Vektor – Wikipédia

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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">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</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">Bevezető</div> </a> </li> <li id="toc-Általános_leírás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Általános_leírás"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Általános leírás</span> </div> </a> <ul id="toc-Általános_leírás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definíció" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definíció"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definíció</span> </div> </a> <button aria-controls="toc-Definíció-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>A(z) Definíció alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Definíció-sublist" class="vector-toc-list"> <li id="toc-Lineáris_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lineáris_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Lineáris algebra</span> </div> </a> <ul id="toc-Lineáris_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometria"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Geometria</span> </div> </a> <ul id="toc-Geometria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analízis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analízis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Analízis</span> </div> </a> <ul id="toc-Analízis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vektorműveletek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorműveletek"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vektorműveletek</span> </div> </a> <button aria-controls="toc-Vektorműveletek-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>A(z) Vektorműveletek alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Vektorműveletek-sublist" class="vector-toc-list"> <li id="toc-Összeadás" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Összeadás"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Összeadás</span> </div> </a> <ul id="toc-Összeadás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Szorzatok" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Szorzatok"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Szorzatok</span> </div> </a> <ul id="toc-Szorzatok-sublist" class="vector-toc-list"> <li id="toc-Skalárral_szorzás" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Skalárral_szorzás"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Skalárral szorzás</span> </div> </a> <ul id="toc-Skalárral_szorzás-sublist" class="vector-toc-list"> <li id="toc-Lineáris_kombináció" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Lineáris_kombináció"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1.1</span> <span>Lineáris kombináció</span> </div> </a> <ul id="toc-Lineáris_kombináció-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Skaláris_szorzat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Skaláris_szorzat"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Skaláris szorzat</span> </div> </a> <ul id="toc-Skaláris_szorzat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vektoriális_szorzat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vektoriális_szorzat"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Vektoriális szorzat</span> </div> </a> <ul id="toc-Vektoriális_szorzat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tenzori_(diadikus)_szorzat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tenzori_(diadikus)_szorzat"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Tenzori (diadikus) szorzat</span> </div> </a> <ul id="toc-Tenzori_(diadikus)_szorzat-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Alkalmazások" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alkalmazások"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Alkalmazások</span> </div> </a> <button aria-controls="toc-Alkalmazások-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>A(z) Alkalmazások alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Alkalmazások-sublist" class="vector-toc-list"> <li id="toc-A_matematikában" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_matematikában"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>A matematikában</span> </div> </a> <ul id="toc-A_matematikában-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_fizikában" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_fizikában"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>A fizikában</span> </div> </a> <ul id="toc-A_fizikában-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_számítástechnikában" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_számítástechnikában"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>A számítástechnikában</span> </div> </a> <ul id="toc-A_számítástechnikában-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Források</span> </div> </a> <ul id="toc-Források-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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="Ugrás egy más nyelvű szócikkre. Elérhető 95 nyelven" > <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 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclidean_vector" title="Euclidean vector – angol" lang="en" hreflang="en" data-title="Euclidean vector" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektor_(Wiskunde)" title="Vektor (Wiskunde) – afrikaans" lang="af" hreflang="af" data-title="Vektor (Wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" 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 – svájci német" lang="gsw" hreflang="gsw" data-title="Vektor" data-language-autonym="Alemannisch" data-language-local-name="svájci német" 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 – amhara" lang="am" hreflang="am" data-title="ጨረር" data-language-autonym="አማርኛ" data-language-local-name="amhara" 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 – arab" lang="ar" hreflang="ar" data-title="متجه" data-language-autonym="العربية" data-language-local-name="arab" 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 – asztúr" lang="ast" hreflang="ast" data-title="Vector" data-language-autonym="Asturianu" data-language-local-name="asztúr" 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) – azerbajdzsáni" lang="az" hreflang="az" data-title="Vektor (həndəsə)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" 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) – baskír" lang="ba" hreflang="ba" data-title="Вектор (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="baskír" 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) – belarusz" lang="be" hreflang="be" data-title="Вектар (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarusz" 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 – bolgár" lang="bg" hreflang="bg" data-title="Вектор" data-language-autonym="Български" data-language-local-name="bolgár" 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 – bangla" lang="bn" hreflang="bn" data-title="সদিক রাশি" data-language-autonym="বাংলা" data-language-local-name="bangla" 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 – bosnyák" lang="bs" hreflang="bs" data-title="Euklidski vektor" data-language-autonym="Bosanski" data-language-local-name="bosnyák" 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) – katalán" lang="ca" hreflang="ca" data-title="Vector (matemàtiques)" data-language-autonym="Català" data-language-local-name="katalán" 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 – közép-ázsiai kurd" lang="ckb" hreflang="ckb" data-title="ئاڕاستەبڕی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="közép-ázsiai kurd" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektor" title="Vektor – cseh" lang="cs" hreflang="cs" data-title="Vektor" data-language-autonym="Čeština" data-language-local-name="cseh" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Вектор (геометри) – csuvas" lang="cv" hreflang="cv" data-title="Вектор (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" 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 – walesi" lang="cy" hreflang="cy" data-title="Fector" data-language-autonym="Cymraeg" data-language-local-name="walesi" 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) – dán" lang="da" hreflang="da" data-title="Vektor (geometri)" data-language-autonym="Dansk" data-language-local-name="dán" 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 – német" lang="de" hreflang="de" data-title="Vektor" data-language-autonym="Deutsch" data-language-local-name="német" 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 – görög" lang="el" hreflang="el" data-title="Ευκλείδειο διάνυσμα" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektoro" title="Vektoro – eszperantó" lang="eo" hreflang="eo" data-title="Vektoro" data-language-autonym="Esperanto" data-language-local-name="eszperantó" 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 – spanyol" lang="es" hreflang="es" data-title="Vector" data-language-autonym="Español" data-language-local-name="spanyol" 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 – észt" lang="et" hreflang="et" data-title="Vektor" data-language-autonym="Eesti" data-language-local-name="észt" 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) – baszk" lang="eu" hreflang="eu" data-title="Bektore (matematika)" data-language-autonym="Euskara" data-language-local-name="baszk" 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 – perzsa" lang="fa" hreflang="fa" data-title="بردار اقلیدسی" data-language-autonym="فارسی" data-language-local-name="perzsa" 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 – finn" lang="fi" hreflang="fi" data-title="Vektori" data-language-autonym="Suomi" data-language-local-name="finn" 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 – francia" lang="fr" hreflang="fr" data-title="Vecteur euclidien" data-language-autonym="Français" data-language-local-name="francia" 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 – északi fríz" lang="frr" hreflang="frr" data-title="Vektor" data-language-autonym="Nordfriisk" data-language-local-name="északi fríz" 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 – ír" lang="ga" hreflang="ga" data-title="Veicteoir" data-language-autonym="Gaeilge" data-language-local-name="ír" 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 – skóciai kelta" lang="gd" hreflang="gd" data-title="Bheactor" data-language-autonym="Gàidhlig" data-language-local-name="skóciai kelta" 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 – gallego" lang="gl" hreflang="gl" data-title="Vector" data-language-autonym="Galego" data-language-local-name="gallego" 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 – héber" lang="he" hreflang="he" data-title="וקטור אוקלידי" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%B0%E0%A4%BE%E0%A4%B6%E0%A4%BF" title="सदिश राशि – hindi" lang="hi" hreflang="hi" data-title="सदिश राशि" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vektor" title="Vektor – horvát" lang="hr" hreflang="hr" data-title="Vektor" data-language-autonym="Hrvatski" data-language-local-name="horvát" 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 – haiti kreol" lang="ht" hreflang="ht" data-title="Vektè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haiti kreol" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Vektor_Euklides" title="Vektor Euklides – indonéz" lang="id" hreflang="id" data-title="Vektor Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" 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 – idó" lang="io" hreflang="io" data-title="Vektoro" data-language-autonym="Ido" data-language-local-name="idó" 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) – izlandi" lang="is" hreflang="is" data-title="Vigur (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="izlandi" 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) – olasz" lang="it" hreflang="it" data-title="Vettore (matematica)" data-language-autonym="Italiano" data-language-local-name="olasz" 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 – japán" lang="ja" hreflang="ja" data-title="空間ベクトル" data-language-autonym="日本語" data-language-local-name="japán" 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 – grúz" lang="ka" hreflang="ka" data-title="ვექტორი" data-language-autonym="ქართული" data-language-local-name="grúz" 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 – kazah" lang="kk" hreflang="kk" data-title="Вектор" data-language-autonym="Қазақша" data-language-local-name="kazah" 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 – koreai" lang="ko" hreflang="ko" data-title="유클리드 벡터" data-language-autonym="한국어" data-language-local-name="koreai" 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) – latin" lang="la" hreflang="la" data-title="Vector (mathematica)" data-language-autonym="Latina" data-language-local-name="latin" 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) – lombard" lang="lmo" hreflang="lmo" data-title="Vettor (matematega)" data-language-autonym="Lombard" data-language-local-name="lombard" 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 – litván" lang="lt" hreflang="lt" data-title="Vektorius" data-language-autonym="Lietuvių" data-language-local-name="litván" 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 – lett" lang="lv" hreflang="lv" data-title="Vektors" data-language-autonym="Latviešu" data-language-local-name="lett" 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="kiemelt szócikk"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор – macedón" lang="mk" hreflang="mk" data-title="Вектор" data-language-autonym="Македонски" data-language-local-name="macedón" 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) – malajálam" lang="ml" hreflang="ml" data-title="സദിശം (ജ്യാമിതി)" data-language-autonym="മലയാളം" data-language-local-name="malajálam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Евклидийн вектор – mongol" lang="mn" hreflang="mn" data-title="Евклидийн вектор" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Vektor" title="Vektor – maláj" lang="ms" hreflang="ms" data-title="Vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="maláj" 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 – máltai" lang="mt" hreflang="mt" data-title="Vettur ewklidju" data-language-autonym="Malti" data-language-local-name="máltai" 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) – erzjány" lang="myv" hreflang="myv" data-title="Вектор (геометрия)" data-language-autonym="Эрзянь" data-language-local-name="erzjány" 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 – alsónémet" lang="nds" hreflang="nds" data-title="Vekter" data-language-autonym="Plattdüütsch" data-language-local-name="alsónémet" 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) – holland" lang="nl" hreflang="nl" data-title="Vector (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="holland" 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 – norvég (nynorsk)" lang="nn" hreflang="nn" data-title="Vektor" data-language-autonym="Norsk nynorsk" data-language-local-name="norvég (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) – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Vektor (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kalqabee" title="Kalqabee – oromo" lang="om" hreflang="om" data-title="Kalqabee" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektor" title="Wektor – lengyel" lang="pl" hreflang="pl" data-title="Wektor" data-language-autonym="Polski" data-language-local-name="lengyel" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Vetor" title="Vetor – Piedmontese" lang="pms" hreflang="pms" data-title="Vetor" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D8%A7%D9%82%D9%84%D9%8A%D8%AF%D8%B3_%D9%84%D9%88%D8%B1%DB%8C" title="د اقليدس لوری – pastu" lang="ps" hreflang="ps" data-title="د اقليدس لوری" data-language-autonym="پښتو" data-language-local-name="pastu" 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) – portugál" lang="pt" hreflang="pt" data-title="Vetor (matemática)" data-language-autonym="Português" data-language-local-name="portugál" 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 – román" lang="ro" hreflang="ro" data-title="Vector euclidian" data-language-autonym="Română" data-language-local-name="román" 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) – orosz" lang="ru" hreflang="ru" data-title="Вектор (геометрия)" data-language-autonym="Русский" data-language-local-name="orosz" 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) – szaha" lang="sah" hreflang="sah" data-title="Вектор (геометрия)" data-language-autonym="Саха тыла" data-language-local-name="szaha" 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 – szicíliai" lang="scn" hreflang="scn" data-title="Vettura euclideu" data-language-autonym="Sicilianu" data-language-local-name="szicíliai" 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 – szerbhorvát" lang="sh" hreflang="sh" data-title="Vektor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="szerbhorvát" 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 – szingaléz" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දෛශිකය" data-language-autonym="සිංහල" data-language-local-name="szingaléz" 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) – szlovák" lang="sk" hreflang="sk" data-title="Vektor (matematika)" data-language-autonym="Slovenčina" data-language-local-name="szlovák" 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) – szlovén" lang="sl" hreflang="sl" data-title="Vektor (matematika)" data-language-autonym="Slovenščina" data-language-local-name="szlovén" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Vektor" title="Vektor – inari számi" lang="smn" hreflang="smn" data-title="Vektor" data-language-autonym="Anarâškielâ" data-language-local-name="inari számi" 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 – albán" lang="sq" hreflang="sq" data-title="Vektori" data-language-autonym="Shqip" data-language-local-name="albán" 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 – szerb" lang="sr" hreflang="sr" data-title="Вектор" data-language-autonym="Српски / srpski" data-language-local-name="szerb" 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) – szundanéz" lang="su" hreflang="su" data-title="Véktor (rohangan)" data-language-autonym="Sunda" data-language-local-name="szundanéz" 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 – svéd" lang="sv" hreflang="sv" data-title="Vektor" data-language-autonym="Svenska" data-language-local-name="svéd" 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 – sziléziai" lang="szl" hreflang="szl" data-title="Wektůr" data-language-autonym="Ślůnski" data-language-local-name="sziléziai" 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 – tamil" lang="ta" hreflang="ta" data-title="திசையன்" data-language-autonym="தமிழ்" data-language-local-name="tamil" 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 – thai" lang="th" hreflang="th" data-title="เวกเตอร์" data-language-autonym="ไทย" data-language-local-name="thai" 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 – türkmén" lang="tk" hreflang="tk" data-title="Wektor ululyklar" data-language-autonym="Türkmençe" data-language-local-name="türkmén" 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 – török" lang="tr" hreflang="tr" data-title="Vektör" data-language-autonym="Türkçe" data-language-local-name="török" 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 – ukrán" lang="uk" hreflang="uk" data-title="Евклідів вектор" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C_%D8%B3%D9%85%D8%AA%DB%8C%DB%81" title="اقلیدسی سمتیہ – urdu" lang="ur" hreflang="ur" data-title="اقلیدسی سمتیہ" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) – üzbég" lang="uz" hreflang="uz" data-title="Vektor (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="üzbég" 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 – vietnámi" lang="vi" hreflang="vi" data-title="Vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 – wu kínai" lang="wuu" hreflang="wuu" data-title="向量" data-language-autonym="吴语" data-language-local-name="wu kínai" 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 – jiddis" lang="yi" hreflang="yi" data-title="וועקטאר" data-language-autonym="ייִדיש" data-language-local-name="jiddis" 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 – kínai" lang="zh" hreflang="zh" data-title="向量" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng" title="Hiòng-liōng – min nan kínai" lang="nan" hreflang="nan" data-title="Hiòng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan kínai" 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 – kantoni" lang="yue" hreflang="yue" data-title="向量" data-language-autonym="粵語" data-language-local-name="kantoni" 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="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</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="Névterek"> <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="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:Vektor" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</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="Nyelvvariáns váltása" > <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">magyar</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="Nézetek"> <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>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <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="Eszközök" > <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">Eszközök</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">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </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>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</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>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </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%A1lis:Mi_hivatkozik_erre/Vektor" title="Az erre a lapra hivatkozó más lapok listája [j]" accesskey="j"><span>Mi hivatkozik erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/Vektor" rel="nofollow" title="Az erről a lapról hivatkozott lapok utolsó változtatásai [k]" accesskey="k"><span>Kapcsolódó változtatások</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Speci%C3%A1lis_lapok" title="Az összes speciális lap listája [q]" accesskey="q"><span>Speciális lapok</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Vektor&oldid=26883192" title="Állandó hivatkozás ezen lap ezen változatához"><span>Hivatkozás erre a változatra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Vektor&action=info" title="További információk erről a lapról"><span>Lapinformációk</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Hivatkoz%C3%A1s&page=Vektor&id=26883192&wpFormIdentifier=titleform" title="Információk a lap idézésével kapcsolatban"><span>Hogyan hivatkozz erre a lapra?</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FVektor"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FVektor"><span>QR-kód letöltése</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"> Nyomtatás/exportálás </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%A1lis:K%C3%B6nyv&bookcmd=book_creator&referer=Vektor"><span>Könyv készítése</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:DownloadAsPdf&page=Vektor&action=show-download-screen"><span>Letöltés PDF-ként</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Vektor&printable=yes" title="A lap nyomtatható változata [p]" accesskey="p"><span>Nyomtatható változat</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"> Társprojektek </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>Wikimédia 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="Kapcsolt adattárelem [g]" accesskey="g"><span>Wikidata-adatlap</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="Oldal eszközök"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <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">Megjelenés</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">elrejtés</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 id="mw-indicator-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--stable"></span><b>Ellenőrzött</b></indicator></div> </div> <div id="siteSub" class="noprint">A Wikipédiából, a szabad enciklopédiából</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Speci%C3%A1lis:Badtitle/Message">ellenőrizve</a>: <i>2024. február 14.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Disambig.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/19px-Disambig.svg.png" decoding="async" width="19" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/29px-Disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/38px-Disambig.svg.png 2x" data-file-width="230" data-file-height="183" /></a></span></td><td><i>Ez a szócikk a matematikai fogalomról szól. Hasonló címmel lásd még: <a href="/wiki/Vektor_(egy%C3%A9rtelm%C5%B1s%C3%ADt%C5%91_lap)" class="mw-disambig" title="Vektor (egyértelműsítő lap)">Vektor (egyértelműsítő lap)</a>.</i></td></tr></tbody></table> <p>A <b>vektor</b> a matematikában használatos fogalom, a <a href="/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra">lineáris algebra</a> egyik alapvető jelentőségű mennyisége. Általában az ember a vektorokkal mint <i>irányított szakaszokkal</i> szokott találkozni, de a matematikában a jelentése ennél lényegesen bőségesebb. A fogalom különböző irányú általánosításai egyes tudományágakban is megjelennek. Így például a biológiában vektornak nevezik a fertőzéseket terjesztő élőlényeket, hatásokat. Az analógia teljesen világos, a hordozótól a fertőzöttig vezető utat pont a köztesgazda jelenti, azaz két pontot egy meghatározott irányban köt össze. </p><p>A matematikában azonban sokkal elvontabb vektorokat is ismerünk. Ezek haszna sokszor a laikusok számára végképp nem nyilvánvaló, és ritkán ismertek, közismertek. Alkalmazásaik azonban széles körűek, különösen a modern tudományokban. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Általános_leírás"><span id=".C3.81ltal.C3.A1nos_le.C3.ADr.C3.A1s"></span>Általános leírás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=1" title="Szakasz szerkesztése: Általános leírás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vektor a <a href="/wiki/Matematika" title="Matematika">matematikában</a> egy felettébb fontos fogalom. Alkalmazásai rendkívül sokrétűek, a geometriától az absztrakt analízisig mindenhol lehet velük találkozni. Ennek megfelelően az értelmezése is többféleképpen történhet. </p><p>Maguk a vektorok a <a href="/wiki/Sz%C3%A1m" title="Szám">számok</a> egyfajta általánosításainak is tekinthetőek. Ezzel a megközelítéssel főleg az algebrai definíciók dolgoznak, és ekkor legjellemzőbb alkalmazásaik az egyenletrendszerek kezelése. </p><p>Mivel a fogalom eredete a fizika, ezért fizikai és geometriai meggondolások is szolgálhatnak alapjául, ekkor főleg a viselkedésük lesz a definíció alapja. A legközkeletűbb értelmezése a fogalomnak is geometriai: olyan szakasz, amit a nagyságán túl az iránya is jellemez. Ez szinte tipizálja a fogalmat, hiszen így olyan mennyiségeket, mint a sebesség vagy az erő, kényelmesen szemléletessé tudunk tenni.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>A vektorok legáltalánosabb, és így legmélyebb definícióját az <a href="/wiki/Matematikai_anal%C3%ADzis" title="Matematikai analízis">analízisben</a> találjuk, ahol a vektorok egy bizonyos típusú halmazhoz rendelhető másik halmaz elemei. Ez felettes értelmezése a fenti két definíciónak, hiszen mindkettőt magába foglalja. </p> <div class="mw-heading mw-heading2"><h2 id="Definíció"><span id="Defin.C3.ADci.C3.B3"></span>Definíció</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=2" title="Szakasz szerkesztése: Definíció"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lineáris_algebra"><span id="Line.C3.A1ris_algebra"></span>Lineáris algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=3" title="Szakasz szerkesztése: Lineáris algebra"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"></span> euklideszi geometriai tér. Ekkor 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 {\mathcal {E}}\times {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86706ccf8a4b031e9912b128f560b1f2fcb38ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}"></span> halmaz elemeit, mint rendezett párokat <i>irányított szakasznak</i> nevezzük. Tekintsük most a térben a párhuzamos eltolásokat. Ezek segítségével <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86706ccf8a4b031e9912b128f560b1f2fcb38ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}"></span> felett egy ekvivalenciarelációt határozhatunk meg. </p><p>Két pontpárt, <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </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/ce67314185650d6f0deba39db7dcec9378f4d4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}"></span>-t és <span class="mwe-math-element"><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,D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C,D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f43d345d78cc3f2e2da26476b0991b2b738926c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.534ex; height:2.843ex;" alt="{\displaystyle (C,D)}"></span>-t ekvivalensnek tekintünk, ha van olyan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> párhuzamos eltolás, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(A)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(A)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de7ca4aa813cad96bad7e2a8e9a4cd6f48b9738c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.676ex; height:2.843ex;" alt="{\displaystyle p(A)=C}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(B)=D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(B)=D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce3520d7a7ad0844ec7a0bfd2a4dc11f370bb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.855ex; height:2.843ex;" alt="{\displaystyle p(B)=D}"></span>. </p><p>Az ekvivalenciarelációk a halmazt faktorhalmazokra bontják. Az <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86706ccf8a4b031e9912b128f560b1f2fcb38ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}\times {\mathcal {E}}}"></span> felett az előbbiekben bevezetett ekvivalenciareláció faktorhalmazait <i>szabadvektoroknak</i>, a faktorhalmazok elemeit a szabadvektor <i>reprezentánsainak</i> nevezzük. </p> <div class="mw-heading mw-heading3"><h3 id="Geometria">Geometria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=4" title="Szakasz szerkesztése: Geometria"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A geometriában a vektorok az eltolások, mint transzformációk meghatározásában játszanak szerepet. Legyen ugyanis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> két párhuzamos sík. Ha a távolságuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95dd6dbd67bfda5e8e2a73c50538849673935dd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.477ex; height:2.676ex;" alt="{\displaystyle d\neq 0}"></span>, akkor a tér bármely <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> pontjához olyan módon rendel hozzá egy <span class="mwe-math-element"><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'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865f8505e90120a535a4ee68ca253dbd8ce7eb6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}"></span> pontot, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |XX'|=2d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <msup> <mi>X</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |XX'|=2d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c214c7d84f15ba6749f9c0b87e66ef2ad852a214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:3.009ex;" alt="{\displaystyle |XX'|=2d}"></span>. Ezen túl az <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> pontok képe olyan, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XX'||YY'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <msup> <mi>X</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <msup> <mi>Y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XX'||YY'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0033f9df2998e4ef8133c6dc676a8fe46385db35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.313ex; height:3.009ex;" alt="{\displaystyle XX'||YY'}"></span> és egyező irányításúak. Ezt a leképezést <i>eltolásnak</i> nevezzük. </p><p>Világos, hogy az eltoláshoz elegendő az <span class="mwe-math-element"><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,X')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,X')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4efde37bc4a5a9b910d11c43ab598b47358e3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.505ex; height:3.009ex;" alt="{\displaystyle (X,X')}"></span> pontpárt megadni, mivel ez bármely pontnak a képét megadja a fentebbiek szerint.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Az <span class="mwe-math-element"><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,X')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,X')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4efde37bc4a5a9b910d11c43ab598b47358e3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.505ex; height:3.009ex;" alt="{\displaystyle (X,X')}"></span> párt ekkor <i>vektornak</i> nevezzük. Eszerint egyébként a konkrét síkokra nincs is szükség, kizárólag a távolságuk és fekvésük lényeges. </p><p>A fentebbiek során a lineáris algebrai definíció szerinti osztályozást is megvalósítottuk, tehát itt is lehet beszélni az adott irányítású és hosszúságú vektorok ekvivalenciaosztályáról, amit ez alapján <i>szabadvektornak</i> nevezünk. Ha rögzítünk egy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> pontot, akkor a szabadvektorok azon reprezentánsait, amik kezdőpontja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>, <i>kötött vektoroknak</i> nevezzük. Ezek például egy koordináta-rendszer pontjait határozhatják meg. </p> <div class="mw-heading mw-heading3"><h3 id="Analízis"><span id="Anal.C3.ADzis"></span>Analízis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=5" title="Szakasz szerkesztése: Analízis"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> test,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> pedig halmaz. Ha értelmezünk két függvényt: </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 +:H\times H\rightarrow H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>:</mo> <mi>H</mi> <mo>×</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +:H\times H\rightarrow H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd89b6ba834d119ab2754c227c604d3c34fdad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.391ex; height:2.343ex;" alt="{\displaystyle +:H\times H\rightarrow H}"></span>, amit általában <i>összeadásnak</i> nevezünk, és</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 .:T\times H\rightarrow H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo>:</mo> <mi>T</mi> <mo>×</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle .:T\times H\rightarrow H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e7f6f7130ce92b827214c18c39e043354fb3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.544ex; height:2.176ex;" alt="{\displaystyle .:T\times H\rightarrow H}"></span>, amit leggyakrabban <i>skalárral való szorzás</i> néven emlegetünk,</dd></dl> <p>úgy, hogy a + asszociatív, kommutatív, invertálható és van neutrális eleme, valamint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \alpha ,\beta \in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha ,\beta \in T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2589a76f5107fdeb0f03e480ebe5622d7082fe26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.623ex; height:2.509ex;" alt="{\displaystyle \forall \alpha ,\beta \in T}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339611e55ee2c426a658792a069a3dc3a739c18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.716ex; height:2.509ex;" alt="{\displaystyle \forall x,y\in H}"></span> esetén </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha .(\beta .x)=(\alpha \cdot \beta ).x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>.</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>.</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>⋅</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha .(\beta .x)=(\alpha \cdot \beta ).x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925318df2104aab4e2a540800562439c99dd73fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.797ex; height:2.843ex;" alt="{\displaystyle \alpha .(\beta .x)=(\alpha \cdot \beta ).x}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha +\beta ).x=\alpha .x+\beta .x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mi>x</mi> <mo>=</mo> <mi>α</mi> <mo>.</mo> <mi>x</mi> <mo>+</mo> <mi>β</mi> <mo>.</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha +\beta ).x=\alpha .x+\beta .x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe682a617dc1594af2c89acf0ba9a8d48c666e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.319ex; height:2.843ex;" alt="{\displaystyle (\alpha +\beta ).x=\alpha .x+\beta .x}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha .(x+y)=\alpha .x+\alpha .y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>.</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo>.</mo> <mi>x</mi> <mo>+</mo> <mi>α</mi> <mo>.</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha .(x+y)=\alpha .x+\alpha .y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e30cd49bf190801e21a28caaa6119f24d08910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.124ex; height:2.843ex;" alt="{\displaystyle \alpha .(x+y)=\alpha .x+\alpha .y}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594582c90224d1b2021f9c4cab8d5cce190077bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.567ex; height:2.176ex;" alt="{\displaystyle 1.x=x}"></span></li></ul> <p>teljesül, akkor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>-t a <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> test feletti vektortérnek</i> nevezzük, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> elemeit pedig <i>vektoroknak</i>. </p><p>Mint látható, az analitikus definíció olyan mértékben absztrakt, hogy egészen furcsa halmazokat is tudunk vektortérként kezelni. Ilyen lehet például 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>-ben haladó konvergens sorozatok halmaza, vagy 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 H\rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29564958a34107b347d3c3535c9cafd222acfd14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.314ex; height:2.176ex;" alt="{\displaystyle H\rightarrow T}"></span> függvények halmaza. </p> <div class="mw-heading mw-heading2"><h2 id="Vektorműveletek"><span id="Vektorm.C5.B1veletek"></span>Vektorműveletek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=6" title="Szakasz szerkesztése: Vektorműveletek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Vector_addition3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/200px-Vector_addition3.svg.png" decoding="async" width="200" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/300px-Vector_addition3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/400px-Vector_addition3.svg.png 2x" data-file-width="190" data-file-height="78" /></a><figcaption>Két vektor összege rajzban a paralelogramma-szabály szerint képezhető</figcaption></figure> <p>A vektorműveletek értelmezése céljából érdemes a szemléletes és áttekinthető geometriai képből kiindulni. Ekkor a vektorokat <b>irányított szakaszként</b>, nyíllal ábrázoljuk, és így a jelentésük is a köznapi fogalmakkal megragadhatóvá válik. Ezeken keresztül néhány egyéb fogalom is világossá válik. </p> <div class="mw-heading mw-heading3"><h3 id="Összeadás"><span id=".C3.96sszead.C3.A1s"></span>Összeadás</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=7" title="Szakasz szerkesztése: Összeadás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Miután a szabadvektorokat a reprezentánsaikkal is jellemezhetjük, két szabadvektor összegét is így tudjuk értelmezni. Az <span class="mwe-math-element"><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> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> vektorok összegzését egy köztes pont felvételével tudjuk meghatározni. Vegyünk fel egy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> pontot, és tekintsük az <span class="mwe-math-element"><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> vektor azon reprezentánsát, aminek végpontja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, és 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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> vektor azon reprezentánsát, aminek kezdőpontja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Ekkor 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 {\vec {c}}={\vec {a}}+{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606f0086204fe0b7db7cd5f7d25baa80de992a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}}"></span> vektor egy reprezentánsát az <span class="mwe-math-element"><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> vektor kezdőpontja és 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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> végpontja jelöli ki. Ezt <i>paralelogramma-szabálynak</i> nevezzük. Az elnevezés a mellékelt ábra alapján válik érthetővé. </p><p>Szemléletesen, ha a vektorokat elmozdulásnak tekintjük, az <span class="mwe-math-element"><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 {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2035df645f8f359159ccd71fed8b4b1ee1b8bfe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:3.676ex;" alt="{\displaystyle {\vec {AB}}}"></span> vektor az <span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> pontból <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> pontba jutást jelenti. Hasonlóan 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 {\vec {BC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {BC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef35b893ba01926ffc83421652c8ec2b36e29706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:3.676ex;" alt="{\displaystyle {\vec {BC}}}"></span> vektor is értelmezhető, ekkor pedig az összegvektoruk az <span class="mwe-math-element"><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>-ból <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>-be jutást jelenti, azaz <span class="mwe-math-element"><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 {AB}}+{\vec {BC}}={\vec {AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {AB}}+{\vec {BC}}={\vec {AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aff18a42d308fb00e279e3e614880f5b77a29de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.486ex; height:3.843ex;" alt="{\displaystyle {\vec {AB}}+{\vec {BC}}={\vec {AC}}}"></span>. </p><p>E definíció alapján ellenőrizhető, hogy a definícióban megkövetelt feltételek az összeadásra hogyan teljesülnek. </p> <div class="mw-heading mw-heading3"><h3 id="Szorzatok">Szorzatok</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=8" title="Szakasz szerkesztése: Szorzatok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vektorokból a szorzásra emlékeztető művelet többféle is definiálható. Ebből egy külső, három pedig belső művelet.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Skalárral_szorzás"><span id="Skal.C3.A1rral_szorz.C3.A1s"></span>Skalárral szorzás</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=9" title="Szakasz szerkesztése: Skalárral szorzás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A skalárral való szorzás<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> ötlete a szorzat, mint ismételt összeadás hasonlóságából ered: </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 \lambda .{\vec {a}}=\underbrace {{\vec {a}}+{\vec {a}}+\dots +{\vec {a}}} _{\lambda {\text{ darab}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> darab</mtext> </mrow> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda .{\vec {a}}=\underbrace {{\vec {a}}+{\vec {a}}+\dots +{\vec {a}}} _{\lambda {\text{ darab}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd958a78b26ec722ef9a9451c776e0b5251c3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-right: -0.028ex; width:21.679ex; height:6.009ex;" alt="{\displaystyle \lambda .{\vec {a}}=\underbrace {{\vec {a}}+{\vec {a}}+\dots +{\vec {a}}} _{\lambda {\text{ darab}}}}"></span>, ha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5070bcd6a4ad284dee37e33eba3f81acfb962747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {N} }"></span>.</dd></dl> <p>Mivel ebben az esetben a vektor hossza nőtt meg, adja magát, hogy 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 \lambda .{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>.</mo> <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 \lambda .{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948d1c6cdb6cd96d216bfd3a922b888969696bf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.619ex; height:2.343ex;" alt="{\displaystyle \lambda .{\vec {a}}}"></span> 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 {\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>-val párhuzamos, és annál <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>-szor hosszabb bármilyen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>-beli elem esetén. </p><p>Mivel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> elemeit nevezzük skalárnak, adja magát az elnevezés is. A skalár szó ugyan számot jelent, de ez nem jelent problémát, mivel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> általában számhalmaz, rendszerint a <a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">valós</a> vagy <a href="/wiki/Komplex_sz%C3%A1mok" title="Komplex számok">komplex számok</a> teste. </p> <div class="mw-heading mw-heading5"><h5 id="Lineáris_kombináció"><span id="Line.C3.A1ris_kombin.C3.A1ci.C3.B3"></span>Lineáris kombináció</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=10" title="Szakasz szerkesztése: Lineáris kombináció"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A skalárral való szorzás eredménye vektor, így a <a href="/wiki/Vektort%C3%A9r" title="Vektortér">vektortér</a> eleme, és az összeadásban tag lehet. Ha adott <span class="mwe-math-element"><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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> és <span class="mwe-math-element"><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 {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"></span>, valamint <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> skalárok, akkor <i>lineáris kombinációnak</i> nevezzük az alábbi vektort: </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 {\vec {r}}=\alpha .{\vec {a}}+\beta .{\vec {b}}+\gamma .{\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>α</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>β</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>γ</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=\alpha .{\vec {a}}+\beta .{\vec {b}}+\gamma .{\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34a7c5a62e60606f7d0abad676bbbfc4b359d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.733ex; height:3.343ex;" alt="{\displaystyle {\vec {r}}=\alpha .{\vec {a}}+\beta .{\vec {b}}+\gamma .{\vec {c}}}"></span></dd></dl> <p>Általános alakban írva a <a href="/wiki/Line%C3%A1ris_kombin%C3%A1ci%C3%B3" title="Lineáris kombináció">lineáris kombináció</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 \sum _{i=1}^{n}\alpha _{i}.{\vec {a_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\alpha _{i}.{\vec {a_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a719cf79ffcbbd425911f1d41baa78ad1da465cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.387ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}\alpha _{i}.{\vec {a_{i}}}}"></span></dd></dl> <p>Ennek segítségével lehet értelmezni a vektortér <a href="/wiki/Dimenzi%C3%B3" title="Dimenzió">dimenzióját</a> is. A nullvektort ugyanis elő lehet állítani </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 {\vec {0}}=0.{\vec {a_{1}}}+0.{\vec {a_{2}}}+\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>0.</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}=0.{\vec {a_{1}}}+0.{\vec {a_{2}}}+\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0d0caff627fd711260493d24c62145420aab2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.931ex; height:3.343ex;" alt="{\displaystyle {\vec {0}}=0.{\vec {a_{1}}}+0.{\vec {a_{2}}}+\dots }"></span></dd></dl> <p>alakban. Ezt <i>triviális előállításnak</i> nevezzük. Előfordulhat azonban, hogy a zérusvektort nem nulla skalárokkal is megkaphatjuk, ekkor az <span class="mwe-math-element"><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_{i}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {a_{i}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf41efa3f62d6f1a839d599531a112438886fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.133ex; height:3.509ex;" alt="{\displaystyle ({\vec {a_{i}}})}"></span> rendszert <i>lineárisan függőnek</i> mondjuk. A legbővebb olyan rendszert, ami lineárisan független a vektortérben, a tér <i>bázisának</i> nevezzük. A bázis elemszáma lesz a vektortér dimenziója. </p><p>Ha pedig van egy bázisunk egy vektortérben, akkor bármely vektor megadható egyértelműen a bázis lineáris kombinációjaként. Az együtthatókat ekkor a vektor <i>koordinátáinak</i> nevezzük. Könnyen belátható, hogy két vektor összegének koordinátái a két vektor koordinátáinak összege. </p> <div class="mw-heading mw-heading4"><h4 id="Skaláris_szorzat"><span id="Skal.C3.A1ris_szorzat"></span>Skaláris szorzat</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=11" title="Szakasz szerkesztése: Skaláris szorzat"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Egyes fizikai összefüggések vektormennyiségekből skaláris értékek létrehozását igénylik. Ilyen például a munka kiszámításának a problémája. A definíció is őrzi a fizikai eredetet: a két vektor szorzata legyen maximális, ha párhuzamosak, és nulla, ha merőlegesek, továbbá a szorzat legyen lineáris függvénye a két vektor hosszának. </p><p>Az első feltétel szerint a szorzat a két vektor által bezárt szögtől függ. A legegyszerűbb függvény, ami ezt a feltételt kielégíti, a <a href="/wiki/Koszinusz" class="mw-redirect" title="Koszinusz">koszinusz</a> függvény. A második feltétellel együtt a szorzat kiszámítására 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 {\vec {a}}{\vec {b}}=a\cdot b\cdot \cos \phi }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}{\vec {b}}=a\cdot b\cdot \cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c262fe1a4eaad0b6b0a62daeeb6ebabdcf1195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.891ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}{\vec {b}}=a\cdot b\cdot \cos \phi }"></span></dd></dl> <p>képlet szolgál, ahol <span class="mwe-math-element"><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> az <span class="mwe-math-element"><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> vektor hossza. </p><p>Könnyen belátható, hogy a <a href="/wiki/Skal%C3%A1ris_szorzat" title="Skaláris szorzat">skaláris szorzat</a> kommutatív és a vektorok összeadására nézve disztributív, de nem asszociatív művelet. </p><p>A skaláris szorzat adott bázis esetén könnyebben is kiszámítható. Az egyszerűség kedvéért tegyük fel, hogy a bázis vektorai egymásra merőlegesek, ez mindig elérhető a <a href="/wiki/Gram%E2%80%93Schmidt-elj%C3%A1r%C3%A1s" title="Gram–Schmidt-eljárás">Gram–Schmidt-eljárás</a> segítségével például. Ekkor az <span class="mwe-math-element"><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> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> vektorok a bázis lineáris kombinációjaként felírva: </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 {\vec {a}}=\sum a_{i}{\vec {e}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=\sum a_{i}{\vec {e}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b8dd48e1cbb224dfc4a5150dbbcd44b393c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.123ex; height:3.843ex;" alt="{\displaystyle {\vec {a}}=\sum a_{i}{\vec {e}}_{i}}"></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 {\vec {b}}=\sum b_{i}{\vec {e}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∑</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=\sum b_{i}{\vec {e}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5156926ecc9216ecefbd7d62c828157d970122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.754ex; height:3.843ex;" alt="{\displaystyle {\vec {b}}=\sum b_{i}{\vec {e}}_{i}}"></span></dd></dl> <p>lesznek. A szorzáskor figyelembe véve a skaláris szorzat disztributivitását és az egységvektorok merőlegességét kapjuk: </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 {\vec {a}}{\vec {b}}=\sum a_{i}b_{i}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}{\vec {b}}=\sum a_{i}b_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8665489d079f4e86c8780b3dccc80555ef7c7927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.991ex; height:3.843ex;" alt="{\displaystyle {\vec {a}}{\vec {b}}=\sum a_{i}b_{i}}"></span>.</dd></dl> <p>Például számoljuk ki 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 {\vec {a}}=(2,1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(2,1,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5080f6e92ed7995fdb61d42b42fbffc60cd11290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.693ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(2,1,3)}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}=(4,-2,-3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=(4,-2,-3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20e9793d398e5c6bc4056cb1a4c65cd15cc4e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.173ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}=(4,-2,-3)}"></span> vektorok skaláris szorzatát: </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 {\vec {a}}{\vec {b}}=2\cdot 4+1\cdot (-2)+3\cdot (-3)=8-2-9=-3}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}{\vec {b}}=2\cdot 4+1\cdot (-2)+3\cdot (-3)=8-2-9=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a25eb5fc471014b7f40f9cfcef6a679f710a48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.686ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}{\vec {b}}=2\cdot 4+1\cdot (-2)+3\cdot (-3)=8-2-9=-3}"></span>.</dd></dl> <p>Nem merőleges bázisvektorok esetén a skaláris szorzat vegyes tagokat is fog tartalmazni, valamint egy, a bázist jellemző operátort. </p><p>Ha egy vektort önmagával szorzunk, akkor a definíció értelmében a hosszának a négyzetét kapjuk. Ezt a <a href="/wiki/Koszinuszt%C3%A9tel" title="Koszinusztétel">koszinusztétel</a> bizonyítása során is kihasználjuk. </p><p>A matematikában a vektor fogalmának általánosítása okán a skaláris szorzatot <i>belső szorzat</i> néven szokás emlegetni. </p> <div class="mw-heading mw-heading4"><h4 id="Vektoriális_szorzat"><span id="Vektori.C3.A1lis_szorzat"></span>Vektoriális szorzat</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=12" title="Szakasz szerkesztése: Vektoriális szorzat"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vektoriális szorzat egy kizárólag három és hét<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> dimenzióban értelmezhető belső szorzat. Eredete a skaláris szorzathoz hasonlóan fizikai: elsősorban a forgatónyomaték kezelésében bukkan fel, de később több más területen is kényelmesnek bizonyult a használata. A definíciója is ennek megfelelően elsősorban technikai jellegű. Legyen <span class="mwe-math-element"><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> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> két vektor. Ekkor hozzájuk rendelhető egy harmadik <span class="mwe-math-element"><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 {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"></span> vektor a következő szabályok szerint: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19504826864c3c47fae43e87cd7f82ba78289d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.484ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}=0}"></span>, ha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}||{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}||{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e1a432fd8cfeb31108ae21d3ae969359548c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.617ex; height:3.343ex;" alt="{\displaystyle {\vec {a}}||{\vec {b}}}"></span>;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"></span> maximális, ha a vektorok merőlegesek egymásra;</li> <li>a szorzat egyenesen arányos 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 {\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> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"></span> hosszával.</li> <li>a három vektor úgy helyezkedik el egymáshoz képest, mint a koordináta-rendszer x, y és z tengelyei.</li></ol> <p>Az első két feltételt a vektorok által bezárt szög szinusza is kielégíti, így a harmadik feltétellel együtt a skaláris szorzathoz hasonlóan 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 {\vec {a}}\times {\vec {b}}=a\cdot b\cdot \sin \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\times {\vec {b}}=a\cdot b\cdot \sin \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0370dad3729f921fcafe4ef155f75695ed77b1e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.476ex; height:3.176ex;" alt="{\displaystyle {\vec {a}}\times {\vec {b}}=a\cdot b\cdot \sin \phi }"></span></dd></dl> <p>képletet írhatjuk fel. Ebben nincsen benne a szorzatvektor iránya, úgyhogy azt továbbra is külön fel kell írnunk. </p><p>A vektoriális szorzat nem asszociatív, de ez nem meglepő. Még kevésbé meglepő, hogy nem kommutatív, viszont <i>antikommutatív</i>, azaz <span class="mwe-math-element"><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}}\times {\vec {b}}=-{\vec {b}}\times {\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> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <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}}\times {\vec {b}}=-{\vec {b}}\times {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668ff5336a1740a767812471704c2553294c9434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.235ex; height:3.009ex;" alt="{\displaystyle {\vec {a}}\times {\vec {b}}=-{\vec {b}}\times {\vec {a}}}"></span>. A vektorok összeadására nézve disztributív. </p><p>A tér merőleges bázisvektorai esetén érvényes összefüggések: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}\times {\vec {e}}_{2}={\vec {e}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}\times {\vec {e}}_{2}={\vec {e}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eba2ea5cf1ea712c033d97dfc100418096bccb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.771ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{1}\times {\vec {e}}_{2}={\vec {e}}_{3}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{2}\times {\vec {e}}_{3}={\vec {e}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}\times {\vec {e}}_{3}={\vec {e}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d9ae5e1eb79c1783bf28fe27d8c26d1c5c940a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.771ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{2}\times {\vec {e}}_{3}={\vec {e}}_{1}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{3}\times {\vec {e}}_{1}={\vec {e}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{3}\times {\vec {e}}_{1}={\vec {e}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b630d1ba52e5d98861a6dbed828417e6c8f3dc44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.771ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{3}\times {\vec {e}}_{1}={\vec {e}}_{2}}"></span>,</li></ol> <p>ezeket a koordinátás alak kiszámításakor használjuk ki. </p><p>A fenti tulajdonságok alapján levezethető a vektorok szorzatára vonatkozó számítási módszer: Ha <span class="mwe-math-element"><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}}\times {\vec {b}}={\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\times {\vec {b}}={\vec {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5976682d3867d0c9f0b7e243419b312e81a9e816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.486ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}\times {\vec {b}}={\vec {c}}}"></span>, akkor </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 {\vec {c}}_{1}={\vec {a}}_{2}{\vec {b}}_{3}-{\vec {a}}_{3}{\vec {b}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{1}={\vec {a}}_{2}{\vec {b}}_{3}-{\vec {a}}_{3}{\vec {b}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/561a08913922da004a9359831704fa5c55873c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.081ex; height:3.176ex;" alt="{\displaystyle {\vec {c}}_{1}={\vec {a}}_{2}{\vec {b}}_{3}-{\vec {a}}_{3}{\vec {b}}_{2}}"></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 {\vec {c}}_{2}={\vec {a}}_{3}{\vec {b}}_{1}-{\vec {a}}_{1}{\vec {b}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{2}={\vec {a}}_{3}{\vec {b}}_{1}-{\vec {a}}_{1}{\vec {b}}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061d481633dbe7e5326a9d5301b428647d28c4aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.081ex; height:3.176ex;" alt="{\displaystyle {\vec {c}}_{2}={\vec {a}}_{3}{\vec {b}}_{1}-{\vec {a}}_{1}{\vec {b}}_{3}}"></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 {\vec {c}}_{3}={\vec {a}}_{1}{\vec {b}}_{2}-{\vec {a}}_{2}{\vec {b}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{3}={\vec {a}}_{1}{\vec {b}}_{2}-{\vec {a}}_{2}{\vec {b}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4e856c925d2d23f30e23fadcc523303cac1b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.081ex; height:3.176ex;" alt="{\displaystyle {\vec {c}}_{3}={\vec {a}}_{1}{\vec {b}}_{2}-{\vec {a}}_{2}{\vec {b}}_{1}}"></span>.</dd></dl> <p>Például az <span class="mwe-math-element"><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}}=(2,5,-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}=(2,5,-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/059684bd865c3f1f601623280c0885f9ced62b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.501ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}=(2,5,-2)}"></span> és <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}=(3,-4,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}=(3,-4,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caeeb837bbadad77446da49945513c5465cdf19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.173ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}=(3,-4,-1)}"></span> vektorok vektoriális szorzata: </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 {\vec {c}}_{1}=5\cdot (-1)-(-2)\cdot (-4)=-5-8=-13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mo>−</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{1}=5\cdot (-1)-(-2)\cdot (-4)=-5-8=-13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7f257a8055edf9b18ce3bbaf4a0dd9b89593f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.38ex; height:2.843ex;" alt="{\displaystyle {\vec {c}}_{1}=5\cdot (-1)-(-2)\cdot (-4)=-5-8=-13}"></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 {\vec {c}}_{2}=(-2)\cdot 3-2\cdot (-1)=-6+2=-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{2}=(-2)\cdot 3-2\cdot (-1)=-6+2=-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5e4ce581180524bded294511a8cb6079377354" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.6ex; height:2.843ex;" alt="{\displaystyle {\vec {c}}_{2}=(-2)\cdot 3-2\cdot (-1)=-6+2=-4}"></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 {\vec {c}}_{3}=2\cdot (-4)-5\cdot 3=-8-15=-23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mo>−</mo> <mn>8</mn> <mo>−</mo> <mn>15</mn> <mo>=</mo> <mo>−</mo> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}_{3}=2\cdot (-4)-5\cdot 3=-8-15=-23}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179bc5c3d40e62945fd4b17384d83d3f7e28be4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.308ex; height:2.843ex;" alt="{\displaystyle {\vec {c}}_{3}=2\cdot (-4)-5\cdot 3=-8-15=-23}"></span>.</dd></dl> <p>Skaláris szorzással ellenőrizhet, hogy e vektor mindkettő tényezőre merőleges. </p> <div class="mw-heading mw-heading4"><h4 id="Tenzori_(diadikus)_szorzat"><span id="Tenzori_.28diadikus.29_szorzat"></span>Tenzori (diadikus) szorzat</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=13" title="Szakasz szerkesztése: Tenzori (diadikus) szorzat"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vektorok legáltalánosabb szorzata a fizikában a <a href="/wiki/Tenzor" title="Tenzor">tenzormennyiségek</a> definiálására is szolgál. A tenzorszorzat két vektorhoz egy lineáris leképezést rendel az alábbi definíció szerint: </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 {\hat {L}}{\vec {x}}={\vec {a}}.\left({\vec {b}}{\vec {x}}\right)=\left({\vec {a}}\circ {\vec {b}}\right){\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {L}}{\vec {x}}={\vec {a}}.\left({\vec {b}}{\vec {x}}\right)=\left({\vec {a}}\circ {\vec {b}}\right){\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13295df49e117b97ab958fc523c85c6121f8d0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.583ex; height:4.843ex;" alt="{\displaystyle {\hat {L}}{\vec {x}}={\vec {a}}.\left({\vec {b}}{\vec {x}}\right)=\left({\vec {a}}\circ {\vec {b}}\right){\vec {x}}}"></span></dd></dl> <p>A definíció valamely koordináta-rendszerben kifejtve a tenzorszorzat egy <i>reprezentációját</i> kapjuk: </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 {\hat {L}}_{ij}={\vec {a}}_{i}{\vec {b}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {L}}_{ij}={\vec {a}}_{i}{\vec {b}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df60117cdee5a4228b7aadc3ef1a2d6120a228f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.192ex; height:3.509ex;" alt="{\displaystyle {\hat {L}}_{ij}={\vec {a}}_{i}{\vec {b}}_{j}}"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <p>A vektorok tenzorszorzatával a <a href="/w/index.php?title=Tenzoralgebra&action=edit&redlink=1" class="new" title="Tenzoralgebra (a lap nem létezik)">tenzoralgebra</a> foglalkozik alaposabban. </p><p>Példaként szorozzuk össze 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 {\vec {u}}=\left(3,-2,5\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}=\left(3,-2,5\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cffdad929448d4085f2f8c29d16c49e4ddb5de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.601ex; height:2.843ex;" alt="{\displaystyle {\vec {u}}=\left(3,-2,5\right)}"></span> és 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 {\vec {v}}=\left(4,2,-3\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\left(4,2,-3\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/441bf9dbe3c08cabf873e4de8dfcdfc82d22613b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.446ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=\left(4,2,-3\right)}"></span> vektorokat! </p> <table class="wikitable"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 4=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 4=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9536f1b2f7f3bf7cdf34bd66e150b4a6cb980e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.427ex; height:2.176ex;" alt="{\displaystyle 3\cdot 4=12}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a259e246683231004af67398992f1e890b519a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.265ex; height:2.176ex;" alt="{\displaystyle 3\cdot 2=6}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot (-3)=-9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot (-3)=-9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa693af1e9bc06ac140ae453fff6a55e2ab6ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.691ex; height:2.843ex;" alt="{\displaystyle 3\cdot (-3)=-9}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)\cdot 4=-8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>4</mn> <mo>=</mo> <mo>−</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)\cdot 4=-8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673fdd9993583cf7130be3a718dae78db313d77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.691ex; height:2.843ex;" alt="{\displaystyle (-2)\cdot 4=-8}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)\cdot 2=-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)\cdot 2=-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a3d22fd04e5d41b92fe70d7389c26c03d35935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.691ex; height:2.843ex;" alt="{\displaystyle (-2)\cdot 2=-4}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-2)\cdot (-3)=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-2)\cdot (-3)=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0896aae5944c7e17d248921b702e7c2b7291bee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.5ex; height:2.843ex;" alt="{\displaystyle (-2)\cdot (-3)=6}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 4=20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>4</mn> <mo>=</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 4=20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10376fc8207a3e5dc4276291e9f60e5c63e72b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.427ex; height:2.176ex;" alt="{\displaystyle 5\cdot 4=20}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 2=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 2=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e5367bc27090f5d7e60e97371f4ac98609a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.427ex; height:2.176ex;" alt="{\displaystyle 5\cdot 2=10}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot (-3)=-15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot (-3)=-15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b7ca3befd710d138229d1a4bb247311b2fb0d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.853ex; height:2.843ex;" alt="{\displaystyle 5\cdot (-3)=-15}"></span> </td></tr></tbody></table> <p>A szorzat tehát: </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}12&6&-9\\-8&-4&6\\20&10&-15\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> <mn>12</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mo>−</mo> <mn>9</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>20</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mo>−</mo> <mn>15</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}12&6&-9\\-8&-4&6\\20&10&-15\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6df8009fb530e2690750505158e4255e4423e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.538ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}12&6&-9\\-8&-4&6\\20&10&-15\end{pmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Alkalmazások"><span id="Alkalmaz.C3.A1sok"></span>Alkalmazások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=14" title="Szakasz szerkesztése: Alkalmazások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vektoroknak számtalan alkalmazása van. Ezek főleg a matematikához, fizikához és informatikához kötődnek. Legtöbbször egyfajta általánosítása a hagyományos vektorfogalomnak az egyes tudományágak saját fogalomalkotása, de az analógia legtöbbször nyilvánvaló. </p> <div class="mw-heading mw-heading3"><h3 id="A_matematikában"><span id="A_matematik.C3.A1ban"></span>A matematikában</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=15" title="Szakasz szerkesztése: A matematikában"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jellemzően a lineáris algebrában fordulnak elő vektorok, illetve ehhez kapcsolódóan az <a href="/wiki/Koordin%C3%A1tageometria" title="Koordinátageometria">analitikus geometriában</a>. </p><p>A lineáris algebra az <a href="/wiki/Egyenletrendszer" title="Egyenletrendszer">egyenletrendszereket</a> lineáris egyenletekhez hasonló módon kezelő eszközöket kap, ha az ismeretleneket és az egyenletek jobb oldalát egy-egy vektorként kezeljük, ekkor az együtthatók ugyanis egy leképezés mátrixának alakját fogja ölteni. Például </p> <table class="wikitable"> <tbody><tr> <th>Egyenletrendszer</th> <th>Egyenlet </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}&=b_{2}\\&\vdots \\a_{n1}x_{1}+a_{n2}x_{2}+\dots +a_{nn}x_{n}&=b_{n}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>b</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>21</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}&=b_{2}\\&\vdots \\a_{n1}x_{1}+a_{n2}x_{2}+\dots +a_{nn}x_{n}&=b_{n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd53ec1fb70ef0a5a65b4725b10a5a850a3e4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:34.601ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}&=b_{2}\\&\vdots \\a_{n1}x_{1}+a_{n2}x_{2}+\dots +a_{nn}x_{n}&=b_{n}\end{aligned}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b82cb3720d07ecbcab36145af8fee0b7a3519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.656ex; height:2.176ex;" alt="{\displaystyle ax=b}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}{\vec {x}}={\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}{\vec {x}}={\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d7b431bb130ae303c86a15634d94393ec583e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.298ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}{\vec {x}}={\vec {b}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b82cb3720d07ecbcab36145af8fee0b7a3519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.656ex; height:2.176ex;" alt="{\displaystyle ax=b}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\hat {A}}^{-1}{\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\hat {A}}^{-1}{\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db0dc008817c6604525a5cbb8e9db0a4ef08106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.63ex; height:3.343ex;" alt="{\displaystyle {\vec {x}}={\hat {A}}^{-1}{\vec {b}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a^{-1}b={\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a^{-1}b={\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60233e988b64712560f33659fcbd3109468d865f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.153ex; height:5.343ex;" alt="{\displaystyle x=a^{-1}b={\frac {b}{a}}}"></span> </td></tr></tbody></table> <p>A koordinátageometriában a vektorok tulajdonképpen a pontokat jelölik ki, így <i>helyvektorként</i> funkcionálnak, a vektorokból pedig a fentebb említett műveletek segítségével a geometria legalapvetőbb ponthalmazai építhetőek fel. Ez egyben az összekötő kapocs is az algebra és a geometria között, aminek segítségével igazolható e két nagy terület egyenértékűsége. </p><p>Ezen túl a vektor fogalom általánosabb, elvontabb formája segítségével az analízis sok fogalma is kényelmesebben kezelhetővé válik. Ilyen például az intervallum felett korlátos függvények halmaza, vagy egy test felett értelmezett operátorok tere. </p><p>A vektorok segítségével lehet a Hilbert-tereket, mint speciális, skalárszorzatos tereket definiálni, amik a fizikai alkalmazások során nyernek jelentőséget, mivel a fizikai mennyiségeket a Hilbert-tereken ható operátorokkal jellemezzük. </p> <div class="mw-heading mw-heading3"><h3 id="A_fizikában"><span id="A_fizik.C3.A1ban"></span>A fizikában</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=16" title="Szakasz szerkesztése: A fizikában"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A fizikában a vektorokat más szemlélettel definiáljuk, mint a matematikában. Mivel a fizika a világot koordináta-rendszerekben írja le, a legfontosabb mennyiségeket a koordináta-rendszerek közötti átmenet szempontjából, azaz a transzformációk során mutatott viselkedésük szerint írjuk le. Az alaptípus a koordináta-rendszert kifeszítő, egységvektorokból álló generátorhalmaz. Ekkor vektornak nevezünk minden olyan fizikai mennyiséget, amelyek úgy transzformálódnak, mint a koordináta-rendszert kifeszítő egységvektorok. </p><p>A vektorok azonban nem feltétlenül viselkednek azonosan a fentebbi meghatározással. A vektori szorzat esetén például a tükrözés során a két tényező előjelet vált, a szorzatvektor viszont nem: </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 \left(-{\vec {x}}\right)\times \left(-{\vec {y}}\right)=\left(-1\right)^{2}\left({\vec {x}}\times {\vec {y}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-{\vec {x}}\right)\times \left(-{\vec {y}}\right)=\left(-1\right)^{2}\left({\vec {x}}\times {\vec {y}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b6b171dfbabec6b78fd9a973583decf41566c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.233ex; height:3.343ex;" alt="{\displaystyle \left(-{\vec {x}}\right)\times \left(-{\vec {y}}\right)=\left(-1\right)^{2}\left({\vec {x}}\times {\vec {y}}\right)}"></span></dd></dl> <p>Az ilyen vektort, mivel tulajdonképpen egy tengelyt jelöl ki, <i>axiálvektornak</i> nevezünk. </p><p>A köznapi szemlélet szempontjából a fizikai vektorok legfontosabb tulajdonsága, hogy a nagyságuk mellett irányításuk is van. Ez alapján definiálhatóak a <i>vektormennyiségek</i>: olyan fizikai mennyiség, amit két mennyiség, irány és nagyság jellemez. Ez egyben három paramétert igényel, azonban speciális esetekben a koordináta-rendszer megfelelő megválasztásával egy vagy kettő zérussá tehető. Ez az oka, hogy a bevezető fizikai tanulmányok során az egyenes vonalú mozgások, állandó irányú hatások játszanak elsődleges szerepet. </p><p>A klasszikus fizika háromdimenziós vektorokkal foglalkozik, a relativisztikus fizika azonban a téridő leírására már négy paramétert alkalmaz, így itt a vektorok viselkedése teljesen váratlanná válhat a laikus szemlélő számára. A vektorokat három nagy csoportba sorolhatjuk: időszerű, fényszerű és térszerű vektorok. </p><p>Az időszerű vektorok hosszának négyzete pozitív, ezen vektorokhoz mindig találunk olyan koordináta-rendszert, hogy a vektor az időtengellyel párhuzamos lesz. Ha két esemény időszerű kapcsolatban van egymással, akkor mindig van olyan megfigyelő, aki számára a két esemény ugyanott, de egymás után történik. A legegyszerűbb időszerű kapcsolat a kauzalitás, azaz az egyik pont, mint esemény, oka a másiknak. Ha egy vektor időszerű, akkor minden megfigyelő számára időszerű a kapcsolat, ez fejezi ki a relativitáselmélet determinisztikusságát. </p><p>A térszerű vektorok hossznégyzete negatív,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> azaz ezekhez mindig találunk olyan koordináta-rendszert, aminek egy térbeli koordinátatengelyével párhuzamosak. A gyakorlatban a térszerű kapcsolat két esemény között azt jelenti, hogy van olyan megfigyelő, aki számára a két esemény egyszerre történik, de eltérő helyeken. </p><p>A fényszerű vektorok hossznégyzete nulla, ezek tehát a koordináta-rendszerek transzformációja során nem változnak. Az elnevezés tükrözi szerepüket: a fényszerű vektorokat a fénysugarak jelölik ki, azaz a fény sebessége minden megfigyelő számára egyenlő. </p> <div class="mw-heading mw-heading3"><h3 id="A_számítástechnikában"><span id="A_sz.C3.A1m.C3.ADt.C3.A1stechnik.C3.A1ban"></span>A számítástechnikában</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=17" title="Szakasz szerkesztése: A számítástechnikában"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A számítástechnikában általában az egydimenziós tömböket nevezik vektornak, ez megfelel ugyanis a vektorkoordináták mátrixreprezentációjának. Ugyanakkor azonban egy tömb elemei nem csak számok lehetnek, ennyiben a matematikai vektorfogalomtól el is tér. A konkrét értelmezés általában a programozási nyelv része. Például a <a href="/wiki/C%2B%2B" title="C++">C++</a>-ban a <i>Vector</i> egy konténer osztály, aminek elemei tetszőleges, akár több különböző típusba tartozó adatok lehetnek. </p> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=18" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Ez ugyanakkor a fogalom elmélyítését már jelentősen megnehezítheti!</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Konkrétan megadható a pár alapján bármely pont képének a <i>szerkesztése</i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Fontos feltétel, hogy test legyen, ellenkező esetben a feltételeink "elromlanak".</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Belső művelet alatt azt értjük, hogy a szorzat tényezői mind a vektortérből erednek.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Vigyázzunk, ne keverjük a később értelmezett skaláris szorzással!</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://betterexplained.com/articles/cross-product/">https://betterexplained.com/articles/cross-product/</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Ha jól megnézzük, feltűnő lesz, hogy a tenzor <i>nyoma</i> a két vektor skaláris szorzata. Három dimenzióban a főátló feletti és alatti elemek pedig a vektoriális szorzat tagjai lesznek.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Ezért beszélünk a vektorok hosszának <i>négyzetéről</i> - ennek ugyanis van fizikai értelme.</span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&action=edit&section=19" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Jánossy Lajos, Tasnádi Péter – Vektorszámítás I (Vektor és tenzoralgebra), Nemzeti tankönyvkiadó, Budapest 2002 <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9631939529" title="Speciális:Könyvforrások/9631939529">ISBN 963 19 3952 9</a></li> <li>Király Bertalan – Lineáris algebra, EKTF Líceum kiadó, Eger 2004, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9639417149" title="Speciális:Könyvforrások/9639417149">ISBN 963 941714 9</a></li> <li>Fényes Imre – Modern fizikai kisenciklopédia, Gondolat könyvkiadó, Budapest, 1971</li> <li><span class="citation"><a href="/w/index.php?title=Eric_W._Weisstein&action=edit&redlink=1" class="new" title="Eric W. Weisstein (a lap nem létezik)">Weisstein, Eric W.</a>: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Vector.html"><i>Vektor</i></a> (angol nyelven). <a href="/wiki/MathWorld" title="MathWorld">Wolfram MathWorld</a></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130301040741/http://www.sulinet.hu/tanar/kompetenciateruletek/2_matematika/3_modulleirasok-tanar-tanulo-eszkoz/2_a_tipus/9-evfolyam/2_tanari_modulok/amat9_14_2modulleiras.pdf">Sulinetes anyag a vektorokról</a></li> <li><a rel="nofollow" class="external text" href="http://phet.colorado.edu/hu/simulation/vector-addition">Magyarított, letölthető interaktív Flash szimuláció síkvektorok összeadásáról</a> a PhET-től. Grafikus megjelenítés mellett a polár- és derékszögű koordinátákat is megadja.</li> <li><a rel="nofollow" class="external text" href="http://nagysandor.eu/harrisonia/UnitVectors_HU.html">Magyarított Flash szimuláció a derékszögű koordináták és a megfelelő egységvektorok kapcsolatáról</a>. Szerző: David M. Harrison</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r26593303">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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