto misto"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Prod_misto.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Prod_misto.png/300px-Prod_misto.png" decoding="async" width="300" height="317" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Prod_misto.png/450px-Prod_misto.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4e/Prod_misto.png 2x" data-file-width="568" data-file-height="600" /></a><figcaption>Produto misto</figcaption></figure> <p>Sejam os vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc19e1f57588a10e647a1851a53fd02c1a10e3b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"></span> em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span> sobre estes definimos o produto misto como: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd06b33f6cdc9aa366f6508518e9e740222ebae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:28.503ex; height:9.843ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"></span> </p><p>Em síntese, a operação do produto vetorial nos fornece um novo vetor perpendicular aos dois que lhe deram origem, por outro lado, o produto escalar deste vetor por outro nos fornece um escalar, que representa o produto misto. Observando estas operações mais detalhadamente, quando operamos o produto vetorial temos um vetor definido com os versores primários, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\times {\vec {u}}={\vec {p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\times {\vec {u}}={\vec {p}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2373a63ba3134fdd9051607bea05de833ad9d1f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.416ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}\times {\vec {u}}={\vec {p}},}"></span> temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}i+p_{y}j+p_{z}k)\cdot (w_{x}i+w_{y}j+w_{z}k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>j</mi> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>k</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>j</mi> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}i+p_{y}j+p_{z}k)\cdot (w_{x}i+w_{y}j+w_{z}k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f48d5e0597e951d707aff8b990ce4c81c7d24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:45.407ex; height:3.009ex;" alt="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}i+p_{y}j+p_{z}k)\cdot (w_{x}i+w_{y}j+w_{z}k)}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})|i|^{2}+(p_{y}w_{y})|j|^{2}+(p_{z}w_{z})|k|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})|i|^{2}+(p_{y}w_{y})|j|^{2}+(p_{z}w_{z})|k|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ba17049cce2ac4b66260cd9823506059305956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:43.928ex; height:3.509ex;" alt="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})|i|^{2}+(p_{y}w_{y})|j|^{2}+(p_{z}w_{z})|k|^{2}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})+(p_{y}w_{y})+(p_{z}w_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})+(p_{y}w_{y})+(p_{z}w_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5467999d629d2a3cf8f5699dc9089fcae763de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:33.913ex; height:3.009ex;" alt="{\displaystyle {\vec {p}}\cdot {\vec {w}}=(p_{x}w_{x})+(p_{y}w_{y})+(p_{z}w_{z})}"></span> </p><p>logo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}\cdot {\vec {w}}={\begin{vmatrix}1&1&1\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}(w_{x}+w_{y}+w_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}\cdot {\vec {w}}={\begin{vmatrix}1&1&1\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}(w_{x}+w_{y}+w_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf16ee4d925c112028c3cb43d7613396581ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; margin-left: -0.089ex; width:37.465ex; height:9.843ex;" alt="{\displaystyle {\vec {p}}\cdot {\vec {w}}={\begin{vmatrix}1&1&1\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}(w_{x}+w_{y}+w_{z})}"></span> </p><p>Que nos dá: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd06b33f6cdc9aa366f6508518e9e740222ebae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:28.503ex; height:9.843ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Propriedades_do_produto_misto">Propriedades do produto misto</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=21" title="Editar secção: Propriedades do produto misto" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=21" title="Editar código-fonte da secção: Propriedades do produto misto"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As propriedades do produto misto são análogas às dos determinantes em geral, apenas uma operação algébrica entre produtos devemos destacar: </p><p>Comutativa entre produto escalar e produto vetorial em um produto misto: </p><p>Dados três vetores: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc19e1f57588a10e647a1851a53fd02c1a10e3b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {u}},{\vec {w}}}"></span> em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span> podemos comutar os vetores e produtos tais que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})=({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})=({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3055bed8da4b4fdcfc8a5d8ba030956239f39a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.094ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})=({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}}"></span> </p><p>No que se refere à operação em determinantes, a operação: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})={\begin{vmatrix}v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\\w_{x}&w_{y}&w_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})={\begin{vmatrix}v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\\w_{x}&w_{y}&w_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fc81b42553d42099ed830382d464fce36e3ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:28.503ex; height:9.843ex;" alt="{\displaystyle {\vec {v}}\cdot ({\vec {u}}\times {\vec {w}})={\begin{vmatrix}v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\\w_{x}&w_{y}&w_{z}\end{vmatrix}}}"></span> </p><p>enquanto que: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd06b33f6cdc9aa366f6508518e9e740222ebae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:28.503ex; height:9.843ex;" alt="{\displaystyle ({\vec {v}}\times {\vec {u}})\cdot {\vec {w}}={\begin{vmatrix}w_{x}&w_{y}&w_{z}\\v_{x}&v_{y}&v_{z}\\u_{x}&u_{y}&u_{z}\end{vmatrix}}}"></span> </p><p>Para fazer com que o primeiro determinante se torne o segundo basta permutar a mesma linha duas vezes dentro do determinante, ou seja, inverter o sinal do mesmo duas vezes, o que faz com que este retorne ao valor original. Algebricamente, os dois determinantes definem o mesmo valor quando operados. Isto define a operação como válida. </p> <div class="mw-heading mw-heading2"><h2 id="Vetores_na_Física"><span id="Vetores_na_F.C3.ADsica"></span>Vetores na Física</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=22" title="Editar secção: Vetores na Física" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=22" title="Editar código-fonte da secção: Vetores na Física"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Vetores_velocidade_e_aceleração"><span id="Vetores_velocidade_e_acelera.C3.A7.C3.A3o"></span>Vetores velocidade e aceleração</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=23" title="Editar secção: Vetores velocidade e aceleração" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=23" title="Editar código-fonte da secção: Vetores velocidade e aceleração"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A trajetória de um ponto em movimento pode ser definida em cada instante <i>t</i> através do vetor de <a href="/wiki/Posi%C3%A7%C3%A3o" title="Posição">posição</a> do ponto: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}(t)=x(t){\vec {e}}_{x}+y(t){\vec {e}}_{y}+z(t){\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}(t)=x(t){\vec {e}}_{x}+y(t){\vec {e}}_{y}+z(t){\vec {e}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/839187de742bc46b5e0b98ba4d778cb2a3f4c3ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.064ex; height:3.009ex;" alt="{\displaystyle {\vec {r}}(t)=x(t){\vec {e}}_{x}+y(t){\vec {e}}_{y}+z(t){\vec {e}}_{z}}"></span> </p><p>Cada uma das três componentes, x(t), y(t) e z(t), é uma <a href="/wiki/Fun%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função (matemática)">função</a> do tempo. Num intervalo de tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t=t_{2}-t_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t=t_{2}-t_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9b1a219ffab416f464a3dca3c4f74205c5241c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.149ex; height:2.509ex;" alt="{\displaystyle \Delta t=t_{2}-t_{1},}"></span> o <a href="/wiki/Deslocamento" title="Deslocamento">deslocamento</a> do ponto é: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88d68b1edd47687087cd9c54f7a468543489587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.652ex; height:2.676ex;" alt="{\displaystyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}"></span> </p><p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3846ad020fcaf0f97523dc407977b9f4f3e87e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{1}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c026fd0d6f679bda24121c0e1fdcfa40fbf82dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{2}}"></span> são os vetores posição nos instantes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92237d6568e2a209e6099272bc530fa1d6b6ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.343ex;" alt="{\displaystyle t_{2}.}"></span> O vetor obtido dividindo o deslocamento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa03f18f88517c4b8f3bba09de0cbe4d60149f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.159ex; height:2.343ex;" alt="{\displaystyle \Delta {\vec {r}}}"></span> por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span> é o vetor velocidade média, com a mesma direção e sentido do deslocamento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\vec {r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb125e407b5d4614941ab5de71a960ba9e4b9b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.806ex; height:2.343ex;" alt="{\displaystyle \Delta {\vec {r}}.}"></span> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetores_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Vetores_2.png/300px-Vetores_2.png" decoding="async" width="300" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/64/Vetores_2.png 1.5x" data-file-width="435" data-file-height="301" /></a><figcaption>Trajetória de um ponto e deslocamento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa03f18f88517c4b8f3bba09de0cbe4d60149f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.159ex; height:2.343ex;" alt="{\displaystyle \Delta {\vec {r}}}"></span> entre dois instantes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92237d6568e2a209e6099272bc530fa1d6b6ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.343ex;" alt="{\displaystyle t_{2}.}"></span></figcaption></figure> <p>Define-se o <i>vetor velocidade</i> em cada instante, igual ao deslocamento dividido por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06fe467038f5ddc05db9f430fc49d5e231d6795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.422ex; height:2.509ex;" alt="{\displaystyle \Delta t,}"></span> no limite em que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span> se aproxima de zero: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}=\lim _{\Delta \to 0}{\frac {\Delta {\vec {v}}}{\Delta t}}={\frac {d{\vec {v}}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=\lim _{\Delta \to 0}{\frac {\Delta {\vec {v}}}{\Delta t}}={\frac {d{\vec {v}}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4200a3f7b1ecbf733366ede4186036c33e6bc739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.823ex; height:5.676ex;" alt="{\displaystyle {\vec {a}}=\lim _{\Delta \to 0}{\frac {\Delta {\vec {v}}}{\Delta t}}={\frac {d{\vec {v}}}{dt}}}"></span> </p><p>e as suas componentes serão as derivadas das componentes da velocidade: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}={\overset {.}{v_{x}}}{\vec {e}}_{x}+{\overset {.}{v_{y}}}{\vec {e}}_{y}+{\overset {.}{v_{z}}}{\vec {e}}_{z}={\overset {..}{x}}{\vec {e}}_{x}+{\overset {..}{y}}{\vec {e}}_{y}+{\overset {..}{z}}{\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mrow> <mo>.</mo> <mo>.</mo> </mrow> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mrow> <mo>.</mo> <mo>.</mo> </mrow> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mrow> <mo>.</mo> <mo>.</mo> </mrow> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\overset {.}{v_{x}}}{\vec {e}}_{x}+{\overset {.}{v_{y}}}{\vec {e}}_{y}+{\overset {.}{v_{z}}}{\vec {e}}_{z}={\overset {..}{x}}{\vec {e}}_{x}+{\overset {..}{y}}{\vec {e}}_{y}+{\overset {..}{z}}{\vec {e}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4b73ce6f7f63c650ea00852144f785db4cc580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.753ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}={\overset {.}{v_{x}}}{\vec {e}}_{x}+{\overset {.}{v_{y}}}{\vec {e}}_{y}+{\overset {.}{v_{z}}}{\vec {e}}_{z}={\overset {..}{x}}{\vec {e}}_{x}+{\overset {..}{y}}{\vec {e}}_{y}+{\overset {..}{z}}{\vec {e}}_{z}}"></span> </p><p>As equações de <i>vetor velocidade</i> e suas componentes da velocidade são as equações de movimento em 3 dimensões, escritas de forma vetorial. Como a igualdade de dois vetores implica a igualdade das suas componentes, temos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x}={\overset {.}{x}},a_{x}={\overset {.}{v_{x}}},v_{x}={\overset {..}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>.</mo> </mover> </mrow> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mover> </mrow> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mrow> <mo>.</mo> <mo>.</mo> </mrow> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x}={\overset {.}{x}},a_{x}={\overset {.}{v_{x}}},v_{x}={\overset {..}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231152003e15daf29079aee731c901e8fb2dd67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.325ex; height:2.843ex;" alt="{\displaystyle v_{x}={\overset {.}{x}},a_{x}={\overset {.}{v_{x}}},v_{x}={\overset {..}{x}}}"></span> e equações semelhantes para as componentes y e z. Portanto, o movimento em 3 dimensões é a sobreposição de 3 movimentos em uma dimensão, ao longo dos eixos x, y e z, e para cada um desses 3 movimentos verificam-se as equações de movimento ao longo de um eixo. </p><p>Para cada uma das componentes cartesianas existe uma equação de movimento que relaciona a aceleração com a velocidade e a posição: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{x}-v_{x}{\frac {dv_{x}}{dx}}\quad a_{y}-v_{y}{\frac {dv_{y}}{dy}}\quad a_{z}-v_{z}{\frac {dv_{z}}{dz}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{x}-v_{x}{\frac {dv_{x}}{dx}}\quad a_{y}-v_{y}{\frac {dv_{y}}{dy}}\quad a_{z}-v_{z}{\frac {dv_{z}}{dz}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5206c00d1560f91b1419a05eb56104ce2fcd20df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.448ex; height:6.176ex;" alt="{\displaystyle a_{x}-v_{x}{\frac {dv_{x}}{dx}}\quad a_{y}-v_{y}{\frac {dv_{y}}{dy}}\quad a_{z}-v_{z}{\frac {dv_{z}}{dz}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Velocidade_e_aceleração_relativas"><span id="Velocidade_e_acelera.C3.A7.C3.A3o_relativas"></span>Velocidade e aceleração relativas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=24" title="Editar secção: Velocidade e aceleração relativas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=24" title="Editar código-fonte da secção: Velocidade e aceleração relativas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A figura abaixo mostra os vetores posição de um mesmo ponto P em dois referenciais diferentes. O primeiro referencial tem eixos x, y, z e origem O. Os eixos e origem do segundo referencial foram designados <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x',y',z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x',y',z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d94981279540ede623b547d2dd05d7025ffe9df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.703ex; height:2.843ex;" alt="{\displaystyle x',y',z'}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44ed431b3db2258962dffe620d0edb99936a009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.509ex;" alt="{\displaystyle O'.}"></span> </p><p>A relação que existe entre o vetor posição <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span>em relação à origem O e o vetor posição <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}\,'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}\,'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68dbd76b6b7b5188666cbdd895cb7a5805897b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.295ex; height:2.509ex;" alt="{\displaystyle {\vec {r}}\,'}"></span> em relação à origem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470d46b4d626e35d82cd0d9f9aadc65ac47ab568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.458ex; height:2.509ex;" alt="{\displaystyle O'}"></span> é a seguinte: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}\,'={\vec {r}}+{\vec {r}}\,'_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}\,'={\vec {r}}+{\vec {r}}\,'_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876248751f27cd828f485cdc6f805a1ef7fd00e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.097ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}\,'={\vec {r}}+{\vec {r}}\,'_{o}}"></span> </p><p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}\,'_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}\,'_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc213331bbbf1989170badda984608feafe8a987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.64ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}\,'_{o}}"></span> é o vetor de posição da primeira origem O em relação à segunda origem. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Vetor_3.png/350px-Vetor_3.png" decoding="async" width="350" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/ea/Vetor_3.png 1.5x" data-file-width="516" data-file-height="331" /></a><figcaption>Relação entre os vetores posição de um ponto em dois referenciais diferentes</figcaption></figure> <p>Derivando essa expressão em relação ao tempo, obtemos a relação entre as velocidades: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d{\vec {r}}\,'}{d_{t}}}={\frac {d{\vec {r}}}{d_{t}}}+{\frac {d{\vec {r}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {v}}\,'={\vec {v}}+{\vec {v}}\,'_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">→</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d{\vec {r}}\,'}{d_{t}}}={\frac {d{\vec {r}}}{d_{t}}}+{\frac {d{\vec {r}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {v}}\,'={\vec {v}}+{\vec {v}}\,'_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a0dabd35201c62c9c2f5a0112ae3e070d40a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.465ex; height:6.009ex;" alt="{\displaystyle {\frac {d{\vec {r}}\,'}{d_{t}}}={\frac {d{\vec {r}}}{d_{t}}}+{\frac {d{\vec {r}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {v}}\,'={\vec {v}}+{\vec {v}}\,'_{o}}"></span> </p><p>e derivando novamente obtemos a relação entre as acelerações: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d{\vec {v}}\,'}{d_{t}}}={\frac {d{\vec {v}}}{d_{t}}}+{\frac {d{\vec {v}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {a}}\,'={\vec {a}}+{\vec {a}}\,'_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="1em" /> <mo stretchy="false">→</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mspace width="thinmathspace" /> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d{\vec {v}}\,'}{d_{t}}}={\frac {d{\vec {v}}}{d_{t}}}+{\frac {d{\vec {v}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {a}}\,'={\vec {a}}+{\vec {a}}\,'_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db07743e1092ff3caf51d1a34e80e5f834fa866b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.486ex; height:6.009ex;" alt="{\displaystyle {\frac {d{\vec {v}}\,'}{d_{t}}}={\frac {d{\vec {v}}}{d_{t}}}+{\frac {d{\vec {v}}\,'_{o}}{d_{t}}}\quad \rightarrow \quad {\vec {a}}\,'={\vec {a}}+{\vec {a}}\,'_{o}}"></span> </p><p>Consequentemente, a velocidade vetorial em relação a um segundo referencial é igual à velocidade vetorial em relação ao primeiro referencial, mais a velocidade vetorial do primeiro referencial em relação ao segundo. O mesmo princípio aplica-se ao vetor aceleração. </p><p>Assim, por exemplo, se nos deslocarmos com velocidade vetorial dentro de um comboio, para obtermos a nossa velocidade vetorial em relação à Terra, teríamos de somar a velocidade vetorial do comboio em relação à Terra.<sup id="cite_ref-Villate_6-0" class="reference"><a href="#cite_note-Villate-6"><span>[</span>6<span>]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_avi%C3%A3o.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Vetor_avi%C3%A3o.png/200px-Vetor_avi%C3%A3o.png" decoding="async" width="200" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Vetor_avi%C3%A3o.png/300px-Vetor_avi%C3%A3o.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/64/Vetor_avi%C3%A3o.png 2x" data-file-width="376" data-file-height="290" /></a><figcaption>A aceleração de um corpo em queda livre, em relação a um referencial que também está em queda livre, é nula.</figcaption></figure> <p>Mas como a Terra se desloca em relação ao Sol, para encontramos a nossa velocidade em relação ao Sol teríamos de somar também a velocidade vetorial do ponto da Terra onde nos encontrarmos, em relação ao Sol. Em relação à Galaxia teríamos de somar a velocidade vetorial do Sol na galaxia, e assim sucessivamente. </p><p>O princípio de adição de acelerações vetoriais relativas é aproveitado para treinar os candidatos a astronautas. Se o astronauta, a bordo de um avião, tropeça e cai para o chão, a sua aceleração vetorial durante a queda, em relação à Terra, é o vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aad928c73fda5199478a151663f0ce3a57a8027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.174ex; height:2.676ex;" alt="{\displaystyle {\vec {g}}}"></span> que aponta para o centro da Terra e tem módulo igual à aceleração da gravidade. Mas se o avião também estiver a cair livremente, a sua aceleração vetorial em relação à Terra será o mesmo vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {g}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {g}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7f42728549f586911e40526d8e0e9020fbf9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.821ex; height:2.676ex;" alt="{\displaystyle {\vec {g}}.}"></span> Portanto, a aceleração vetorial do astronauta em relação ao avião será a diferença entre essas duas acelerações em relação à Terra, que é zero. Em relação ao avião, o astronauta não acelera em nenhuma direção, mas flutua no meio do avião, durante os segundos que o piloto conseguir manter o avião em queda livre.<sup id="cite_ref-Villate_6-1" class="reference"><a href="#cite_note-Villate-6"><span>[</span>6<span>]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Produto_escalar_2">Produto escalar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=25" title="Editar secção: Produto escalar" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=25" title="Editar código-fonte da secção: Produto escalar"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_cosceno.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Vetor_cosceno.png/290px-Vetor_cosceno.png" decoding="async" width="290" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/31/Vetor_cosceno.png 1.5x" data-file-width="315" data-file-height="269" /></a><figcaption>Dois vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> e o ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> entre as suas direções.</figcaption></figure> <p>O produto escalar entre dois vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> e o ângulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> formado pelas duas direções. O produto a cos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> é igual à componente do vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> a direção paralela ao vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> e o produto b cos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> é igual à componente do vetor <i>b</i> na direção paralela ao vetor <i>a</i>. Assim, o produto escalar é igual ao produto do módulo de um dos vetores vezes a componente do segundo vetor na direção paralela ao primeiro. </p><p>É designado de produto escalar, porque os módulos dos dois vetores e o ângulo entre as direções são grandezas escalares, que não dependem do referencial usado para os medir; consequentemente, o produto ab cos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> é também um escalar, independente do sistema de eixos usado. </p><p>Duas retas que se cruzam num ponto definem dois ângulos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 180^{\circ }-\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180^{\circ }-\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1af4f5d34f94825a488d7ae3bfefb03f65c5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.119ex; height:2.509ex;" alt="{\displaystyle 180^{\circ }-\theta .}"></span> </p><p>No caso dos vetores, não há ambiguidade na definição do ângulo, porque se deslocarmos os vetores para um vértice comum, o ângulo será a região dos pontos que estão deslocados no sentido dos dois vetores em relação ao vértice. </p><p>O produto escalar entre dois vetores com módulos a e b estará sempre dentro do intervalo [-ab, ab]. Se o ângulo entre os vetores for agudo, cos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta >\theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>></mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta >\theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e13e1435bef4ebe0acd0d8f354817a93dfdc6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.926ex; height:2.509ex;" alt="{\displaystyle \theta >\theta ,}"></span> o produto será positivo, no contrário será obtuso o produto sendo negativo. Se os vetores forem perpendiculares, o produto será nulo. </p><p>O valor mínimo do produto, - ab, obtém-se no caso em que os vetores tenham a mesma direção mas sentidos opostos. O valor máximo, ab, é obtido no caso em que os vetores tenham a mesma direção e sentido. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_cos_escalar.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Vetor_cos_escalar.png/380px-Vetor_cos_escalar.png" decoding="async" width="380" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Vetor_cos_escalar.png/570px-Vetor_cos_escalar.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/98/Vetor_cos_escalar.png 2x" data-file-width="670" data-file-height="213" /></a><figcaption>O produto escalar entre dois vetores é positivo se o ângulo entre os vetoresfor agudo, nulo se os vetores forem perpendiculares, ou negativo,se o ângulo for obtuso.</figcaption></figure> <p>Como os versores têm todos módulo igual a 1, o produto entre dois versores é sempre igual ao cosseno do ângulo entre as suas direções. Assim, o ângulo entre duas direções no espaço é igual ao arco cosseno do produto escalar entre dois versores nessas direções: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{ab}=arccos({\vec {e}}_{a}.{\vec {e}}_{b})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>.</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{ab}=arccos({\vec {e}}_{a}.{\vec {e}}_{b})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9fd2db848076aaabe8a7702bae14a319052351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.835ex; height:2.843ex;" alt="{\displaystyle \theta _{ab}=arccos({\vec {e}}_{a}.{\vec {e}}_{b})}"></span> </p><p>No caso dos três versores cartesianos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7939002929d83432b4e6bc5a0295386ac3920db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.607ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z},}"></span> o produto escalar entre dois versores diferentes é zero, por serem perpendiculares, e o produto de um dos versores consigo próprio é 1. Esse resultado pode ser usado para obter outra expressão para o cálculo do produto escalar entre dois vetores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> Usando a propriedade distributiva do produto escalar temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}{\vec {e}}_{x}+a_{y}{\vec {e}}_{y}+a_{z}{\vec {e}}_{z}).(b_{x}{\vec {e}}_{x}+b_{y}{\vec {e}}_{y}+b_{z}{\vec {e}}_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}{\vec {e}}_{x}+a_{y}{\vec {e}}_{y}+a_{z}{\vec {e}}_{z}).(b_{x}{\vec {e}}_{x}+b_{y}{\vec {e}}_{y}+b_{z}{\vec {e}}_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766b1733939348acd98240e77b0a8af79c24b49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.385ex; height:3.509ex;" alt="{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}{\vec {e}}_{x}+a_{y}{\vec {e}}_{y}+a_{z}{\vec {e}}_{z}).(b_{x}{\vec {e}}_{x}+b_{y}{\vec {e}}_{y}+b_{z}{\vec {e}}_{z})}"></span> </p><p>ou seja: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17cdb9d62494f86b3ed395235f2a5b00d9226cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.075ex; height:3.509ex;" alt="{\displaystyle {\vec {a}}.{\vec {b}}=(a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z})}"></span> </p><p>As componentes dos dois vetores são diferentes em diferentes referenciais, mas o produto (ax bx + ay by + az bz)deverá dar o mesmo resultado em qualquer referencial, já que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {a}};{\vec {b}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {a}};{\vec {b}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39b7d29d61dbb7ebbc71d3a3db09d9173ebb43b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.167ex; height:3.343ex;" alt="{\displaystyle ({\vec {a}};{\vec {b}})}"></span> é um escalar. </p><p>Para calcularmos o produto escalar de um vetor consigo próprio, temos que elevar ao quadrado todos seus componentes vetoriais: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}.{\vec {a}}=a^{2}=a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}.{\vec {a}}=a^{2}=a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e77aa0af6b367f6102f168c5825a79e3ccaec84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.626ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}.{\vec {a}}=a^{2}=a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}"></span> </p><p>Assim, para calcular o módulo de um vetor com componentes (ax , ay , az) usa-se a expressão: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {a^{2}}}={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {a^{2}}}={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593d28c3bbd518a82bd7b81772e7b5d0eaad1552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.009ex; height:4.843ex;" alt="{\displaystyle a={\sqrt {a^{2}}}={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Vetores_deslizantes">Vetores deslizantes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=26" title="Editar secção: Vetores deslizantes" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=26" title="Editar código-fonte da secção: Vetores deslizantes"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_deslizante.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vetor_deslizante.png/280px-Vetor_deslizante.png" decoding="async" width="280" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/ae/Vetor_deslizante.png 1.5x" data-file-width="358" data-file-height="249" /></a><figcaption>Três forças com o mesmo módulo, direção e sentido. F1 E F2 são equivalentes,mas são diferentes de F3</figcaption></figure> <p>Os vetores que são mais comumente estudados são denominados de <b>vetores livres</b>, que são considerados iguais se tiverem o mesmo módulo, direção e sentido, independentemente do ponto do espaço onde se encontrem. No caso das forças, não basta saber o módulo, direção e sentido. Se fixarmos o módulo, direção e sentido de uma força que vai ser aplicada numa porta para fechá-la, a forma como a porta será fechada dependerá também do ponto de aplicação dessa força. Quanto mais longe das dobradiças for aplicada a força, mais fácil será fechar a porta. </p><p>Assim, as forças são realmente vetores deslizantes, que produzem o mesmo efeito em qualquer ponto da linha de ação (a linha reta que passa pelo ponto onde a força é aplicada, seguindo a direção da força) mas produzem efeitos diferentes quando aplicadas em diferentes linhas paralelas. No exemplo apresentado na figura , as três forças <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{1},{\vec {F}}_{2};{\vec {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{1},{\vec {F}}_{2};{\vec {F}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf01bd057cbb1e873b0e8db7cf4288bd543bf3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.543ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{1},{\vec {F}}_{2};{\vec {F}}_{3}}"></span> têm o mesmo módulo, direção e sentido; <b>F1</b> e <b>F2</b> são iguais, por terem também a mesma linha de ação, mas são diferentes de <b>F3</b> que atua noutra linha de ação diferente. </p> <div class="mw-heading mw-heading3"><h3 id="Adição_de_forças"><span id="Adi.C3.A7.C3.A3o_de_for.C3.A7as"></span>Adição de forças</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=27" title="Editar secção: Adição de forças" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=27" title="Editar código-fonte da secção: Adição de forças"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_deslizante_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Vetor_deslizante_2.png/450px-Vetor_deslizante_2.png" decoding="async" width="450" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Vetor_deslizante_2.png/675px-Vetor_deslizante_2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/cd/Vetor_deslizante_2.png 2x" data-file-width="684" data-file-height="269" /></a><figcaption>Adição de forças com linhas de ação que se cruzam num ponto comum.</figcaption></figure> <p>Duas forças <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{1};{\vec {F}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{1};{\vec {F}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25430a3708f3de0094439c0f5240bf30886c01b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.684ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{1};{\vec {F}}_{2}}"></span> com a mesma linha de ação podem ser deslocadas para um ponto comum e somadas nesse ponto. A força resultante estará na mesma linha de ação. Se as linhas de ação das duas forças forem diferentes, mas tiverem um ponto em comum, <b>R</b>, como acontece com as forças na figura a seguir, podemos somá-las como se mostra no lado direito da figura: deslocam-se as duas forças para o ponto de <a href="/wiki/Interse%C3%A7%C3%A3o" title="Interseção">interseção</a> <b>R</b> e nesse ponto aplica-se a regra do paralelogramo; a linha de ação da força resultante será a reta que passa por esse ponto de interseção. </p><p>Quando as duas linhas de ação são paralelas, como é o caso da próxima figura, podemos usar o seguinte procedimento, ilustrado no lado direito da figura: desloca-se a força <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cab4803a9a02bcf62c430e8ac4ac9271675e325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{2}}"></span> na sua linha de ação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"></span> com a perpendicular que passa pelo ponto <b>P</b>. Nos pontos <b>P</b> e <b>R</b> podemos adicionar duas forças <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaa45e732c1a80126c3173155407698ed1566b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{3}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\vec {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {F}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34913d22e6a7371083675ae1d59c6848249ddcfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.633ex; height:3.176ex;" alt="{\displaystyle -{\vec {F}}_{3}}"></span> com a mesma linha de ação, já que a soma dessas duas forças é nula. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Vetor_deslizante_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Vetor_deslizante_3.png/550px-Vetor_deslizante_3.png" decoding="async" width="550" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/93/Vetor_deslizante_3.png 1.5x" data-file-width="751" data-file-height="223" /></a><figcaption>Adição de forças paralelas.</figcaption></figure> <p>No ponto P somamos as forças <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d86118c3c34637a40ccf9a925d8f83c0b4c8066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{1}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaa45e732c1a80126c3173155407698ed1566b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{3}}"></span> sendo substituídas pela resultante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a0c7e9b63a75d75934ea7017a4b910f4f4bc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{4}}"></span> No ponto <b>R</b> somamos as forças <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cab4803a9a02bcf62c430e8ac4ac9271675e325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{2}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\vec {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\vec {F}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34913d22e6a7371083675ae1d59c6848249ddcfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.633ex; height:3.176ex;" alt="{\displaystyle -{\vec {F}}_{3}}"></span> substituindo-as pela resultante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{5}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{5}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895ef0b8090d2ef18fb8e3c10be6958088caa1e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.472ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{5}.}"></span> As linhas de ação das forças <b>F4</b> e <b>F5</b> terão sempre um ponto de interseção <b>S</b>, onde podemos somá-las obtendo o resultado final <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa2b15f1b6e9eed088ac5705b54ad2168450cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{6}}"></span> no ponto <b>S</b>. </p><p>Observe na figura que, sempre que as direções e sentidos das forças forem iguais, o módulo da força resultante será igual à soma dos módulos das forças somadas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}=F_{1}+F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}=F_{1}+F_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f504b2601660cb99ba848ec5b5d01c3c3163c9ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.585ex; height:2.509ex;" alt="{\displaystyle F_{6}=F_{1}+F_{2}}"></span> Para calcular as distâncias <b>d1</b> e <b>d2</b>, entre as linhas de ação das forças somadas e a linha de ação <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216f2bbfb0444710e69c8e2a8997e43a0f729728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{6}}"></span> da força resultante, vemos na figura que h pode ser calculada nos dois triângulos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=d_{1}tan\theta ={\frac {d_{1}F_{1}}{F_{3}}}\qquad \qquad \qquad h=d_{2}\;tan\beta ={\frac {d_{2}F_{2}}{F_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mi>h</mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thickmathspace" /> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=d_{1}tan\theta ={\frac {d_{1}F_{1}}{F_{3}}}\qquad \qquad \qquad h=d_{2}\;tan\beta ={\frac {d_{2}F_{2}}{F_{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb0f3ee0deab0fe631e233aff9365f67a42390b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.826ex; height:5.843ex;" alt="{\displaystyle h=d_{1}tan\theta ={\frac {d_{1}F_{1}}{F_{3}}}\qquad \qquad \qquad h=d_{2}\;tan\beta ={\frac {d_{2}F_{2}}{F_{3}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Versor">Versor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=28" title="Editar secção: Versor" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=28" title="Editar código-fonte da secção: Versor"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um versor é um vetor de norma unitária, utilizado para indicar uma dada direcção, sentido e o ângulo formado com o eixo referencial. </p><p>Dado um vetor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> (de norma arbitrária) com a direção e sentido que pretendemos representar, o versor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {u}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {u}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adccb18b3b18c193af9f9ca2b0c0c500103e3ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {u}} }"></span> com as mesmas caraterísticas obtém-se fazendo:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {v} }{||\mathbf {v} ||}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">u</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {v} }{||\mathbf {v} ||}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5a164ea407c377179a0b019ddb33334a095579" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.418ex; height:5.509ex;" alt="{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {v} }{||\mathbf {v} ||}}}"></span> </p><p>Versores podem ser utilizados como <a href="/wiki/Base_(%C3%A1lgebra_linear)" title="Base (álgebra linear)">bases</a> de um dado <a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">espaço vetorial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} ^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f8fb78e535f4c19696c64e18c049e14f643020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {V} ^{n}.}"></span> A condição necessária e suficiente para tanto, é que tais versores sejam <a href="/wiki/Linearmente_independentes" class="mw-redirect" title="Linearmente independentes">linearmente independentes</a> entre si. Uma propriedade altamente conveniente é que todo vetor pertencente ao espaço vetorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {V} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {V} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376a8d173c8a56b6c993d77885c535cf945d1f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {V} ^{n}}"></span> de base <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a_{1}} ,\mathbf {a_{2}} ,...,\mathbf {a_{n}} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a_{1}} ,\mathbf {a_{2}} ,...,\mathbf {a_{n}} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd50320c74f345eda28302b1bbb379708329c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.549ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a_{1}} ,\mathbf {a_{2}} ,...,\mathbf {a_{n}} )}"></span> pode ser expresso como uma <a href="/wiki/Combina%C3%A7%C3%A3o_linear" title="Combinação linear">combinação linear</a> dos versores base. Assim, dado um vetor genérico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3654ae7aa05aad4aa7a3432e5d4169dfb0be078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.132ex; height:2.509ex;" alt="{\displaystyle \mathbf {b} ,}"></span> temos que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =k_{1}\mathbf {a_{1}} +k_{2}\mathbf {a_{2}} +...+k_{n}\mathbf {a_{n}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msub> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =k_{1}\mathbf {a_{1}} +k_{2}\mathbf {a_{2}} +...+k_{n}\mathbf {a_{n}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320c190c4a38fd8e5bf190a50462f0aed94f32c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.286ex; height:2.509ex;" alt="{\displaystyle \mathbf {b} =k_{1}\mathbf {a_{1}} +k_{2}\mathbf {a_{2}} +...+k_{n}\mathbf {a_{n}} ,}"></span> em que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"></span> são números reais. </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=29" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=29" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><a href="/wiki/Versor" class="mw-redirect" title="Versor">Versor</a></li> <li><a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">Espaço vetorial</a></li> <li><a href="/wiki/Espa%C3%A7o_afim" title="Espaço afim">Espaço afim</a>, que distingue vetores e <a href="/wiki/Ponto_(geometria)" class="mw-redirect" title="Ponto (geometria)">pontos</a></li> <li><a href="/wiki/Quadrivetor" title="Quadrivetor">Quadrivetor</a>, a especialização para o espaço-tempo na <a href="/wiki/Teoria_da_relatividade" title="Teoria da relatividade">teoria da relatividade</a></li> <li><a href="/wiki/Vetor_normal" class="mw-redirect" title="Vetor normal">Vetor normal</a></li> <li><a href="/wiki/Vetor_nulo" class="mw-redirect" title="Vetor nulo">Vetor nulo</a></li> <li><a href="/wiki/Vetor_unit%C3%A1rio" title="Vetor unitário">Vetor unitário</a></li> <li><a href="/wiki/C%C3%A1lculo_vetorial" title="Cálculo vetorial">Cálculo vetorial</a></li> <li><a href="/wiki/Fibrado_vectorial" class="mw-redirect" title="Fibrado vectorial">Fibrado vetorial</a></li> <li><a href="/wiki/Fasor" title="Fasor">Fasor</a> (vetor de rotação)</li></ul> </div> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-santos-1"><span class="mw-cite-backlink"><a href="#cite_ref-santos_1-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mat.ufmg.br/~regi/gaalt/gaalt1.pdf">SANTOS, Reginaldo J. <i>Matrizes, Vetores e Geometria Analítica</i>. Minas Gerais: UFMG, 2010. 709 págs</a>. <a href="/wiki/Especial:Fontes_de_livros/8574700142" class="internal mw-magiclink-isbn">ISBN 85-7470-014-2</a>. Acesso em 17 jan. 2011.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Beirigo, Thiago (2016). <i>Números complexos: uma metodologia baseada na história para obtenção de conceito</i>. Confresa: Clube de Autores</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AVetor+%28matem%C3%A1tica%29&rft.aufirst=Thiago&rft.aulast=Beirigo&rft.btitle=N%C3%BAmeros+complexos%3A+uma+metodologia+baseada+na+hist%C3%B3ria+para+obten%C3%A7%C3%A3o+de+conceito&rft.date=2016&rft.genre=book&rft.place=Confresa&rft.pub=Clube+de+Autores&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-cruz-3"><span class="mw-cite-backlink"><a href="#cite_ref-cruz_3-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://wwwp.fc.unesp.br/~lfcruz/GA_cap_01.pdf">CRUZ, Luiz Francisco da. <i>Cálculo Vetorial e Geometria Analítica</i></a>. Departamento de <a href="/wiki/Matem%C3%A1tica" title="Matemática">Matemática</a> da Faculdade de Ciências da <a href="/wiki/UNESP" class="mw-redirect" title="UNESP">UNESP</a>, Cap. 1, pág. 5. Acesso em 09/05/2011.</span> </li> <li id="cite_note-:0-4"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-:0_4-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-:0_4-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-:0_4-2">c</a></b></i></sup></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.ufrgs.br/reamat/AlgebraLinear/livro/s12-comprimento_x00e2ngulos_e_o_produto_escalar.html">«Comprimento, ângulos e o produto escalar»</a>. <i>REAMAT, Recursos Educacionais Abertos de Matemática</i>. Instituto de Matemática e Estatística da Universidade Federal do Rio Grande do Sul<span class="reference-accessdate">. Consultado em 18 de julho de 2018</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AVetor+%28matem%C3%A1tica%29&rft.atitle=Comprimento%2C+%C3%A2ngulos+e+o+produto+escalar&rft.genre=unknown&rft.jtitle=REAMAT%2C+Recursos+Educacionais+Abertos+de+Matem%C3%A1tica&rft_id=https%3A%2F%2Fwww.ufrgs.br%2Freamat%2FAlgebraLinear%2Flivro%2Fs12-comprimento_x00e2ngulos_e_o_produto_escalar.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-:1-5"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-:1_5-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-:1_5-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-:1_5-2">c</a></b></i></sup></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.ufrgs.br/reamat/AlgebraLinear/livro/s2-vetores.html">«Vetores»</a>. <i>REAMAT, Recursos Educacionais Abertos de Matemática</i>. Instituto de Matemática e Estatística da Universidade Federal do Rio Grande do Sul<span class="reference-accessdate">. Consultado em 18 de julho de 2018</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AVetor+%28matem%C3%A1tica%29&rft.atitle=Vetores&rft.genre=unknown&rft.jtitle=REAMAT%2C+Recursos+Educacionais+Abertos+de+Matem%C3%A1tica&rft_id=https%3A%2F%2Fwww.ufrgs.br%2Freamat%2FAlgebraLinear%2Flivro%2Fs2-vetores.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Villate-6"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-Villate_6-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-Villate_6-1">b</a></b></i></sup></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://villate.org/doc/fisica1/dinamica_20130320.pdf">Jaime E. Villate. <i>Dinâmica e Sistemas Dinâmicos</i>. Porto: Jaime E. Villate, 20 de março de 2013. 267 págs</a>. <a href="/wiki/Creative_Commons" title="Creative Commons">Creative Commons</a> Atribuição-Partilha (versão 3.0) <a href="/wiki/Especial:Fontes_de_livros/9789729939617" class="internal mw-magiclink-isbn">ISBN 978-972-99396-1-7</a>. Acesso em 06 jun. 2013.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&veaction=edit&section=30" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vetor_(matem%C3%A1tica)&action=edit&section=30" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.lizardonunes.pro.br/PDFs/Cinematica_Aula5.pdf">Apostila on-line da Fundação CECIERJ</a></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://efisica.if.usp.br/mecanica/universitario/vetores/">«Módulo 'vetores<span style="padding-right:0.2em;">'</span>»</a>. no sistema e-física do <a href="/wiki/Instituto_de_F%C3%ADsica_da_Universidade_de_S%C3%A3o_Paulo" title="Instituto de Física da Universidade de São Paulo">Instituto de Física da USP</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AVetor+%28matem%C3%A1tica%29&rft.btitle=M%C3%B3dulo+%27vetores%27&rft.genre=unknown&rft_id=http%3A%2F%2Fefisica.if.usp.br%2Fmecanica%2Funiversitario%2Fvetores%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://cwx.prenhall.com/bookbind/pubbooks/thomas_br/chapter1/medialib/custom3/topics/vectors.htm">História dos vetores</a></li> <li><a rel="nofollow" class="external text" href="http://www.vetorizando.com.br/tutor_oqevetor/oqueevetor.htm">O que é vetor (arte vetorial)</a></li></ul> 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href="/wiki/Produto_interno" title="Produto interno">Espaço com produto interno</a></li> <li><a href="/wiki/Espa%C3%A7o_de_Hilbert" title="Espaço de Hilbert">Espaço de Hilbert</a></li></ul> </div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="Tópicos_relacionados_com_álgebra_linear" style="padding:3px"><table class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:%C3%81lgebra_linear" title="Predefinição:Álgebra linear"><abbr title="Ver esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">v</abbr></a></li><li class="nv-discutir"><a href="/w/index.php?title=Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:%C3%81lgebra_linear&action=edit&redlink=1" class="new" title="Predefinição Discussão:Álgebra linear (página não existe)"><abbr title="Discutir esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:%C3%81lgebra_linear&action=edit"><abbr title="Editar esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">e</abbr></a></li></ul></div><div id="Tópicos_relacionados_com_álgebra_linear" style="font-size:114%;margin:0 4em">Tópicos relacionados com <a href="/wiki/%C3%81lgebra_linear" title="Álgebra linear">álgebra linear</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conceitos básicos</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Grandeza_escalar" title="Grandeza escalar">Escalar</a></li> <li><a class="mw-selflink selflink">Vetor</a></li> <li><a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">Espaço vetorial</a></li> <li><a href="/wiki/Proje%C3%A7%C3%A3o_de_um_vetor" title="Projeção de um vetor">Projeção de um vetor</a></li> <li><a href="/wiki/Espa%C3%A7o_vectorial_gerado" title="Espaço vectorial gerado">Espaço vetorial gerado</a></li> <li><a href="/wiki/Transforma%C3%A7%C3%A3o_linear" title="Transformação linear">Transformação linear</a></li> <li><a href="/wiki/Proje%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Projeção (matemática)">Projeção</a></li> <li><a href="/wiki/Independ%C3%AAncia_linear" title="Independência linear">Independência linear</a></li> <li><a href="/wiki/Combina%C3%A7%C3%A3o_linear" title="Combinação linear">Combinação linear</a></li> <li><a href="/wiki/Base_(%C3%A1lgebra_linear)" title="Base (álgebra linear)">Base</a></li> <li><a href="/wiki/Espa%C3%A7o_coluna" class="mw-redirect" title="Espaço coluna">Espaço coluna</a></li> <li><a href="/wiki/Espa%C3%A7o_linha" class="mw-redirect" title="Espaço linha">Espaço linha</a></li> <li><a href="/wiki/Espa%C3%A7o_dual" title="Espaço dual">Espaço dual</a></li> <li><a href="/wiki/Perpendicularidade" title="Perpendicularidade">Ortogonalidade</a></li> <li><a href="/wiki/N%C3%BAcleo_(%C3%A1lgebra_linear)" title="Núcleo (álgebra linear)">Núcleo</a></li> <li><a href="/wiki/Valor_pr%C3%B3prio" class="mw-redirect" title="Valor próprio">Valor próprio</a></li> <li><a href="/wiki/M%C3%A9todo_dos_m%C3%ADnimos_quadrados" title="Método dos mínimos quadrados">Método dos mínimos quadrados</a></li> <li><a href="/wiki/Produto_di%C3%A1dico" title="Produto diádico">Produto diádico</a></li> <li><a href="/wiki/Produto_interno" title="Produto interno">Espaço com produto interno</a></li> <li><a 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LU">Decomposição LU</a></li> <li><a href="/wiki/Menor_(%C3%A1lgebra_linear)" title="Menor (álgebra linear)">Menor</a></li> <li><a href="/wiki/Posto_matricial" title="Posto matricial">Posto matricial</a></li> <li><a href="/wiki/Regra_de_Cramer" title="Regra de Cramer">Regra de Cramer</a></li> <li><a href="/wiki/Matriz_inversa" title="Matriz inversa">Matriz inversa</a></li> <li><a href="/wiki/Elimina%C3%A7%C3%A3o_de_Gauss" title="Eliminação de Gauss">Eliminação de Gauss</a></li> <li><a href="/wiki/Matriz_de_transforma%C3%A7%C3%A3o" title="Matriz de transformação">Matriz de transformação</a></li> <li><a href="/wiki/Matriz_em_bloco" title="Matriz em bloco">Matriz em bloco</a></li> <li><a href="/wiki/Matriz_unimodular" title="Matriz unimodular">Matriz unimodular</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%C3%81lgebra_linear_num%C3%A9rica" title="Álgebra linear numérica">Álgebra linear numérica</a></th><td class="navbox-list 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