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Vektor – Wikipedie
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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_cs.wikipedia.org&uselang=cs" class=""><span>Podpořte Wikipedii</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1ln%C3%AD:Vytvo%C5%99it_%C3%BA%C4%8Det&returnto=Vektor&returntoquery=section%3D13%26veaction%3Dedit" title="Doporučujeme vytvořit si účet a přihlásit se, ovšem není to povinné" class=""><span>Vytvoření účtu</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1ln%C3%AD:P%C5%99ihl%C3%A1sit&returnto=Vektor&returntoquery=section%3D13%26veaction%3Dedit" title="Doporučujeme vám přihlásit se, ovšem není to povinné. 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class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Obsah" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li 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">(úvod)</div> </a> </li> <li id="toc-Definice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definice</span> </div> </a> <button aria-controls="toc-Definice-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>Přepnout podsekci Definice</span> </button> <ul id="toc-Definice-sublist" class="vector-toc-list"> <li id="toc-Pravý_a_axiální_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pravý_a_axiální_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Pravý a axiální vektor</span> </div> </a> <ul id="toc-Pravý_a_axiální_vektor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Reprezentace_vektoru" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reprezentace_vektoru"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Reprezentace vektoru</span> </div> </a> <ul id="toc-Reprezentace_vektoru-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operace_s_vektory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operace_s_vektory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operace s vektory</span> </div> </a> <button aria-controls="toc-Operace_s_vektory-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>Přepnout podsekci Operace s vektory</span> </button> <ul id="toc-Operace_s_vektory-sublist" class="vector-toc-list"> <li id="toc-Sčítání_vektorů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sčítání_vektorů"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Sčítání vektorů</span> </div> </a> <ul id="toc-Sčítání_vektorů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Násobení_vektoru_číslem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Násobení_vektoru_číslem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Násobení vektoru číslem</span> </div> </a> <ul id="toc-Násobení_vektoru_číslem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Součin_vektorů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Součin_vektorů"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Součin vektorů</span> </div> </a> <ul id="toc-Součin_vektorů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti_vektorových_operací" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vlastnosti_vektorových_operací"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Vlastnosti vektorových operací</span> </div> </a> <ul id="toc-Vlastnosti_vektorových_operací-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invariance_operací" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invariance_operací"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Invariance operací</span> </div> </a> <ul id="toc-Invariance_operací-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Úhel_dvou_vektorů" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Úhel_dvou_vektorů"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Úhel dvou vektorů</span> </div> </a> <ul id="toc-Úhel_dvou_vektorů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Další_vektorové_operace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Další_vektorové_operace"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Další vektorové operace</span> </div> </a> <ul id="toc-Další_vektorové_operace-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Zvláštní_druhy_vektorů" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zvláštní_druhy_vektorů"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Zvláštní druhy vektorů</span> </div> </a> <button aria-controls="toc-Zvláštní_druhy_vektorů-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>Přepnout podsekci Zvláštní druhy vektorů</span> </button> <ul id="toc-Zvláštní_druhy_vektorů-sublist" class="vector-toc-list"> <li id="toc-Jednotkový_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jednotkový_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Jednotkový vektor</span> </div> </a> <ul id="toc-Jednotkový_vektor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nulový_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nulový_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Nulový vektor</span> </div> </a> <ul id="toc-Nulový_vektor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tečný_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tečný_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Tečný vektor</span> </div> </a> <ul id="toc-Tečný_vektor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermitovsky_sdružený_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermitovsky_sdružený_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Hermitovsky sdružený vektor</span> </div> </a> <ul id="toc-Hermitovsky_sdružený_vektor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-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="Obsah" 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="Přepnout obsah" > <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">Přepnout obsah</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">Vektor</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="Přejděte k článku v jiném jazyce. Je dostupný v 95 jazycích" > <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 jazyků</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) – afrikánština" lang="af" hreflang="af" data-title="Vektor (Wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" 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 – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Vektor" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" 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 – amharština" lang="am" hreflang="am" data-title="ጨረር" data-language-autonym="አማርኛ" data-language-local-name="amharština" 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 – arabština" lang="ar" hreflang="ar" data-title="متجه" data-language-autonym="العربية" data-language-local-name="arabština" 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 – asturština" lang="ast" hreflang="ast" data-title="Vector" data-language-autonym="Asturianu" data-language-local-name="asturština" 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) – ázerbájdžánština" lang="az" hreflang="az" data-title="Vektor (həndəsə)" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" 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) – baškirština" lang="ba" hreflang="ba" data-title="Вектор (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="baškirština" 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) – běloruština" lang="be" hreflang="be" data-title="Вектар (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="běloruština" 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 – bulharština" lang="bg" hreflang="bg" data-title="Вектор" data-language-autonym="Български" data-language-local-name="bulharština" 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 – bengálština" lang="bn" hreflang="bn" data-title="সদিক রাশি" data-language-autonym="বাংলা" data-language-local-name="bengálština" 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 – bosenština" lang="bs" hreflang="bs" data-title="Euklidski vektor" data-language-autonym="Bosanski" data-language-local-name="bosenština" 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) – katalánština" lang="ca" hreflang="ca" data-title="Vector (matemàtiques)" data-language-autonym="Català" data-language-local-name="katalánština" 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 – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="ئاڕاستەبڕی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</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) – čuvaština" lang="cv" hreflang="cv" data-title="Вектор (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" 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 – velština" lang="cy" hreflang="cy" data-title="Fector" data-language-autonym="Cymraeg" data-language-local-name="velština" 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) – dánština" lang="da" hreflang="da" data-title="Vektor (geometri)" data-language-autonym="Dansk" data-language-local-name="dánština" 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 – němčina" lang="de" hreflang="de" data-title="Vektor" data-language-autonym="Deutsch" data-language-local-name="němčina" 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 – řečtina" lang="el" hreflang="el" data-title="Ευκλείδειο διάνυσμα" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" 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 – angličtina" lang="en" hreflang="en" data-title="Euclidean vector" data-language-autonym="English" data-language-local-name="angličtina" 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 – španělština" lang="es" hreflang="es" data-title="Vector" data-language-autonym="Español" data-language-local-name="španělština" 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 – estonština" lang="et" hreflang="et" data-title="Vektor" data-language-autonym="Eesti" data-language-local-name="estonština" 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) – baskičtina" lang="eu" hreflang="eu" data-title="Bektore (matematika)" data-language-autonym="Euskara" data-language-local-name="baskičtina" 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 – perština" lang="fa" hreflang="fa" data-title="بردار اقلیدسی" data-language-autonym="فارسی" data-language-local-name="perština" 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 – finština" lang="fi" hreflang="fi" data-title="Vektori" data-language-autonym="Suomi" data-language-local-name="finština" 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 – francouzština" lang="fr" hreflang="fr" data-title="Vecteur euclidien" data-language-autonym="Français" data-language-local-name="francouzština" 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 – fríština (severní)" lang="frr" hreflang="frr" data-title="Vektor" data-language-autonym="Nordfriisk" data-language-local-name="fríština (severní)" 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 – irština" lang="ga" hreflang="ga" data-title="Veicteoir" data-language-autonym="Gaeilge" data-language-local-name="irština" 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 – skotská gaelština" lang="gd" hreflang="gd" data-title="Bheactor" data-language-autonym="Gàidhlig" data-language-local-name="skotská gaelština" 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 – galicijština" lang="gl" hreflang="gl" data-title="Vector" data-language-autonym="Galego" data-language-local-name="galicijština" 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 – hebrejština" lang="he" hreflang="he" data-title="וקטור אוקלידי" data-language-autonym="עברית" data-language-local-name="hebrejština" 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 – hindština" lang="hi" hreflang="hi" data-title="सदिश राशि" data-language-autonym="हिन्दी" data-language-local-name="hindština" 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 – chorvatština" lang="hr" hreflang="hr" data-title="Vektor" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" 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 – haitština" lang="ht" hreflang="ht" data-title="Vektè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitština" 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 – maďarština" lang="hu" hreflang="hu" data-title="Vektor" data-language-autonym="Magyar" data-language-local-name="maďarština" 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 – indonéština" lang="id" hreflang="id" data-title="Vektor Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" 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) – islandština" lang="is" hreflang="is" data-title="Vigur (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandština" 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) – italština" lang="it" hreflang="it" data-title="Vettore (matematica)" data-language-autonym="Italiano" data-language-local-name="italština" 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 – japonština" lang="ja" hreflang="ja" data-title="空間ベクトル" data-language-autonym="日本語" data-language-local-name="japonština" 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 – gruzínština" lang="ka" hreflang="ka" data-title="ვექტორი" data-language-autonym="ქართული" data-language-local-name="gruzínština" 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 – kazaština" lang="kk" hreflang="kk" data-title="Вектор" data-language-autonym="Қазақша" data-language-local-name="kazaština" 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 – korejština" lang="ko" hreflang="ko" data-title="유클리드 벡터" data-language-autonym="한국어" data-language-local-name="korejština" 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) – latina" lang="la" hreflang="la" data-title="Vector (mathematica)" data-language-autonym="Latina" data-language-local-name="latina" 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) – lombardština" lang="lmo" hreflang="lmo" data-title="Vettor (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardština" 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 – litevština" lang="lt" hreflang="lt" data-title="Vektorius" data-language-autonym="Lietuvių" data-language-local-name="litevština" 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 – lotyština" lang="lv" hreflang="lv" data-title="Vektors" data-language-autonym="Latviešu" data-language-local-name="lotyština" 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="nejlepší článek"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор – makedonština" lang="mk" hreflang="mk" data-title="Вектор" data-language-autonym="Македонски" data-language-local-name="makedonština" 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) – malajálamština" lang="ml" hreflang="ml" data-title="സദിശം (ജ്യാമിതി)" data-language-autonym="മലയാളം" data-language-local-name="malajálamština" 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 – mongolština" lang="mn" hreflang="mn" data-title="Евклидийн вектор" data-language-autonym="Монгол" data-language-local-name="mongolština" 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 – malajština" lang="ms" hreflang="ms" data-title="Vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" 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 – maltština" lang="mt" hreflang="mt" data-title="Vettur ewklidju" data-language-autonym="Malti" data-language-local-name="maltština" 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) – erzjanština" lang="myv" hreflang="myv" data-title="Вектор (геометрия)" data-language-autonym="Эрзянь" data-language-local-name="erzjanština" 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 – dolnoněmčina" lang="nds" hreflang="nds" data-title="Vekter" data-language-autonym="Plattdüütsch" data-language-local-name="dolnoněmčina" 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) – nizozemština" lang="nl" hreflang="nl" data-title="Vector (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemština" 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 – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Vektor" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (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) – norština (bokmål)" lang="nb" hreflang="nb" data-title="Vektor (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norština (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 – oromština" lang="om" hreflang="om" data-title="Kalqabee" data-language-autonym="Oromoo" data-language-local-name="oromština" 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 – polština" lang="pl" hreflang="pl" data-title="Wektor" data-language-autonym="Polski" data-language-local-name="polština" 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 – piemonština" lang="pms" hreflang="pms" data-title="Vetor" data-language-autonym="Piemontèis" data-language-local-name="piemonština" 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 – paštština" lang="ps" hreflang="ps" data-title="د اقليدس لوری" data-language-autonym="پښتو" data-language-local-name="paštština" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática) – portugalština" lang="pt" hreflang="pt" data-title="Vetor (matemática)" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vector_euclidian" title="Vector euclidian – rumunština" lang="ro" hreflang="ro" data-title="Vector euclidian" data-language-autonym="Română" data-language-local-name="rumunština" 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) – ruština" lang="ru" hreflang="ru" data-title="Вектор (геометрия)" data-language-autonym="Русский" data-language-local-name="ruština" 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) – jakutština" lang="sah" hreflang="sah" data-title="Вектор (геометрия)" data-language-autonym="Саха тыла" data-language-local-name="jakutština" 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 – sicilština" lang="scn" hreflang="scn" data-title="Vettura euclideu" data-language-autonym="Sicilianu" data-language-local-name="sicilština" 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 – srbochorvatština" lang="sh" hreflang="sh" data-title="Vektor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" 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 – sinhálština" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දෛශිකය" data-language-autonym="සිංහල" data-language-local-name="sinhálština" 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) – slovenština" lang="sk" hreflang="sk" data-title="Vektor (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovenština" 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) – slovinština" lang="sl" hreflang="sl" data-title="Vektor (matematika)" data-language-autonym="Slovenščina" data-language-local-name="slovinština" 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 – sámština (inarijská)" lang="smn" hreflang="smn" data-title="Vektor" data-language-autonym="Anarâškielâ" data-language-local-name="sámština (inarijská)" 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 – albánština" lang="sq" hreflang="sq" data-title="Vektori" data-language-autonym="Shqip" data-language-local-name="albánština" 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 – srbština" lang="sr" hreflang="sr" data-title="Вектор" data-language-autonym="Српски / srpski" data-language-local-name="srbština" 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) – sundština" lang="su" hreflang="su" data-title="Véktor (rohangan)" data-language-autonym="Sunda" data-language-local-name="sundština" 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 – švédština" lang="sv" hreflang="sv" data-title="Vektor" data-language-autonym="Svenska" data-language-local-name="švédština" 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 – slezština" lang="szl" hreflang="szl" data-title="Wektůr" data-language-autonym="Ślůnski" data-language-local-name="slezština" 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 – tamilština" lang="ta" hreflang="ta" data-title="திசையன்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" 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 – thajština" lang="th" hreflang="th" data-title="เวกเตอร์" data-language-autonym="ไทย" data-language-local-name="thajština" 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 – turkmenština" lang="tk" hreflang="tk" data-title="Wektor ululyklar" data-language-autonym="Türkmençe" data-language-local-name="turkmenština" 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 – tagalog" lang="tl" hreflang="tl" data-title="Euclidyanong bektor" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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 – turečtina" lang="tr" hreflang="tr" data-title="Vektör" data-language-autonym="Türkçe" data-language-local-name="turečtina" 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 – ukrajinština" lang="uk" hreflang="uk" data-title="Евклідів вектор" data-language-autonym="Українська" data-language-local-name="ukrajinština" 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 – urdština" lang="ur" hreflang="ur" data-title="اقلیدسی سمتیہ" data-language-autonym="اردو" data-language-local-name="urdština" 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) – uzbečtina" lang="uz" hreflang="uz" data-title="Vektor (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" 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 – vietnamština" lang="vi" hreflang="vi" data-title="Vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" 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 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="向量" data-language-autonym="吴语" data-language-local-name="čínština (dialekty 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 – jidiš" lang="yi" hreflang="yi" data-title="וועקטאר" data-language-autonym="ייִדיש" data-language-local-name="jidiš" 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 – čínština" lang="zh" hreflang="zh" data-title="向量" data-language-autonym="中文" data-language-local-name="čínština" 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 – čínština (dialekty Minnan)" lang="nan" hreflang="nan" data-title="Hiòng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="čínština (dialekty Minnan)" 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 – kantonština" lang="yue" hreflang="yue" data-title="向量" data-language-autonym="粵語" data-language-local-name="kantonština" 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="Editovat mezijazykové odkazy" class="wbc-editpage">Upravit odkazy</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="Jmenné prostory"> <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/Vektor" title="Zobrazit obsahovou stránku [c]" accesskey="c"><span>Článek</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskuse:Vektor" rel="discussion" title="Diskuse ke stránce [t]" accesskey="t"><span>Diskuse</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Změnit variantu jazyka" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">čeština</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Zobrazení"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Vektor"><span>Číst</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&veaction=edit" title="Editovat tuto stránku [v]" accesskey="v"><span>Editovat</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Editovat zdrojový kód této stránky [e]" accesskey="e"><span>Editovat zdroj</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=history" title="Starší verze této stránky. [h]" accesskey="h"><span>Zobrazit historii</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Nástroje ke stránce"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Nástroje" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Nástroje</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Nástroje</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skrýt</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Další možnosti" > <div class="vector-menu-heading"> Akce </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Vektor"><span>Číst</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&veaction=edit" title="Editovat tuto stránku [v]" accesskey="v"><span>Editovat</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Editovat zdrojový kód této stránky [e]" accesskey="e"><span>Editovat zdroj</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&action=history"><span>Zobrazit historii</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Obecné </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Co_odkazuje_na/Vektor" title="Seznam všech wikistránek, které sem odkazují [j]" accesskey="j"><span>Odkazuje sem</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Souvisej%C3%ADc%C3%AD_zm%C4%9Bny/Vektor" rel="nofollow" title="Nedávné změny stránek, na které je odkazováno [k]" accesskey="k"><span>Související změny</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=cs" title="Nahrát obrázky či jiná multimédia [u]" accesskey="u"><span>Načíst soubor</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Speci%C3%A1ln%C3%AD_str%C3%A1nky" title="Seznam všech speciálních stránek [q]" accesskey="q"><span>Speciální stránky</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Vektor&oldid=24051191" title="Trvalý odkaz na současnou verzi této stránky"><span>Trvalý odkaz</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Vektor&action=info" title="Více informací o této stránce"><span>Informace o stránce</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:Citovat&page=Vektor&id=24051191&wpFormIdentifier=titleform" title="Informace o tom, jak citovat tuto stránku"><span>Citovat stránku</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:UrlShortener&url=https%3A%2F%2Fcs.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DVektor%26section%3D13%26veaction%3Dedit"><span>Získat zkrácené URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:QrCode&url=https%3A%2F%2Fcs.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DVektor%26section%3D13%26veaction%3Dedit"><span>Stáhnout QR kód</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Tisk/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:Kniha&bookcmd=book_creator&referer=Vektor"><span>Vytvořit knihu</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:DownloadAsPdf&page=Vektor&action=show-download-screen"><span>Stáhnout jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Vektor&printable=yes" title="Tato stránka v podobě vhodné k tisku [p]" accesskey="p"><span>Verze k tisku</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Na jiných projektech </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Vectors" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44528" title="Odkaz na propojenou položku datového úložiště [g]" accesskey="g"><span>Položka Wikidat</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Nástroje ke stránce"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhled"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Vzhled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skrýt</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><div class="uvodni-upozorneni hatnote noprint">Tento článek je o matematickému pojmu. Další významy jsou uvedeny na stránce <a href="/wiki/Vektor_(rozcestn%C3%ADk)" class="mw-disambig" title="Vektor (rozcestník)">Vektor (rozcestník)</a>.</div> <p>V <a href="/wiki/Matematika" title="Matematika">matematice</a> je <b>vektor</b> definován abstraktně jako prvek <a href="/wiki/Vektorov%C3%BD_prostor" title="Vektorový prostor">vektorového prostoru</a>. Vektory se dají spolu sčítat a dále násobit prvky komutativního algebraického tělesa, tzv. skaláry, např. reálnými čísly. </p><p>V každém vektorovém prostoru lze díky <a href="/wiki/Axiom_v%C3%BDb%C4%9Bru" title="Axiom výběru">axiomu výběru</a> najít bázi, která určuje <a href="/wiki/Soustava_sou%C5%99adnic" title="Soustava souřadnic">souřadnice</a> daného vektoru vzhledem k této bázi. Pokud je vektorový prostor konečněrozměrný, souřadnice vektoru tvoří <a href="/wiki/Uspo%C5%99%C3%A1dan%C3%A1_n-tice" title="Uspořádaná n-tice">uspořádané <i>n</i>-tice</a> <a href="/wiki/%C4%8C%C3%ADslo" title="Číslo">čísel</a>, označovaných jako <i>složky</i> (též <i>komponenty</i>) <i>vektoru</i>. Speciálně, pokud se za vektorový prostor volí <a href="/wiki/Kart%C3%A9zsk%C3%BD_sou%C4%8Din" title="Kartézský součin">kartézský součin</a> množin <a href="/wiki/Re%C3%A1ln%C3%A1_%C4%8D%C3%ADsla" class="mw-redirect" title="Reálná čísla">reálných</a> či <a href="/wiki/Komplexn%C3%AD_%C4%8D%C3%ADsla" class="mw-redirect" title="Komplexní čísla">komplexních čísel</a>, tj. pokud je za vektorový prostor bráno <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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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 {R} ^{n}}"></span> či <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 {C} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358e78277499c3d29fe16a3f62f2f9e4915720ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{m}}"></span> pro nějaká <a href="/wiki/P%C5%99irozen%C3%A9_%C4%8D%C3%ADslo" title="Přirozené číslo">přirozená čísla</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, tak se jeho prvky nazývají <b>aritmetické vektory</b>. </p><p><b>Vektor</b> představuje ve <a href="/wiki/Vektorov%C3%BD_po%C4%8Det" class="mw-redirect" title="Vektorový počet">vektorovém počtu</a> a <a href="/wiki/Fyzika" title="Fyzika">fyzice</a> veličinu, která má kromě <a href="/w/index.php?title=D%C3%A9lka_vektoru&action=edit&redlink=1" class="new" title="Délka vektoru (stránka neexistuje)">velikosti</a> i <a href="/w/index.php?title=Sm%C4%9Br_(geometrie)&action=edit&redlink=1" class="new" title="Směr (geometrie) (stránka neexistuje)">směr</a> a orientaci. </p><p>Příkladem vektoru je <a href="/wiki/S%C3%ADla" title="Síla">síla</a> — má velikost a směr, a více sil se skládá dohromady podle <a href="/wiki/Skl%C3%A1d%C3%A1n%C3%AD_sil" class="mw-redirect" title="Skládání sil">zákona o skládání sil</a> – <a href="/wiki/Rovnob%C4%9B%C5%BEn%C3%ADk" title="Rovnoběžník">rovnoběžníkového</a> pravidla. Vektory se ve fyzice obvykle popisují pomocí složek (souřadnic), které ovšem závisí na volbě souřadnicových os. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definice">Definice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=1" title="Editace sekce: Definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=1" title="Editovat zdrojový kód sekce Definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neformálně je <b>vektor</b> veličina charakterizovaná <a href="/w/index.php?title=Velikost&action=edit&redlink=1" class="new" title="Velikost (stránka neexistuje)">velikostí</a> (v matematice číslem, ve fyzice počtem jednotek) a <a href="/wiki/Sm%C4%9Br" class="mw-disambig" title="Směr">směrem</a>. Často je <a href="/wiki/Reprezentace_vektoru" class="mw-redirect" title="Reprezentace vektoru">reprezentovaná</a> graficky jako šipka. Příkladem je „Pohyb na sever rychlostí 90 km/hod“ nebo „Přitahován ke středu Země silou 70 <a href="/wiki/Newton" title="Newton">newtonů</a>“. </p><p>Ve fyzice se vektory obvykle zapisují v souřadnicích. Aby byl vektor dobře definován, požaduje se následující vlastnost: jestliže si zvolím novou souřadnicovou soustavu a měřím body v prostoru v novém souřadném systému, pak souřadnice vektoru se změní podle stejného vzorce jak souřadnice bodů v prostorů. Tato vlastnost se nazývá kovariance vůči změně (prostorových) souřadnic ("stejná" změna jeho souřadnic, nové se počítají podle stejného pravidla jako souřadnice polohy). Tedy jestliže systém souřadnic podstoupí lineární transformaci popsanou vztahem <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_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2925bbbbddd3c236721705830cc0bcc9eecf04c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.916ex; height:7.176ex;" alt="{\displaystyle x_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}x_{j}}"></span>, pak složky libovolného vektoru <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> se podobně transformují podle vztahu </p> <dl><dd><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_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}v_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10520bfd57dc3f36b8bc0ee53da60c63aac1da0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.512ex; height:7.176ex;" alt="{\displaystyle v_{i}^{\prime }=\sum _{j=1}^{n}a_{ij}v_{j}}"></span>,</dd></dl> <p>kde <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> jsou složky vektoru <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> v původní soustavě souřadnic 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 v_{i}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5055718020b424211605e20745791032746245e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.927ex; height:2.843ex;" alt="{\displaystyle v_{i}^{\prime }}"></span> jsou složky vektoru <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> v  nové soustavě souřadnic. Tuto transformaci lze vyjádřit v maticovém zápisu jako <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} ^{\prime }=\mathbf {A} \cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ^{\prime }=\mathbf {A} \cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14dd2c309d798af3aed8344572453d452fb2ce88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.304ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} ^{\prime }=\mathbf {A} \cdot \mathbf {v} }"></span>, kde <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/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> je transformační <a href="/wiki/Matice" title="Matice">matice</a> se složkami <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebea6cd2813c330c798921a2894b358f7b643917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}"></span>. Někdy se požaduje invariance ne vůči všem lineárním transformacím, ale jen rotacím a zrcadlením (v klasické mechanice), nebo <a href="/wiki/Lorentzova_transformace" title="Lorentzova transformace">Lorentzovým transformacím</a> (v <a href="/wiki/Speci%C3%A1ln%C3%AD_teorie_relativity" title="Speciální teorie relativity">speciální relativitě</a>). </p><p>Pokud není vektor vázán k žádnému pevnému <a href="/wiki/Bod" title="Bod">bodu</a> prostoru, tzn. pro jeho vyjádření je důležitý pouze jeho směr a velikost, pak hovoříme o <b>volném vektoru</b>. Pokud je daný vektor spojen s určitým bodem prostoru (t.j. má počátek), pak hovoříme o <b>vázaném vektoru</b>. </p><p>Pokud je vektor definován v každém bodě prostoru, pak se hovoří o <a href="/wiki/Vektorov%C3%A9_pole" title="Vektorové pole">vektorovém poli</a>. </p><p>V matematice se pod pojmem vektor obvykle rozumí prvek nějakého <a href="/wiki/Vektorov%C3%BD_prostor" title="Vektorový prostor">vektorového prostoru</a>. Tyto prostory mohou být i nekonečněrozměrné, proto někdy má smysl mluvit, že i <a href="/wiki/Funkce_(matematika)" title="Funkce (matematika)">funkce</a> je vektor, anebo <i>stav fyzikálního systému</i> je vektor (v kvantové mechanice). </p> <div class="mw-heading mw-heading3"><h3 id="Pravý_a_axiální_vektor"><span id="Prav.C3.BD_a_axi.C3.A1ln.C3.AD_vektor"></span>Pravý a axiální vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=2" title="Editace sekce: Pravý a axiální vektor" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=2" title="Editovat zdrojový kód sekce Pravý a axiální vektor"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jako <b>pravý vektor</b> označujeme takovou vektorovou veličinu, která se dá nějakým způsobem měřit nebo počítat za předpokladu pevně zvolené ortonormální souřadnicové soustavy a když se podle stejných pravidel změří nebo spočte v souřadnicové soustavě, která je vůči původní <a href="/wiki/Oto%C4%8Den%C3%AD" title="Otočení">otočená</a> nebo <a href="/wiki/Prostorov%C3%A1_inverze" class="mw-redirect" title="Prostorová inverze">zrcadlená</a>, vyjde nám „stejný“ vektor (t.j. jeho souřadnice se vůči původním změnily podle stejného vzorce jako souřadnice bodů v prostoru). Při zrcadlení os tedy pro pravý vektor platí </p> <dl><dd><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} (-x_{i})=\mathbf {V} (x_{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} (-x_{i})=\mathbf {V} (x_{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b541b5b8a888e96f5f8a0d98f895662e91c7f060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.47ex; height:2.843ex;" alt="{\displaystyle \mathbf {V} (-x_{i})=\mathbf {V} (x_{i}),}"></span></dd></dl> <p>kde <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_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1200abe6f910b7bca6c0e6bdb4099ee0ab180499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.747ex; height:2.843ex;" alt="{\displaystyle (-x_{i})}"></span> označuje souřadnicovou soustavu, která má opačnou <a href="/wiki/Orientace_(matematika)" title="Orientace (matematika)">orientaci</a> jako <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_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65f04ce52a62128b2bcc7f11353b4e77374658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.939ex; height:2.843ex;" alt="{\displaystyle (x_{i})}"></span> . </p><p>Vektorovou veličinu, která se při rotacích transformuje stejně jako souřadnice, avšak při zrcadlení souřadnicových soustav mění znaménko, označujeme jako <b>axiální vektor</b> (<b>nepravý vektor</b> nebo <b>pseudovektor</b>). Při zrcadlení os tedy pro axiální vektor platí </p> <dl><dd><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} (-x_{i})=-\mathbf {V} (x_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} (-x_{i})=-\mathbf {V} (x_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5856680c6dc40af6f2536a6d5ad02a580a9acad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.278ex; height:2.843ex;" alt="{\displaystyle \mathbf {V} (-x_{i})=-\mathbf {V} (x_{i}).}"></span></dd></dl> <p>Matematicky se dá axiální vektor definovat jako prvek druhé <a href="/w/index.php?title=Vn%C4%9Bj%C5%A1%C3%AD_mocnina&action=edit&redlink=1" class="new" title="Vnější mocnina (stránka neexistuje)">vnější mocniny</a> prostoru (v dimenzi 3), resp. obecněji jako prvek <i>(n-1)</i>-ní vnější mocniny <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 \wedge ^{n-1}\mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mo>∧<!-- ∧ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge ^{n-1}\mathbf {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279295745dbd3433f8fca83224f153b24888f42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.889ex; height:2.676ex;" alt="{\displaystyle \wedge ^{n-1}\mathbf {V} }"></span> <i>n</i>-rozměrného vektorového prostoru <b>V</b>. Za předpokladu volby skalárního součinu a orientace na <b>V</b> pak lze takový prvek ztotožnit s vektorem (prvkem <b>V</b>) pomocí <a href="/w/index.php?title=Hodgeova_dualita&action=edit&redlink=1" class="new" title="Hodgeova dualita (stránka neexistuje)">Hodgeovy duality</a>. Znaménko výsledného vektoru pak závisí na volbě orientace. </p><p>Příkladem pravého vektoru je <a href="/wiki/Polohov%C3%BD_vektor" title="Polohový vektor">polohový vektor</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 \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> nebo vektor <a href="/wiki/Rychlost" title="Rychlost">rychlosti</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 \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>, axiálním vektorem je např. vektor <a href="/wiki/%C3%9Ahlov%C3%A1_rychlost" title="Úhlová rychlost">úhlové rychlosti</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 \mathbf {\Omega } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω<!-- Ω --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\Omega } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bbc5c501b899d658ddaa37ae734fe62c91f6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Omega } }"></span>. Pseudovektory se často konstruují z pravých vektorů pomocí <a href="/wiki/Vektorov%C3%BD_sou%C4%8Din" title="Vektorový součin">vektorového součinu</a> (je invariantní vůči rotacím, ale ne zrcadlením). </p> <div class="mw-heading mw-heading2"><h2 id="Reprezentace_vektoru">Reprezentace vektoru</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=3" title="Editace sekce: Reprezentace vektoru" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=3" title="Editovat zdrojový kód sekce Reprezentace vektoru"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Symboly pro vektory jsou obvykle tištěny tučně, jako <b>a</b>; to je také konvence použitá v této encyklopedii. Mezi další zvyklosti označování patří <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> nebo <span style="text-decoration: underline"><i>a</i></span>, zvlášť při ručním psaní. Alternativně lze použít i <i>ã</i>. </p><p> Vektory se obvykle v grafech nebo jiných <a href="/wiki/Diagram" title="Diagram">diagramech</a> označují jako orientované úsečky: </p><center><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Soubor:VectorAB.svg" class="mw-file-description" title="Grafická reprezentace vektoru."><img alt="Grafická reprezentace vektoru." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/169px-VectorAB.svg.png" decoding="async" width="169" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/254px-VectorAB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/VectorAB.svg/338px-VectorAB.svg.png 2x" data-file-width="169" data-file-height="81" /></a></span></center> <p>Zde bod <i>A</i> se nazývá <i>počáteční</i>, bod <i>B</i> <i>koncový bod</i>. Délka šipky představuje velikost vektoru, šipka určuje jeho (orientovaný) směr. </p><p>Vektory jsou také často vyjadřovány pomocí svých <a href="/wiki/Sou%C5%99adnicov%C3%BD_z%C3%A1pis_vektor%C5%AF" class="mw-redirect" title="Souřadnicový zápis vektorů">složek</a>, např. <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> pro vektor <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>. </p><p>V pokročilejší matematické či fyzikální literatuře se pro vektory žádné speciální značení nepoužívá a jsou označovány stejně jako ostatní veličiny, popř. se používá složkový zápis. Např. místo <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> se použije <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> nebo pouze <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>. </p><p><a href="/wiki/Kvantov%C3%A1_fyzika" title="Kvantová fyzika">Kvantová fyzika</a> používá pro zápis vektoru tzv. <a href="/wiki/Diracova_symbolika" class="mw-redirect" title="Diracova symbolika">Diracovu symboliku</a>. </p><p>V <a href="/wiki/Diferenci%C3%A1ln%C3%AD_geometrie" title="Diferenciální geometrie">diferenciální geometrii</a> se vektor v dané <a href="/wiki/Soustava_sou%C5%99adnic" title="Soustava souřadnic">souřadné soustavě</a> často vyjadřuje pomocí operátorů <a href="/wiki/Parci%C3%A1ln%C3%AD_derivace" title="Parciální derivace">parciálních derivací</a>, tedy např. jako </p> <dl><dd><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} =a_{x}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}x}}+a_{y}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}y}}+a_{z}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}z}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold">∂<!-- ∂ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∂<!-- ∂ --></mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold">∂<!-- ∂ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∂<!-- ∂ --></mi> </mrow> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold">∂<!-- ∂ --></mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∂<!-- ∂ --></mi> </mrow> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =a_{x}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}x}}+a_{y}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}y}}+a_{z}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}z}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12bb8c230335fda4d1379a9f36db29ba4f7772b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.667ex; height:6.009ex;" alt="{\displaystyle \mathbf {A} =a_{x}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}x}}+a_{y}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}y}}+a_{z}{\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}z}}\;.}"></span></dd></dl> <p>S výhodou se využívá faktu, že při obecných transformacích souřadnic se vektory transformují stejně jako parciální derivace – pomocí <a href="/wiki/%C5%98et%C3%ADzkov%C3%A9_pravidlo" title="Řetízkové pravidlo">řetízkového pravidla</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Operace_s_vektory">Operace s vektory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=4" title="Editace sekce: Operace s vektory" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=4" title="Editovat zdrojový kód sekce Operace s vektory"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sčítání_vektorů"><span id="S.C4.8D.C3.ADt.C3.A1n.C3.AD_vektor.C5.AF"></span>Sčítání vektorů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=5" title="Editace sekce: Sčítání vektorů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=5" title="Editovat zdrojový kód sekce Sčítání vektorů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pro dva vektory <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} ,\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ,\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88643597666fbf6379537d6b2c2fa05a006fb77e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.954ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} ,\mathbf {B} }"></span> ze stejného vektorového prostoru je definován jejich <a href="/wiki/Sou%C4%8Det" class="mw-redirect" title="Součet">součet</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 \mathbf {C} =\mathbf {A} +\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =\mathbf {A} +\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0dfbd22dd86467004abef69c956157b42fc662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.791ex; height:2.343ex;" alt="{\displaystyle \mathbf {C} =\mathbf {A} +\mathbf {B} }"></span>. Pro složky vektorů platí <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_{i}=A_{i}+B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}=A_{i}+B_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e868dbdef536e9097fac70417915bc446d0529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.507ex; height:2.509ex;" alt="{\displaystyle C_{i}=A_{i}+B_{i}}"></span> </p><p>Pokud jsou dva vektory na sebe <a href="/wiki/Ortogonalita" title="Ortogonalita">kolmé</a>, lze velikost výsledného vektoru určit <a href="/wiki/Pythagorova_v%C4%9Bta" title="Pythagorova věta">Pythagorovou větou</a>. Výsledný vektor je možno reprezentovat graficky a to doplněním do vektorového rovnoběžníku (nechť je cíl výsledného vektoru bod C, počátek bod A, cíl vektoru 1 bod B a cíl vektoru 2 bod D. Úhlopříčka AC vektorového rovnoběžníku ABCD pak představuje výsledný vektor. Délka této vektorové úsečky je rovna velikosti výsledného vektoru.) </p> <div class="mw-heading mw-heading3"><h3 id="Násobení_vektoru_číslem"><span id="N.C3.A1soben.C3.AD_vektoru_.C4.8D.C3.ADslem"></span>Násobení vektoru číslem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=6" title="Editace sekce: Násobení vektoru číslem" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=6" title="Editovat zdrojový kód sekce Násobení vektoru číslem"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pro libovolný vektor <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/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> a číslo <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> je definován vektor <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\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d0a628cd9cb32fe91198252b0e3b8552e73396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.176ex;" alt="{\displaystyle k\mathbf {A} }"></span> se složkami </p> <dl><dd><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\cdot A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅<!-- ⋅ --></mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b04076ca5bbeb7f5cfc6fa6dc068153c01b55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.433ex; height:2.509ex;" alt="{\displaystyle k\cdot A_{i}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Součin_vektorů"><span id="Sou.C4.8Din_vektor.C5.AF"></span>Součin vektorů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=7" title="Editace sekce: Součin vektorů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=7" title="Editovat zdrojový kód sekce Součin vektorů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Sou%C4%8Din" class="mw-redirect" title="Součin">Součin</a> vektorů lze definovat různým způsobem. Používané součiny vektorů jsou </p> <ul><li><a href="/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din" title="Skalární součin">skalární součin</a></li> <li><a href="/wiki/Vektorov%C3%BD_sou%C4%8Din" title="Vektorový součin">vektorový součin</a></li> <li><a href="/wiki/Sm%C3%AD%C5%A1en%C3%BD_sou%C4%8Din" title="Smíšený součin">smíšený součin</a></li> <li><a href="/wiki/Tenzorov%C3%BD_sou%C4%8Din" title="Tenzorový součin">tenzorový součin</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Vlastnosti_vektorových_operací"><span id="Vlastnosti_vektorov.C3.BDch_operac.C3.AD"></span>Vlastnosti vektorových operací</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=8" title="Editace sekce: Vlastnosti vektorových operací" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=8" title="Editovat zdrojový kód sekce Vlastnosti vektorových operací"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mějme vektory <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} ,\mathbf {B} ,\mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ,\mathbf {B} ,\mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9478e59b25b50d1eda7374d54780c6f1eb041196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.92ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} ,\mathbf {B} ,\mathbf {C} }"></span> a <a href="/wiki/Skal%C3%A1r" title="Skalár">skaláry</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 a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span>. Pak platí <a href="/wiki/Komutativita" title="Komutativita">komutativní zákon</a> pro sčítání vektorů </p> <dl><dd><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} +\mathbf {B} =\mathbf {B} +\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11267b1fc3acb7c47ecff8bfa77debcba71eaa4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.62ex; height:2.343ex;" alt="{\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }"></span></dd></dl> <p>Pro sčítání dvou vektorů platí <a href="/wiki/Asociativita" title="Asociativita">asociativní zákon</a>, tzn. </p> <dl><dd><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} +(\mathbf {B} +\mathbf {C} )=(\mathbf {A} +\mathbf {B} )+\mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +(\mathbf {B} +\mathbf {C} )=(\mathbf {A} +\mathbf {B} )+\mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c339ef0e6021aecc62ee40e60d040c8cb480d90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.782ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} +(\mathbf {B} +\mathbf {C} )=(\mathbf {A} +\mathbf {B} )+\mathbf {C} }"></span></dd></dl> <p>Platí také asociativní zákon pro násobení číslem, tedy </p> <dl><dd><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(b\mathbf {A} )=(ab)\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b\mathbf {A} )=(ab)\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75e2253fee7b73867c92e463a324e56ea082723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.211ex; height:2.843ex;" alt="{\displaystyle a(b\mathbf {A} )=(ab)\mathbf {A} }"></span></dd></dl> <p>Dále platí <a href="/wiki/Distributivita" title="Distributivita">distributivní zákony</a> </p> <dl><dd><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+b)\mathbf {A} =a\mathbf {A} +b\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)\mathbf {A} =a\mathbf {A} +b\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9764bee65c8c3c1f3de631c6fcc5f55622627f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.102ex; height:2.843ex;" alt="{\displaystyle (a+b)\mathbf {A} =a\mathbf {A} +b\mathbf {A} }"></span></dd> <dd><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(\mathbf {A} +\mathbf {B} )=a\mathbf {A} +a\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(\mathbf {A} +\mathbf {B} )=a\mathbf {A} +a\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c131ef613643da82d354569450378938415f85b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.119ex; height:2.843ex;" alt="{\displaystyle a(\mathbf {A} +\mathbf {B} )=a\mathbf {A} +a\mathbf {B} }"></span></dd></dl> <p>Existuje <a href="#Nulový_vektor">nulový vektor</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 \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"></span> splňující následující vztahy </p> <dl><dd><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} +\mathbf {0} =\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> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +\mathbf {0} =\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7088a7f3b80546564566f10b1f8b6d02cc48bbd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle \mathbf {A} +\mathbf {0} =\mathbf {A} }"></span></dd> <dd><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 {0} \cdot \mathbf {A} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} \cdot \mathbf {A} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7a596b02926167d0dfb00b2d4541ec7542e078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.47ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} \cdot \mathbf {A} =\mathbf {0} }"></span></dd> <dd><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\mathbf {0} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbf {0} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34261b750e37cff85f3145e5e21d7f79fb00e624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.002ex; height:2.176ex;" alt="{\displaystyle a\mathbf {0} =\mathbf {0} }"></span></dd></dl> <p>Ke každému vektoru <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/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> existuje <i>opačný vektor</i> <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"> <mo>−<!-- − --></mo> <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/76956e020d29720b81be504c1ec33a92ba027776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.828ex; height:2.343ex;" alt="{\displaystyle -\mathbf {A} }"></span>, pro nějž platí </p> <dl><dd><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} +(-\mathbf {A} )=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +(-\mathbf {A} )=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb22e37ca076c06e305f3346653d335bfa87978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.932ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} +(-\mathbf {A} )=\mathbf {0} }"></span></dd> <dd><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\mathbf {A} )=(-a)\mathbf {A} =a(-\mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(a\mathbf {A} )=(-a)\mathbf {A} =a(-\mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6929c19d34c56ed08abe080238bd210b3f00157e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.797ex; height:2.843ex;" alt="{\displaystyle -(a\mathbf {A} )=(-a)\mathbf {A} =a(-\mathbf {A} )}"></span></dd></dl> <p>Pokud <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} =\mathbf {A} +\mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {A} +\mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a2e8f4925e60b23d662fb11a12040a89c41af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.791ex; height:2.343ex;" alt="{\displaystyle \mathbf {B} =\mathbf {A} +\mathbf {C} }"></span>, pak </p> <dl><dd><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 {C} =\mathbf {B} +(-\mathbf {A} )=\mathbf {B} -\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =\mathbf {B} +(-\mathbf {A} )=\mathbf {B} -\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86d4825e062ba5c560c724e4b7e72ae4d92b1d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.267ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} =\mathbf {B} +(-\mathbf {A} )=\mathbf {B} -\mathbf {A} }"></span></dd></dl> <p>Za <a href="/wiki/Line%C3%A1rn%C3%AD_kombinace" title="Lineární kombinace">lineární kombinaci</a> dvou vektorů <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} ,\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ,\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88643597666fbf6379537d6b2c2fa05a006fb77e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.954ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} ,\mathbf {B} }"></span> je považován vektor <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 {C} =a\mathbf {A} +b\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =a\mathbf {A} +b\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4529a09ac81c8558528d064ad6b4114131817f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.018ex; height:2.343ex;" alt="{\displaystyle \mathbf {C} =a\mathbf {A} +b\mathbf {B} }"></span>, kde <i>a</i>, <i>b</i> jsou libovolná <a href="/wiki/%C4%8C%C3%ADslo" title="Číslo">čísla</a>, jehož složky jsou </p> <dl><dd><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_{i}=aA_{i}+bB_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}=aA_{i}+bB_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84584d4f67dfcf0ad33a6568a59fb4327960571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.734ex; height:2.509ex;" alt="{\displaystyle C_{i}=aA_{i}+bB_{i}}"></span></dd></dl> <p><br /> Dva <a href="/wiki/Line%C3%A1rn%C3%AD_z%C3%A1vislost" class="mw-redirect" title="Lineární závislost">lineárně závislé</a> vektory označujeme jako <i>kolineární</i> (<i>rovnoběžné</i>). Jsou-li dva vektory lineárně závislé, je jeden z nich násobkem druhého, oba tedy určují stejný směr v prostoru a jsou tedy rovnoběžné. <a href="/wiki/Vektorov%C3%BD_sou%C4%8Din" title="Vektorový součin">Vektorový součin</a> dvou kolineárních vektorů v <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> </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/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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 {R} ^{3}}"></span> je nulový. </p><p>Tři vzájemně lineárně závislé vektory označujeme jako <i>komplanární</i>. Komplanární vektory leží v jedné rovině. <a href="/wiki/Sm%C3%AD%C5%A1en%C3%BD_sou%C4%8Din" title="Smíšený součin">Smíšený součin</a> komplanárních vektorů v <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> </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/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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 {R} ^{3}}"></span>je nulový. </p><p>Pro součiny vektorů v <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> </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/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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 {R} ^{3}}"></span> platí důležité vztahy, jako je např. <a href="/wiki/Jacobiho_identita" class="mw-redirect" title="Jacobiho identita">Jacobiho identita</a> pro dvojitý vektorový součin, tzn. </p> <dl><dd><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} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×<!-- × --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>×<!-- × --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1941d61d2536babf6d243e3c277ef63d542af047" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.967ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0}"></span>.</dd> <dd>Tato rovnost mj. ukazuje, že vektorové násobení není <a href="/wiki/Asociativita" title="Asociativita">asociativní</a>.</dd></dl> <p>Dále platí tzv. <i>Lagrangeova identita</i> </p> <dl><dd><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} \times \mathbf {B} )\cdot (\mathbf {C} \times \mathbf {D} )=(\mathbf {A} \cdot \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \cdot \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {C} \times \mathbf {D} )=(\mathbf {A} \cdot \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \cdot \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb3aa0d7980bf0e550b5a745f1ab31477dcf47e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.575ex; height:2.843ex;" alt="{\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {C} \times \mathbf {D} )=(\mathbf {A} \cdot \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \cdot \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}"></span>.</dd></dl> <p>Jejím speciálním případem je vztah </p> <dl><dd><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} \times \mathbf {B} )}^{2}={\mathbf {A} }^{2}{\mathbf {B} }^{2}-{(\mathbf {A} \cdot \mathbf {B} )}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(\mathbf {A} \times \mathbf {B} )}^{2}={\mathbf {A} }^{2}{\mathbf {B} }^{2}-{(\mathbf {A} \cdot \mathbf {B} )}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c22fd84858b5582e5888f7388c3ca3c41dba862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.056ex; height:3.343ex;" alt="{\displaystyle {(\mathbf {A} \times \mathbf {B} )}^{2}={\mathbf {A} }^{2}{\mathbf {B} }^{2}-{(\mathbf {A} \cdot \mathbf {B} )}^{2}}"></span>.</dd></dl> <p><br /> Dalšími užívanými vztahy jsou </p> <dl><dd><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} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} (\mathbf {B} \times \mathbf {D} )]\mathbf {C} -[\mathbf {A} (\mathbf {B} \times \mathbf {C} )]\mathbf {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>×<!-- × --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>−<!-- − --></mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} (\mathbf {B} \times \mathbf {D} )]\mathbf {C} -[\mathbf {A} (\mathbf {B} \times \mathbf {C} )]\mathbf {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcca9d9127d57ceebc0c89bae0e3426b0c08356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.67ex; height:2.843ex;" alt="{\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} (\mathbf {B} \times \mathbf {D} )]\mathbf {C} -[\mathbf {A} (\mathbf {B} \times \mathbf {C} )]\mathbf {D} }"></span></dd> <dd><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} \times [\mathbf {B} \times (\mathbf {C} \times \mathbf {D} )]=(\mathbf {A} \times \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \times \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×<!-- × --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \times [\mathbf {B} \times (\mathbf {C} \times \mathbf {D} )]=(\mathbf {A} \times \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \times \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0caea467cff6a89c86a5e791d1577d47aa736c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.544ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} \times [\mathbf {B} \times (\mathbf {C} \times \mathbf {D} )]=(\mathbf {A} \times \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \times \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Invariance_operací"><span id="Invariance_operac.C3.AD"></span>Invariance operací</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=9" title="Editace sekce: Invariance operací" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=9" title="Editovat zdrojový kód sekce Invariance operací"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sčítání vektorů je invariantní vůči všem lineárním zobrazením, tj. <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} (\mathbf {x} +\mathbf {y} )=\mathbf {A} \mathbf {x} +\mathbf {A} \mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {x} +\mathbf {y} )=\mathbf {A} \mathbf {x} +\mathbf {A} \mathbf {y} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13eb8b8b8ad5d6c59f7a0ff9fb4d9582e79726bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.291ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} (\mathbf {x} +\mathbf {y} )=\mathbf {A} \mathbf {x} +\mathbf {A} \mathbf {y} }"></span> pro nějakou lineární transformaci <b>A</b>, přičemž <b>x</b> a <b>y</b> označují vektory. </p><p>Vektorový součin dvou vektorů z <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> </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/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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 {R} ^{3}}"></span> je invariantní vůči rotacím (ale ne zrcadlením). To znamená <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} (\mathbf {v} \times \mathbf {w} )=\mathbf {A} (\mathbf {v} )\times \mathbf {A} (\mathbf {w} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {v} \times \mathbf {w} )=\mathbf {A} (\mathbf {v} )\times \mathbf {A} (\mathbf {w} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0dac074ccc5ee38ef7239a83435f9414ebbb001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.95ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} (\mathbf {v} \times \mathbf {w} )=\mathbf {A} (\mathbf {v} )\times \mathbf {A} (\mathbf {w} )}"></span> pro libovolnou rotaci <b>A</b>. Znamená to, že vektorový součin je dobře definován i na abstraktním třírozměrném reálném vektorovém prostorů, pokud je na něm definován skalární součin a <a href="/wiki/Orientace_(matematika)" title="Orientace (matematika)">orientace</a>. Vektorový součin dvou vektorů v prostoru je tedy dobře definován i „fyzikálně“, až na znaménko (je to <i>pseudovektor</i>). </p><p>Skalárni součin je invariantní vůči všem rotacím, ale navíc i zrcadlením (a nejen u třirozměrných reálných prostorových vektorů, ale i obecně.) </p><p>Smíšený součin tří vektorů z <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> </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/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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 {R} ^{3}}"></span> je invariantní vůči všem lineárním zobrazením, které zachovávají objem a nemění orientaci prostoru (množina takových zobrazení se standardně značí <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 SL(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SL(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316cd3b9142ca1e5626b96575497f00fd0d097ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.054ex; height:2.843ex;" alt="{\displaystyle SL(3)}"></span>). Znamená to opět, že při dané volbě orientace (fyzikálního) třírozměrného prostorů je smíšený součin 3 vektorů dobře definován, obecně jeho znaménko závisí na orientaci prostoru (je to <a href="/wiki/Skal%C3%A1r" title="Skalár">pseudoskalár</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Úhel_dvou_vektorů"><span id=".C3.9Ahel_dvou_vektor.C5.AF"></span>Úhel dvou vektorů</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=10" title="Editace sekce: Úhel dvou vektorů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=10" title="Editovat zdrojový kód sekce Úhel dvou vektorů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>lze určit ze znalosti <a href="/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din" title="Skalární součin">skalárního součinu</a> a <a href="/wiki/Norma_vektoru" class="mw-redirect" title="Norma vektoru">norem</a> obou nenulových vektorů (<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 {\mathbf {A} \cdot \mathbf {A} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2e1eb28399c7179030293ad5775dd056b11543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.097ex; height:3.176ex;" alt="{\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }}}"></span>) pomocí vztahu: </p> <dl><dd><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 \varphi ={\frac {\mathbf {A} \cdot \mathbf {B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo fence="false" stretchy="false">‖<!-- ‖ --></mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi ={\frac {\mathbf {A} \cdot \mathbf {B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6a642a292834fee7239712ba4ae2c67d529da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.523ex; height:6.176ex;" alt="{\displaystyle \cos \varphi ={\frac {\mathbf {A} \cdot \mathbf {B} }{\|\mathbf {A} \|\|\mathbf {B} \|}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Další_vektorové_operace"><span id="Dal.C5.A1.C3.AD_vektorov.C3.A9_operace"></span>Další vektorové operace</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=11" title="Editace sekce: Další vektorové operace" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=11" title="Editovat zdrojový kód sekce Další vektorové operace"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Operace na vektorech: </p> <ul><li><a href="/wiki/Norma_vektoru" class="mw-redirect" title="Norma vektoru">norma vektoru</a></li> <li><a href="/wiki/Divergence" class="mw-disambig" title="Divergence">divergence</a></li> <li><a href="/wiki/Rotace_(oper%C3%A1tor)" title="Rotace (operátor)">rotace</a></li> <li><a href="/wiki/Gradient_(matematika)" title="Gradient (matematika)">gradient</a> <a href="/wiki/Vektorov%C3%A9_pole" title="Vektorové pole">vektorového pole</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Zvláštní_druhy_vektorů"><span id="Zvl.C3.A1.C5.A1tn.C3.AD_druhy_vektor.C5.AF"></span>Zvláštní druhy vektorů</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=12" title="Editace sekce: Zvláštní druhy vektorů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=12" title="Editovat zdrojový kód sekce Zvláštní druhy vektorů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Jednotkový_vektor"><span id="Jednotkov.C3.BD_vektor"></span>Jednotkový vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=13" title="Editace sekce: Jednotkový vektor" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=13" title="Editovat zdrojový kód sekce Jednotkový vektor"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Jednotkovým vektorem</b> označujeme vektor <b>e</b> s jednotkovou normou, tzn. <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 {e} |=1}"> <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"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {e} |=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6962a168f245566872986a3be825554f93094d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.78ex; height:2.843ex;" alt="{\displaystyle |\mathbf {e} |=1}"></span>. </p><p>Jednotkový vektor ve směru libovolného nenulového vektoru <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> je určen vztahem </p> <dl><dd><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 {e} ={\frac {\mathbf {v} }{|\mathbf {v} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </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"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} ={\frac {\mathbf {v} }{|\mathbf {v} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749083c15be4299399ed57620acb8d65fbd7583f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.864ex; height:5.509ex;" alt="{\displaystyle \mathbf {e} ={\frac {\mathbf {v} }{|\mathbf {v} |}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Nulový_vektor"><span id="Nulov.C3.BD_vektor"></span>Nulový vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=14" title="Editace sekce: Nulový vektor" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=14" title="Editovat zdrojový kód sekce Nulový vektor"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Nulový vektor</b> <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 {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"></span> je zvláštním případem vektoru, který lze zapsat jako uspořádanou <i>n</i>-tici <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 (0,0,\cdots ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,\cdots ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aee6cb6ade1c8f881c3d3a8748e4d14ee96e79f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.509ex; height:2.843ex;" alt="{\displaystyle (0,0,\cdots ,0)}"></span>, tzn. všechny složky vektoru jsou <a href="/wiki/Nula" title="Nula">nulové</a>. </p><p>Norma nulového vektoru je rovna nule. </p><p>Z hlediska fyzikálního nemá nulový vektor směr ani orientaci. </p> <div class="mw-heading mw-heading3"><h3 id="Tečný_vektor"><span id="Te.C4.8Dn.C3.BD_vektor"></span>Tečný vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=15" title="Editace sekce: Tečný vektor" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=15" title="Editovat zdrojový kód sekce Tečný vektor"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Je vektor vyskytující se na <a href="/wiki/Varieta_(matematika)" title="Varieta (matematika)">varietách</a>, který má počátek (t.j. pevný bod, z kterého vychází) a určuje rychlost pohybujícího se objektu, který daným bodem prochází. (Formálně se definuje tak, že hladké funkci přiřadí příslušnou směrovou derivaci). Ve fyzice se často pracuje s vektorovými poli na varietách. </p> <div class="mw-heading mw-heading3"><h3 id="Hermitovsky_sdružený_vektor"><span id="Hermitovsky_sdru.C5.BEen.C3.BD_vektor"></span>Hermitovsky sdružený vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=16" title="Editace sekce: Hermitovsky sdružený vektor" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=16" title="Editovat zdrojový kód sekce Hermitovsky sdružený vektor"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vektor je obvykle vyjadřován jako sloupec s komponentami </p> <dl><dd><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 {\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd76e3e0f0609f813d9497433c658dcf9a07d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:7.267ex; height:13.843ex;" alt="{\displaystyle {\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}}"></span></dd></dl> <p>Hermitovské sdružení představuje aplikaci <a href="/wiki/Transponovan%C3%A1_matice" class="mw-redirect" title="Transponovaná matice">transpozice</a> a <a href="/wiki/Komplexn%C4%9B_sdru%C5%BEen%C3%A9_%C4%8D%C3%ADslo" title="Komplexně sdružené číslo">komplexního sdružení</a>, čímž získáme <i>hermitovsky sdružený vektor</i> se složkami </p> <dl><dd><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_{1}^{*},a_{2}^{*},\cdots ,a_{n}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1}^{*},a_{2}^{*},\cdots ,a_{n}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287db3e887a73ae15cf6f6d9943f147c42673980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.038ex; height:3.009ex;" alt="{\displaystyle (a_{1}^{*},a_{2}^{*},\cdots ,a_{n}^{*})}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=17" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=17" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Vektorov%C3%BD_prostor" title="Vektorový prostor">Vektorový prostor</a></li> <li><a href="/wiki/Vektorov%C3%A9_pole" title="Vektorové pole">Vektorové pole</a></li> <li><a href="/wiki/Vektorov%C3%BD_po%C4%8Det" class="mw-redirect" title="Vektorový počet">Vektorový počet</a></li> <li><a href="/wiki/Vektorov%C3%BD_sou%C4%8Din" title="Vektorový součin">Vektorový součin</a></li> <li><a href="/wiki/Vektorov%C3%A1_grafika" title="Vektorová grafika">Vektorová grafika</a></li> <li><a href="/wiki/Skal%C3%A1r" title="Skalár">Skalár</a></li> <li><a href="/wiki/Tenzor" title="Tenzor">Tenzor</a></li> <li><a href="/wiki/%C4%8Cty%C5%99vektor" title="Čtyřvektor">Čtyřvektor</a></li> <li><a href="/wiki/Diracova_notace" title="Diracova notace">Diracova notace</a></li> <li><a href="/wiki/Sou%C5%99adnicov%C3%BD_z%C3%A1pis_vektor%C5%AF" class="mw-redirect" title="Souřadnicový zápis vektorů">Souřadnicový zápis vektorů</a></li> <li><a href="/wiki/%C5%98%C3%ADdk%C3%BD_vektor" title="Řídký vektor">Řídký vektor</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=18" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=18" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons"><img alt="Logo Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://commons.wikimedia.org/wiki/Category:Vectors" class="extiw" title="c:Category:Vectors">vektor</a></span> na <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></span></span></span><i> </i></li></ul> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output .navbox2{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox2 .navbox2{margin-top:0}.mw-parser-output .navbox2+.navbox2{margin-top:-1px}.mw-parser-output .navbox2-inner,.mw-parser-output .navbox2-subgroup{width:100%}.mw-parser-output .navbox2-group,.mw-parser-output .navbox2-title,.mw-parser-output .navbox2-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output th.navbox2-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox2,.mw-parser-output .navbox2-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox2-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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