Vektor – Wikipedia

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data-event-name="pinnable-header.vector-toc.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">dölj</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">Inledning</div> </a> </li> <li id="toc-Historik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historik"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historik</span> </div> </a> <ul id="toc-Historik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vektorbeteckningar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorbeteckningar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vektorbeteckningar</span> </div> </a> <ul id="toc-Vektorbeteckningar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_av_vektorer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representation_av_vektorer"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Representation av vektorer</span> </div> </a> <ul id="toc-Representation_av_vektorer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Egenskaper" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Egenskaper"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Egenskaper</span> </div> </a> <button aria-controls="toc-Egenskaper-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>Växla underavsnittet Egenskaper</span> </button> <ul id="toc-Egenskaper-sublist" class="vector-toc-list"> <li id="toc-Identiska_vektorer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identiska_vektorer"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Identiska vektorer</span> </div> </a> <ul id="toc-Identiska_vektorer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Addition_och_subtraktion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition_och_subtraktion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Addition och subtraktion</span> </div> </a> <ul id="toc-Addition_och_subtraktion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skalär_multiplikation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skalär_multiplikation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Skalär multiplikation</span> </div> </a> <ul id="toc-Skalär_multiplikation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Längd" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Längd"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Längd</span> </div> </a> <ul id="toc-Längd-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skalärprodukt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skalärprodukt"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Skalärprodukt</span> </div> </a> <ul id="toc-Skalärprodukt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skalär_trippelprodukt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skalär_trippelprodukt"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Skalär trippelprodukt</span> </div> </a> <ul id="toc-Skalär_trippelprodukt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kryssprodukt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kryssprodukt"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Kryssprodukt</span> </div> </a> <ul id="toc-Kryssprodukt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vektoriell_trippelprodukt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vektoriell_trippelprodukt"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Vektoriell trippelprodukt</span> </div> </a> <ul id="toc-Vektoriell_trippelprodukt-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vektorprojektion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorprojektion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Vektorprojektion</span> </div> </a> <button aria-controls="toc-Vektorprojektion-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>Växla underavsnittet Vektorprojektion</span> </button> <ul id="toc-Vektorprojektion-sublist" class="vector-toc-list"> <li id="toc-Exempel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Exempel</span> </div> </a> <ul id="toc-Exempel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vektorer_i_ℝ2_och_komplexa_tal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorer_i_ℝ2_och_komplexa_tal"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Vektorer i ℝ<sup>2</sup> och komplexa tal</span> </div> </a> <ul id="toc-Vektorer_i_ℝ2_och_komplexa_tal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vektorfält" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vektorfält"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Vektorfält</span> </div> </a> <ul id="toc-Vektorfält-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_även" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_även"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Se även</span> </div> </a> <ul id="toc-Se_även-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referenser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referenser</span> </div> </a> <button aria-controls="toc-Referenser-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>Växla underavsnittet Referenser</span> </button> <ul id="toc-Referenser-sublist" class="vector-toc-list"> <li id="toc-Noter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noter"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Noter</span> </div> </a> <ul id="toc-Noter-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vidare_läsning" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vidare_läsning"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Vidare läsning</span> </div> </a> <ul id="toc-Vidare_läsning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externa_länkar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externa_länkar"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Externa länkar</span> </div> </a> <ul id="toc-Externa_länkar-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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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="Gå till en artikel på ett annat språk. Tillgänglig på 95 språk" > <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 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektor_(Wiskunde)" title="Vektor (Wiskunde) – 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 – schweizertyska" lang="gsw" hreflang="gsw" data-title="Vektor" data-language-autonym="Alemannisch" data-language-local-name="schweizertyska" 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 – amhariska" lang="am" hreflang="am" data-title="ጨረር" data-language-autonym="አማርኛ" data-language-local-name="amhariska" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Vektor" title="Vektor – enaresamiska" lang="smn" hreflang="smn" data-title="Vektor" data-language-autonym="Anarâškielâ" data-language-local-name="enaresamiska" class="interlanguage-link-target"><span>Anarâškielâ</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 – arabiska" lang="ar" hreflang="ar" data-title="متجه" data-language-autonym="العربية" data-language-local-name="arabiska" 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 – asturiska" lang="ast" hreflang="ast" data-title="Vector" data-language-autonym="Asturianu" data-language-local-name="asturiska" 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) – azerbajdzjanska" lang="az" hreflang="az" data-title="Vektor (həndəsə)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzjanska" 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-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 – bengali" lang="bn" hreflang="bn" data-title="সদিক রাশি" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng" title="Hiòng-liōng – min nan" lang="nan" hreflang="nan" data-title="Hiòng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-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) – basjkiriska" lang="ba" hreflang="ba" data-title="Вектор (геометрия)" data-language-autonym="Башҡортса" data-language-local-name="basjkiriska" 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) – belarusiska" lang="be" hreflang="be" data-title="Вектар (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarusiska" 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 – belarusiska (tarasjkevitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Вэктар" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusiska (tarasjkevitsa)" 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 – bulgariska" lang="bg" hreflang="bg" data-title="Вектор" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Euklidski_vektor" title="Euklidski vektor – bosniska" lang="bs" hreflang="bs" data-title="Euklidski vektor" data-language-autonym="Bosanski" data-language-local-name="bosniska" 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) – katalanska" lang="ca" hreflang="ca" data-title="Vector (matemàtiques)" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</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) – tjuvasjiska" lang="cv" hreflang="cv" data-title="Вектор (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektor" title="Vektor – tjeckiska" lang="cs" hreflang="cs" data-title="Vektor" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Fector" title="Fector – walesiska" lang="cy" hreflang="cy" data-title="Fector" data-language-autonym="Cymraeg" data-language-local-name="walesiska" 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) – danska" lang="da" hreflang="da" data-title="Vektor (geometri)" data-language-autonym="Dansk" data-language-local-name="danska" 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 – tyska" lang="de" hreflang="de" data-title="Vektor" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vektor" title="Vektor – estniska" lang="et" hreflang="et" data-title="Vektor" data-language-autonym="Eesti" data-language-local-name="estniska" class="interlanguage-link-target"><span>Eesti</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 – grekiska" lang="el" hreflang="el" data-title="Ευκλείδειο διάνυσμα" data-language-autonym="Ελληνικά" data-language-local-name="grekiska" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclidean_vector" title="Euclidean vector – engelska" lang="en" hreflang="en" data-title="Euclidean vector" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</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) – erjya" lang="myv" hreflang="myv" data-title="Вектор (геометрия)" data-language-autonym="Эрзянь" data-language-local-name="erjya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector" title="Vector – spanska" lang="es" hreflang="es" data-title="Vector" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektoro" title="Vektoro – esperanto" lang="eo" hreflang="eo" data-title="Vektoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore_(matematika)" title="Bektore (matematika) – baskiska" lang="eu" hreflang="eu" data-title="Bektore (matematika)" data-language-autonym="Euskara" data-language-local-name="baskiska" 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 – persiska" lang="fa" hreflang="fa" data-title="بردار اقلیدسی" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vecteur_euclidien" title="Vecteur euclidien – franska" lang="fr" hreflang="fr" data-title="Vecteur euclidien" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Veicteoir" title="Veicteoir – iriska" lang="ga" hreflang="ga" data-title="Veicteoir" data-language-autonym="Gaeilge" data-language-local-name="iriska" 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 – skotsk gäliska" lang="gd" hreflang="gd" data-title="Bheactor" data-language-autonym="Gàidhlig" data-language-local-name="skotsk gäliska" 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 – galiciska" lang="gl" hreflang="gl" data-title="Vector" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</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 – koreanska" lang="ko" hreflang="ko" data-title="유클리드 벡터" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</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 – kroatiska" lang="hr" hreflang="hr" data-title="Vektor" data-language-autonym="Hrvatski" data-language-local-name="kroatiska" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vektoro" title="Vektoro – ido" lang="io" hreflang="io" data-title="Vektoro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Vektor_Euklides" title="Vektor Euklides – indonesiska" lang="id" hreflang="id" data-title="Vektor Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</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) – isländska" lang="is" hreflang="is" data-title="Vigur (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="isländska" 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) – italienska" lang="it" hreflang="it" data-title="Vettore (matematica)" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</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 – hebreiska" lang="he" hreflang="he" data-title="וקטור אוקלידי" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</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 – georgiska" lang="ka" hreflang="ka" data-title="ვექტორი" data-language-autonym="ქართული" data-language-local-name="georgiska" 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 – kazakiska" lang="kk" hreflang="kk" data-title="Вектор" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Vekt%C3%A8" title="Vektè – haitiska" lang="ht" hreflang="ht" data-title="Vektè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiska" class="interlanguage-link-target"><span>Kreyòl ayisyen</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-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vektors" title="Vektors – lettiska" lang="lv" hreflang="lv" data-title="Vektors" data-language-autonym="Latviešu" data-language-local-name="lettiska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorius" title="Vektorius – litauiska" lang="lt" hreflang="lt" data-title="Vektorius" data-language-autonym="Lietuvių" data-language-local-name="litauiska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Vettor_(matematega)" title="Vettor (matematega) – lombardiska" lang="lmo" hreflang="lmo" data-title="Vettor (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardiska" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektor" title="Vektor – ungerska" lang="hu" hreflang="hu" data-title="Vektor" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="utmärkt artikel"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80" title="Вектор – makedonska" lang="mk" hreflang="mk" data-title="Вектор" data-language-autonym="Македонски" data-language-local-name="makedonska" 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) – malayalam" lang="ml" hreflang="ml" data-title="സദിശം (ജ്യാമിതി)" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Vettur_ewklidju" title="Vettur ewklidju – maltesiska" lang="mt" hreflang="mt" data-title="Vettur ewklidju" data-language-autonym="Malti" data-language-local-name="maltesiska" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Vektor" title="Vektor – malajiska" lang="ms" hreflang="ms" data-title="Vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="malajiska" class="interlanguage-link-target"><span>Bahasa Melayu</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-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 – mongoliska" lang="mn" hreflang="mn" data-title="Евклидийн вектор" data-language-autonym="Монгол" data-language-local-name="mongoliska" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vector_(wiskunde)" title="Vector (wiskunde) – nederländska" lang="nl" hreflang="nl" data-title="Vector (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</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 – japanska" lang="ja" hreflang="ja" data-title="空間ベクトル" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Vektor" title="Vektor – nordfrisiska" lang="frr" hreflang="frr" data-title="Vektor" data-language-autonym="Nordfriisk" data-language-local-name="nordfrisiska" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektor_(matematikk)" title="Vektor (matematikk) – norskt bokmål" lang="nb" hreflang="nb" data-title="Vektor (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektor" title="Vektor – nynorska" lang="nn" hreflang="nn" data-title="Vektor" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorska" class="interlanguage-link-target"><span>Norsk nynorsk</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-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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Vektor_(matematika)" title="Vektor (matematika) – uzbekiska" lang="uz" hreflang="uz" data-title="Vektor (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekiska" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 – pashto" lang="ps" hreflang="ps" data-title="د اقليدس لوری" data-language-autonym="پښتو" data-language-local-name="pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Vetor" title="Vetor – piemontesiska" lang="pms" hreflang="pms" data-title="Vetor" data-language-autonym="Piemontèis" data-language-local-name="piemontesiska" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Vekter" title="Vekter – lågtyska" lang="nds" hreflang="nds" data-title="Vekter" data-language-autonym="Plattdüütsch" data-language-local-name="lågtyska" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektor" title="Wektor – polska" lang="pl" hreflang="pl" data-title="Wektor" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</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) – portugisiska" lang="pt" hreflang="pt" data-title="Vetor (matemática)" data-language-autonym="Português" data-language-local-name="portugisiska" 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 – rumänska" lang="ro" hreflang="ro" data-title="Vector euclidian" data-language-autonym="Română" data-language-local-name="rumänska" 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) – ryska" lang="ru" hreflang="ru" data-title="Вектор (геометрия)" data-language-autonym="Русский" data-language-local-name="ryska" 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) – jakutiska" lang="sah" hreflang="sah" data-title="Вектор (геометрия)" data-language-autonym="Саха тыла" data-language-local-name="jakutiska" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vektori" title="Vektori – albanska" lang="sq" hreflang="sq" data-title="Vektori" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Vettura_euclideu" title="Vettura euclideu – sicilianska" lang="scn" hreflang="scn" data-title="Vettura euclideu" data-language-autonym="Sicilianu" data-language-local-name="sicilianska" class="interlanguage-link-target"><span>Sicilianu</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 – singalesiska" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දෛශිකය" data-language-autonym="සිංහල" data-language-local-name="singalesiska" 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) – slovakiska" lang="sk" hreflang="sk" data-title="Vektor (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" 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) – slovenska" lang="sl" hreflang="sl" data-title="Vektor (matematika)" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Wekt%C5%AFr" title="Wektůr – silesiska" lang="szl" hreflang="szl" data-title="Wektůr" data-language-autonym="Ślůnski" data-language-local-name="silesiska" class="interlanguage-link-target"><span>Ślůnski</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 – centralkurdiska" lang="ckb" hreflang="ckb" data-title="ئاڕاستەبڕی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="centralkurdiska" class="interlanguage-link-target"><span>کوردی</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 – serbiska" lang="sr" hreflang="sr" data-title="Вектор" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vektor" title="Vektor – serbokroatiska" lang="sh" hreflang="sh" data-title="Vektor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatiska" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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) – sundanesiska" lang="su" hreflang="su" data-title="Véktor (rohangan)" data-language-autonym="Sunda" data-language-local-name="sundanesiska" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektori" title="Vektori – finska" lang="fi" hreflang="fi" data-title="Vektori" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</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-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 – thailändska" lang="th" hreflang="th" data-title="เวกเตอร์" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Vekt%C3%B6r" title="Vektör – turkiska" lang="tr" hreflang="tr" data-title="Vektör" data-language-autonym="Türkçe" data-language-local-name="turkiska" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Wektor_ululyklar" title="Wektor ululyklar – turkmeniska" lang="tk" hreflang="tk" data-title="Wektor ululyklar" data-language-autonym="Türkmençe" data-language-local-name="turkmeniska" class="interlanguage-link-target"><span>Türkmençe</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 – ukrainska" lang="uk" hreflang="uk" data-title="Евклідів вектор" data-language-autonym="Українська" data-language-local-name="ukrainska" 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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Vect%C6%A1" title="Vectơ – vietnamesiska" lang="vi" hreflang="vi" data-title="Vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F" title="向量 – wu" lang="wuu" hreflang="wuu" data-title="向量" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%A2%D7%A7%D7%98%D7%90%D7%A8" title="וועקטאר – jiddisch" lang="yi" hreflang="yi" data-title="וועקטאר" data-language-autonym="ייִדיש" data-language-local-name="jiddisch" class="interlanguage-link-target"><span>ייִדיש</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 – kantonesiska" lang="yue" hreflang="yue" data-title="向量" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</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 – kinesiska" lang="zh" hreflang="zh" data-title="向量" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</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="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</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="Namnrymder"> <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="Visa innehållssidan [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskussion:Vektor" rel="discussion" title="Diskussion om innehållssidan [t]" accesskey="t"><span>Diskussion</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="Ändra språkvariant" > <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">svenska</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="Visningar"> <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>Läs</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&veaction=edit" title="Redigera denna sida [v]" accesskey="v"><span>Redigera</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Redigera wikitexten för den här sidan [e]" accesskey="e"><span>Redigera wikitext</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Vektor&action=history" title="Tidigare versioner av sidan [h]" accesskey="h"><span>Visa historik</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sidverktyg"> <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="Verktyg" > <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">Verktyg</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">Verktyg</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">dölj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Fler alternativ" > <div class="vector-menu-heading"> Åtgärder </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>Läs</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&veaction=edit" title="Redigera denna sida [v]" accesskey="v"><span>Redigera</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Vektor&action=edit" title="Redigera wikitexten för den här sidan [e]" accesskey="e"><span>Redigera wikitext</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>Visa historik</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Allmänt </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:L%C3%A4nkar_hit/Vektor" title="Lista över alla wikisidor som länkar hit [j]" accesskey="j"><span>Sidor som länkar hit</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:Senaste_relaterade_%C3%A4ndringar/Vektor" rel="nofollow" title="Visa senaste ändringarna av sidor som den här sidan länkar till [k]" accesskey="k"><span>Relaterade ändringar</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:Specialsidor" title="Lista över alla specialsidor [q]" accesskey="q"><span>Specialsidor</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Vektor&oldid=55195720" title="Permanent länk till den här versionen av sidan"><span>Permanent länk</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Vektor&action=info" title="Mer information om denna sida"><span>Sidinformation</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:Citera&page=Vektor&id=55195720&wpFormIdentifier=titleform" title="Information om hur den här artikeln kan användas som referens"><span>Använd som referens</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FVektor"><span>Hämta förkortad url</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FVektor"><span>Ladda ner QR-kod</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"> Skriv ut/exportera </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=Special:Bok&bookcmd=book_creator&referer=Vektor"><span>Skapa en bok</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Vektor&action=show-download-screen"><span>Ladda ned som PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Vektor&printable=yes" title="Utskriftsvänlig version av den här sidan [p]" accesskey="p"><span>Utskriftsvänlig version</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"> På andra projekt </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>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="Länk till anslutet databasobjekt [g]" accesskey="g"><span>Wikidata-objekt</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="Sidverktyg"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Utseende"> <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">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Från Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sv" dir="ltr"><div class="noexcerpt noprint hanvisning_bas"> <dl><dd><i>Den här artikeln handlar om vektorer inom matematiken.  För andra betydelser, se <a href="/wiki/Vektor_(olika_betydelser)" class="mw-disambig" title="Vektor (olika betydelser)">Vektor (olika betydelser)</a>.  </i></dd></dl></div> <p><b>Vektorer</b> är <a href="/wiki/Matematik" title="Matematik">matematiska</a> <a href="/wiki/Storhet" title="Storhet">storheter</a> som har både storlek (magnitud) och <a href="/wiki/Riktningskoefficient" title="Riktningskoefficient">riktning</a>. De används därför ofta för att beskriva fysikaliska storheter med magnitud och riktning i rummet, som till exempel <a href="/wiki/Kraft" title="Kraft">kraft</a>, <a href="/wiki/Hastighet" title="Hastighet">hastighet</a>, <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, <a href="/wiki/Elektriskt_f%C3%A4lt" title="Elektriskt fält">elektriskt fält</a> och <a href="/wiki/Magnetf%C3%A4lt" title="Magnetfält">magnetfält</a>. Sådana vektorer kallas även rumsvektorer eller geometriska vektorer. Ibland studeras rumsvektorer även i två dimensioner. I motsats till vektorstorheter är storheter som <a href="/wiki/Temperatur" title="Temperatur">temperatur</a> och <a href="/wiki/Ljusstyrka" title="Ljusstyrka">ljusstyrka</a> <a href="/wiki/Skal%C3%A4r" title="Skalär">skalärer</a> då de <i>saknar</i> riktning. </p><p>Inom <a href="/wiki/Matematik" title="Matematik">matematiken</a> generaliseras vektorer till att vara element i ett <a href="/wiki/Vektorrum" class="mw-redirect" title="Vektorrum">vektorrum</a> av godtycklig dimension. En sådan generaliserad vektor kan ha en <a href="/wiki/Norm_(matematik)" title="Norm (matematik)">norm</a> som anknyter till längdbegreppet. För vektorrummet kan en <a href="/wiki/Inre_produkt" class="mw-redirect" title="Inre produkt">inre produkt</a> vara definierad vilken kan sägas mäta vinklar mellan vektorerna. Med denna definition kan många typer av objekt anses vara vektorer. Det enda kravet är att de följer de viktigaste av de räkneregler som gäller för rumsvektorer. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historik">Historik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=1" title="Redigera avsnitt: Historik" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=1" title="Redigera avsnitts källkod: Historik"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vektorbegreppet, såsom vi känner det idag, utvecklades gradvis över en period av mer än 200 år. Omkring ett dussin personer gjorde signifikanta bidrag. <sup id="cite_ref-Crowe_1-0" class="reference"><a href="#cite_note-Crowe-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> </p><p><a href="/wiki/Giusto_Bellavitis" title="Giusto Bellavitis">Giusto Bellavitis</a> abstraherade den grundläggande idén 1835 när han etablerade begreppet ekvipollens. Han studerade det euklidiska planet och definierade som ekvipollenta (likvärdiga) varje par av linjesegment av samma längd och riktning. Väsentligen upptäckte han en ekvipollensrelation för paren av punkter (tvåpunkter) i planet och skapade därmed det första vektorrummet i planet.<sup id="cite_ref-Crowe_1-1" class="reference"><a href="#cite_note-Crowe-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:52–4</sup> </p><p>Termen <i>vektor</i> introducerades av <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> som en del av en <a href="/wiki/Kvaternion" title="Kvaternion">kvaternion</a>, vilken är en summa <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>q</i> = <i>s</i> + <i>v</i></span> av ett <a href="/wiki/Reella_tal" title="Reella tal">reellt tal</a> <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>s</i></span> (också kallat <i>skalär</i>) och en 3-dimensionell <i>vektor</i> <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>v</i></span>. Liksom Bellavitis, betraktade Hamilton vektorer som en representation av <a href="/wiki/Ekvivalensklass" title="Ekvivalensklass">klasser</a> av ekvipollent riktade linjesegment. I analogi med <a href="/wiki/Komplexa_tal" title="Komplexa tal">komplexa tal</a>, som använder en <a href="/wiki/Imagin%C3%A4r_del" class="mw-redirect" title="Imaginär del">imaginär del</a> för att komplettera den reella tallinjen, betraktade Hamilton vektordelen <span style="white-space: nowrap; font-family: times, serif, palatino linotype, new athena unicode, athena, gentium, code2000; font-size: 120%;"><i>v</i></span> som den <i>imaginära delen</i> av en kvaternion.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-reference-link-bracket">[</span>2<span class="cite-reference-link-bracket">]</span></a></sup> </p><p>Flera andra matematiker utvecklade vektorliknande system under 1800-talets mitt, däribland <a href="/wiki/Augustin_Louis_Cauchy" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a>, <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>, <a href="/wiki/August_M%C3%B6bius" class="mw-redirect" title="August Möbius">August Möbius</a>, <a href="/w/index.php?title=Comte_de_Saint-Venant&action=edit&redlink=1" class="new" title="Comte de Saint-Venant [inte skriven än]">Comte de Saint-Venant</a> och <a href="/w/index.php?title=Matthew_O%27Brien&action=edit&redlink=1" class="new" title="Matthew O'Brien [inte skriven än]">Matthew O'Brien </a>. Grassmanns arbete från 1840 <i>Theorie der Ebbe und Flut</i> var det första systemet av rumslig analys som liknade dagens system och presenterade idéer som motsvarar kryssprodukt, skalärprodukt och vektordifferentiering. Grassmanns arbete uppmärksammades först i slutet av 1870-talet.<sup id="cite_ref-Crowe_1-2" class="reference"><a href="#cite_note-Crowe-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> </p><p><a href="/wiki/Peter_Guthrie_Tait" title="Peter Guthrie Tait">Peter Guthrie Tait</a> fortsatte att arbeta med kvaternioner efter Hamilton. Hans <i>Elementary Treatise of Quaternions</i> från 1870 inkluderade en utförlig behandling av nablaoperatorn ∇. </p><p>1878 publicerades <i>Elements of Dynamic</i> av <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">William Kingdon Clifford</a>, ett verk som förenklade studiet av kvaternionen genom att isolera skalärprodukten och kryssprodukten av två vektorer från den kompletta kvaternionprodukten, vilket gjorde vektorberäkningar tillgängliga för ingenjörer och andra som arbetade i tre dimensioner och var skeptiska till den fjärde. </p><p><a href="/wiki/Josiah_Willard_Gibbs" class="mw-redirect" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a>, som stötte på kvaternioner genom <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwells</a> <i>Treatise on Electricity and Magnetism</i>, skilde av deras vektordel för en oberoende behandling. Den första halvan av Gibbs <i>Elements of Vector Analysis</i>, publicerad 1881, presenterade vad som väsentligen är det moderna systemet för vektoranalys.<sup id="cite_ref-Crowe_1-3" class="reference"><a href="#cite_note-Crowe-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> 1901 publicerade <a href="/w/index.php?title=Edwin_Bidwell_Wilson&action=edit&redlink=1" class="new" title="Edwin Bidwell Wilson [inte skriven än]">Edwin Bidwell Wilson</a> <i>Vector Analysis</i>, som i huvudsak var tillämpningar hämtade från Gibbs föreläsningar och som övergav allt omnämnande av kvaternioner. </p> <div class="mw-heading mw-heading2"><h2 id="Vektorbeteckningar">Vektorbeteckningar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=2" title="Redigera avsnitt: Vektorbeteckningar" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=2" title="Redigera avsnitts källkod: Vektorbeteckningar"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:Vektor-beteckningar.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Vektor-beteckningar.png/250px-Vektor-beteckningar.png" decoding="async" width="250" height="231" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Vektor-beteckningar.png/375px-Vektor-beteckningar.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Vektor-beteckningar.png/500px-Vektor-beteckningar.png 2x" data-file-width="800" data-file-height="738" /></a><figcaption></figcaption></figure> <p>Ett vektornamn skrivs vanligen med fet stil, till exempel som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span></dd></dl> <p>I vissa fall kan även notationen </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 {\overrightarrow {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b245e60e48c3c8f577aaf9512a1bdf3049cc6207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.372ex; width:3.637ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {AB}}}"></span></dd></dl> <p>förekomma där <i>A</i> är vektorns startpunkt och <i>B</i> dess ändpunkt. </p><p>Andra vanliga notationer är </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}},\ \mathbf {\hat {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> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">a</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}},\ \mathbf {\hat {a}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a6840eb314f37858db52b483a6fb75dad8fbb72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.181ex; height:2.676ex;" alt="{\displaystyle {\vec {a}},\ \mathbf {\hat {a}} }"></span></dd></dl> <p>där en pil eller "hatt" placerats ovanför namnet. </p> <div class="mw-heading mw-heading2"><h2 id="Representation_av_vektorer">Representation av vektorer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=3" title="Redigera avsnitt: Representation av vektorer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=3" title="Redigera avsnitts källkod: Representation av vektorer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartiklar: <a href="/wiki/Bas_(linj%C3%A4r_algebra)" title="Bas (linjär algebra)">Bas (linjär algebra)</a> och <a href="/wiki/Enhetsvektor" title="Enhetsvektor">Enhetsvektor</a></dd></dl></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:2D-coordinate-system.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/2D-coordinate-system.png/230px-2D-coordinate-system.png" decoding="async" width="230" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/2D-coordinate-system.png/345px-2D-coordinate-system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/2D-coordinate-system.png/460px-2D-coordinate-system.png 2x" data-file-width="715" data-file-height="933" /></a><figcaption>En 2-dimensionell vektor bestämd av positionen av punkten <i>A</i> med koordinaterna (2, 3)</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Ijk-coordinate-system.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Ijk-coordinate-system.png/250px-Ijk-coordinate-system.png" decoding="async" width="250" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Ijk-coordinate-system.png/375px-Ijk-coordinate-system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Ijk-coordinate-system.png/500px-Ijk-coordinate-system.png 2x" data-file-width="920" data-file-height="946" /></a><figcaption>En 3-dimensionell vektor bestämd av basvektorerna <b>i</b>, <b>j</b>, <b>k</b></figcaption></figure> <p>En vektor är inte bunden till en position och det är därför tillåtet att förlägga en vektors startpunkt i <a href="/wiki/Origo" title="Origo">origo</a> i det aktuella <a href="/wiki/Koordinatsystem" title="Koordinatsystem">koordinatsystemet</a>; en konvention som ger en kompakt koordinatlista. Vektorer i ett <i>n</i>-dimensionellt rum ℝ<sup>n</sup> kan då representeras av en lista med koordinaterna för vektorernas ändpunkter enligt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\dots ,\ a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\dots ,\ a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493aa11625093e5347bebf3413563285db29aae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.597ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\dots ,\ a_{n})}"></span></dd></dl> <p>Talen i listan kallas också vektorns <i>komponenter</i>. I enlighet med figuren till höger kan den 2-dimensionella vektorn från <i>O</i> = (0, 0) till <i>A</i> = (2, 3) skrivas som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(2,\ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(2,\ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8501f5482970d6f8e7d3ce85014c93c66510d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.147ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(2,\ 3)}"></span></dd></dl> <p>En vektor kan också beskrivas genom att koordinatlistor anges för både start- och ändpunkter. </p><p>I ℝ<sup>3</sup> identifieras vektorer med tripplar av koordinater: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\ a_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\ a_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74be47bec499b82a739b69e29cfd622d3ea194c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.289ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(a_{1},\ a_{2},\ a_{3})}"></span></dd></dl> <p>eller </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(a_{x},\ a_{y},\ a_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(a_{x},\ a_{y},\ a_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb94452a4decb535871c2adce71e851f2a8e0d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.349ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} =(a_{x},\ a_{y},\ a_{z})}"></span></dd></dl> <p>Ibland arrangeras dessa tripplar till <a href="/wiki/Kolonnvektor" class="mw-redirect" title="Kolonnvektor">kolonnvektorer</a> eller <a href="/wiki/Radvektor" class="mw-redirect" title="Radvektor">radvektorer</a>, särskilt i samband med hantering av <a href="/wiki/Matris" title="Matris">matriser</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 \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76355d8ddd355584968f486a9d3d16b47eda4a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:10.534ex; height:9.176ex;" alt="{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =[a_{1}\ a_{2}\ a_{3}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =[a_{1}\ a_{2}\ a_{3}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693dcadcf595721d81e1d0771c6bcbf625cdb3f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.705ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =[a_{1}\ a_{2}\ a_{3}]}"></span></dd></dl> <p>Ett annat sätt att representera vektorer är att introducera <a href="/wiki/Basvektor" class="mw-redirect" title="Basvektor">standardbasvektorer</a>, vilket i det tredimensionella fallet kräver tre vektorer. Standardbasvektorerna har längden 1 och riktningar som sammanfaller med riktningarna för koordinatsystemets (<a href="/wiki/Kartesiskt_koordinatsystem" title="Kartesiskt koordinatsystem">kartesiskt</a>) tre axlar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {e} }_{1}=(1,\ 0,\ 0),\ {\mathbf {e} }_{2}=(0\ ,1\ ,0),\ {\mathbf {e} }_{3}=(0,\ 0,\ 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {e} }_{1}=(1,\ 0,\ 0),\ {\mathbf {e} }_{2}=(0\ ,1\ ,0),\ {\mathbf {e} }_{3}=(0,\ 0,\ 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b1591ef9be77db0412fb6550fe1c710ffa4ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.94ex; height:2.843ex;" alt="{\displaystyle {\mathbf {e} }_{1}=(1,\ 0,\ 0),\ {\mathbf {e} }_{2}=(0\ ,1\ ,0),\ {\mathbf {e} }_{3}=(0,\ 0,\ 1)}"></span></dd></dl> <p>Med hjälp av standardbasvektorerna kan varje vektor skrivas som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31462a1169d701eca47498c7609db4b9716d2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.61ex; height:2.343ex;" alt="{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}"></span></dd></dl> <p>I elementära läroböcker i <a href="/wiki/Fysik" title="Fysik">fysik</a> betecknas ofta basvektorerna med <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} ,\ \mathbf {j} ,\ \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} ,\ \mathbf {j} ,\ \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7761b569158dea30f3b6d105664cfce6331c7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.199ex; height:2.509ex;" alt="{\displaystyle \mathbf {i} ,\ \mathbf {j} ,\ \mathbf {k} }"></span> (eller <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {x}} ,\ \mathbf {\hat {y}} ,\ \mathbf {\hat {z}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">y</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {x}} ,\ \mathbf {\hat {y}} ,\ \mathbf {\hat {z}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03bece6e7928bf03d0c5d0bfb707be09b6d77eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.388ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {x}} ,\ \mathbf {\hat {y}} ,\ \mathbf {\hat {z}} }"></span>, där <b>^</b> vanligtvis betecknar <a href="/wiki/Enhetsvektor" title="Enhetsvektor">enhetsvektorn</a>). I detta fall betecknas vektorkoordinaterna enligt a<sub>x</sub>, a<sub>y</sub>, a<sub>z</sub>, och <b>a</b><sub>x</sub>, <b>a</b><sub>y</sub>, <b>a</b><sub>z</sub>. Således, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20bdb1085a6b6f39e5cc1f27db0fdb1dfce4725c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.862ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }}"></span></dd></dl> <p>För vektorer kan <a href="/wiki/Basbyte" title="Basbyte">basbyten</a> utföras och nya vektorer kan användas som bas. En vektor kan <a href="/wiki/Transformation_(matematik)" title="Transformation (matematik)">transformeras</a> till att representeras i vilken som helst av dessa nya baser. </p> <div class="mw-heading mw-heading2"><h2 id="Egenskaper">Egenskaper</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=4" title="Redigera avsnitt: Egenskaper" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=4" title="Redigera avsnitts källkod: Egenskaper"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Identiska_vektorer">Identiska vektorer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=5" title="Redigera avsnitt: Identiska vektorer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=5" title="Redigera avsnitts källkod: Identiska vektorer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Två vektorer är identiska om vektorerna har samma storlek och riktning. De två vektorerna </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571d07f851ee25f218183363b7fec7266dfdf2fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.769ex; height:2.343ex;" alt="{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}"></span></dd></dl> <p>och </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46acc2a3f76856b4d526262d5eb74f023ef4cc79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.258ex; height:2.509ex;" alt="{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}"></span></dd></dl> <p>är identiska <a href="/wiki/Om_och_endast_om" title="Om och endast om">om och endast om</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee19c0a882c28c3b99120cf19ef9538440c4ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.016ex; height:2.509ex;" alt="{\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Addition_och_subtraktion">Addition och subtraktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=6" title="Redigera avsnitt: Addition och subtraktion" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=6" title="Redigera avsnitts källkod: Addition och subtraktion"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Summan av två vektorer </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}),\mathbf {b} =(b_{1},b_{2},b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}),\mathbf {b} =(b_{1},b_{2},b_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56528a3fb389fa3d2673ea73f87bef8ef18f29f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.778ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}),\mathbf {b} =(b_{1},b_{2},b_{3})}"></span></dd></dl> <p>är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathbf {a} +\mathbf {b} =(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathbf {a} +\mathbf {b} =(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b364247d9c30817498c89f1e27d5e27984aa3020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.71ex; height:2.843ex;" alt="{\displaystyle \ \mathbf {a} +\mathbf {b} =(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3})}"></span></dd> <dd><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Fil:Vectoraddition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Vectoraddition.svg/160px-Vectoraddition.svg.png" decoding="async" width="160" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Vectoraddition.svg/240px-Vectoraddition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Vectoraddition.svg/320px-Vectoraddition.svg.png 2x" data-file-width="801" data-file-height="581" /></a><figcaption></figcaption></figure><div style="clear:left;"></div></dd></dl> <p>Den resulterande vektorns komponenter är de komponentvisa summorna av vektorernas komponenter vilket kan generaliseras till alla dimensioner. </p><p>Differensen mellan <b>a</b> och <b>b</b> är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1},a_{2}-b_{2},a_{3}-b_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1},a_{2}-b_{2},a_{3}-b_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d822e007e1e739035ec640ca123514261c275383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.13ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1},a_{2}-b_{2},a_{3}-b_{3})}"></span></dd> <dd><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Fil:VectorSubtraction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/VectorSubtraction.svg/160px-VectorSubtraction.svg.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/VectorSubtraction.svg/240px-VectorSubtraction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/VectorSubtraction.svg/320px-VectorSubtraction.svg.png 2x" data-file-width="826" data-file-height="620" /></a><figcaption></figcaption></figure><div style="clear:left;"></div></dd></dl> <p>Subtraktionen <b>a</b> - <b>b</b> kan tolkas som additionen <b>a</b> + -<b>b</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Skalär_multiplikation"><span id="Skal.C3.A4r_multiplikation"></span>Skalär multiplikation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=7" title="Redigera avsnitt: Skalär multiplikation" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=7" title="Redigera avsnitts källkod: Skalär multiplikation"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:ScalarMult.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/ScalarMult.png/250px-ScalarMult.png" decoding="async" width="250" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/ScalarMult.png/375px-ScalarMult.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/ScalarMult.png/500px-ScalarMult.png 2x" data-file-width="800" data-file-height="358" /></a><figcaption>De skalära multiplikationerna −<b>a</b> och 2<b>a</b> av en vektor <b>a</b></figcaption></figure> <p>Om en vektor multipliceras med ett <a href="/wiki/Reella_tal" title="Reella tal">reellt</a> tal <i>r</i> (en skalär) ändras vektorns längd (skalning av vektorn): </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 \ r\mathbf {a} =r(a_{x},a_{y},a_{z})=(ra_{x},ra_{y},ra_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>r</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>r</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ r\mathbf {a} =r(a_{x},a_{y},a_{z})=(ra_{x},ra_{y},ra_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7be418ced01f147751cd5c430d5492675bb948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.9ex; height:3.009ex;" alt="{\displaystyle \ r\mathbf {a} =r(a_{x},a_{y},a_{z})=(ra_{x},ra_{y},ra_{z})}"></span></dd></dl> <p>Om <i>r</i> är negativ kastas vektorns riktning om, det vill säga, vektorn roteras 180°. </p><p>Skalär multiplikation är <a href="/wiki/Distributivitet" title="Distributivitet">distributiv</a> över vektoraddition </p> <dl><dd><i>r</i>(<b>a</b> + <b>b</b>) = <i>r</i><b>a</b> + <i>r</i><b>b</b></dd></dl> <p>för alla vektorer <b>a</b> och <b>b</b> och alla skalärer <i>r</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Längd"><span id="L.C3.A4ngd"></span>Längd</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=8" title="Redigera avsnitt: Längd" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=8" title="Redigera avsnitts källkod: Längd"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Norm_(matematik)" title="Norm (matematik)">Norm (matematik)</a></dd></dl></div> <p><i><a href="/wiki/L%C3%A4ngd" title="Längd">Längden</a></i> eller <i>magnituden</i> eller <i><a href="/wiki/Norm_(matematik)" title="Norm (matematik)">normen</a></i> av vektorn <b>a</b> betecknas ||<b>a</b>||. </p> <dl><dd><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Fil:Vector-length.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Vector-length.png/200px-Vector-length.png" decoding="async" width="200" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Vector-length.png/300px-Vector-length.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Vector-length.png/400px-Vector-length.png 2x" data-file-width="800" data-file-height="665" /></a><figcaption></figcaption></figure><div style="clear:left;"></div></dd></dl> <p>Längden av vektorn <b>a</b> kan i ett vektorrum med euklidisk norm beräknas med <a href="/wiki/Pytagoras_sats" class="mw-redirect" title="Pytagoras sats">Pytagoras sats</a> enligt </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\|\mathbf {a} \right\|={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dfaede1bc16fa79911ecb67db85b74251d2a2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:24.742ex; height:4.843ex;" alt="{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}"></span></dd></dl> <p>då koordinataxlarna är vinkelräta mot varandra i detta vektorrum. </p><p>Normen är även lika med kvadratroten ur <a href="/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt">skalärprodukten</a> (se nedan) av vektorn med sig själv: </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\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55722c0cd12f56a487dc6a61bdfcc4e763a03df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.937ex; height:3.009ex;" alt="{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}}"></span></dd></dl> <p>Vektorer med längden 1 kallas <i><a href="/wiki/Enhetsvektor" title="Enhetsvektor">enhetsvektorer</a></i> och <i><a href="/wiki/Nollvektor" title="Nollvektor">nollvektorn</a></i> har längden noll. <i>Normalisering</i> av en vektor <span style="white-space:nowrap"><b>a</b> = [<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, <i>a</i><sub>3</sub>]</span>, sker genom att vektorn multipliceras med det reciproka värdet av vektorns längd, ||<b>a</b>||: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">a</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a68ef5514253b0c663439482c0559a5f8654c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.895ex; height:5.509ex;" alt="{\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Skalärprodukt"><span id="Skal.C3.A4rprodukt"></span>Skalärprodukt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=9" title="Redigera avsnitt: Skalärprodukt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=9" title="Redigera avsnitts källkod: Skalärprodukt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt">Skalärprodukt</a></dd></dl></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:Scalar-dot-product-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Scalar-dot-product-1.png/250px-Scalar-dot-product-1.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Scalar-dot-product-1.png/375px-Scalar-dot-product-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Scalar-dot-product-1.png/500px-Scalar-dot-product-1.png 2x" data-file-width="806" data-file-height="644" /></a><figcaption></figcaption></figure> <p><i>Skalärprodukten</i> av två vektorer <b>a</b> och <b>b</b> (ibland kallad <i><a href="/wiki/Inre_produkt" class="mw-redirect" title="Inre produkt">inre produkt</a></i>) betecknas <b>a</b> ∙ <b>b</b> och dess resultat är en skalär (ett reellt tal, här en längd multiplicerad med en längd) och är definierad som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f8c962f73f83456742caa95c89970a18a97f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.36ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }"></span></dd></dl> <p>där <i>θ</i> är mätetalet för <a href="/wiki/Vinkel" title="Vinkel">vinkeln</a> mellan <b>a</b> och <b>b</b>. Geometriskt innebär detta att <b>a</b> och <b>b</b> kan antas dragna från en gemensam startpunkt och längden av <a href="/wiki/Projektion_(linj%C3%A4r_algebra)" class="mw-redirect" title="Projektion (linjär algebra)">projektionen</a> av <b>a</b> på <b>b</b> är multiplicerad med <b>b</b>:s längd. </p><p>Skalärprodukten kan i ett <a href="/wiki/Ortonormerad_bas" title="Ortonormerad bas">ortonormerat koordinatsystem</a> definieras som summan av de komponentvisa produkterna enligt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\dots +a_{n}b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <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>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> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</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> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\dots +a_{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb92b2077e8f366cfe1580da48bd485c68cc2df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.81ex; height:6.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\dots +a_{n}b_{n}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Skalär_trippelprodukt"><span id="Skal.C3.A4r_trippelprodukt"></span>Skalär trippelprodukt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=10" title="Redigera avsnitt: Skalär trippelprodukt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=10" title="Redigera avsnitts källkod: Skalär trippelprodukt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Trippelprodukt" title="Trippelprodukt">Trippelprodukt</a></dd></dl></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:3-cross-product.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/3-cross-product.png/250px-3-cross-product.png" decoding="async" width="250" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/3-cross-product.png/375px-3-cross-product.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/3-cross-product.png/500px-3-cross-product.png 2x" data-file-width="817" data-file-height="852" /></a><figcaption></figcaption></figure> <p><i>Skalära trippelprodukten</i> definieras som skalärprodukten av en vektor och kryssprodukten (se nedan) av två andra vektorer: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3eb0e68741b1979a8b6a210462615e383049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.302ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}"></span></dd></dl> <p>Trippelprodukten kan geometriskt tolkas som volymen av en parallellipiped som spänns upp av de tre vektorerna. </p><p>Trippelprodukten kan beräknas enligt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1d6095438e953d75300658518683d98f37c19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.102ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"></span></dd></dl> <p>Om vektorerna i kryssprodukten byter plats negeras trippelprodukten: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fbe44dff91dd0149b297c501594438ea914f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.51ex; height:2.843ex;" alt="{\displaystyle \mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )}"></span></dd></dl> <p>Den skalära trippelprodukten kan också tolkas som <a href="/wiki/Determinant" title="Determinant">determinanten</a> till en 3 × 3 <a href="/wiki/Matris" title="Matris">matris</a> som har tre vektorer som rader eller kolumner (transponering av en matris ändrar inte determinantens värde): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e77e41eb14b0b296a76d6db17319e2af612a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.366ex; height:9.176ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kryssprodukt">Kryssprodukt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=11" title="Redigera avsnitt: Kryssprodukt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=11" title="Redigera avsnitts källkod: Kryssprodukt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Kryssprodukt" title="Kryssprodukt">Kryssprodukt</a></dd></dl></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Cross-product-povray.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Cross-product-povray.png/250px-Cross-product-povray.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Cross-product-povray.png/375px-Cross-product-povray.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Cross-product-povray.png/500px-Cross-product-povray.png 2x" data-file-width="828" data-file-height="828" /></a><figcaption>Kryssproduktens magnitud är lika med arean av parallellogrammen som spänns upp av <b>a</b> och <b>b</b>. <b>n</b> är <a href="/wiki/Normalvektor" title="Normalvektor">normalvektor</a> till både <b>a</b> och <b>b</b></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Right_hand_rule_cross_product.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Right_hand_rule_cross_product.svg/250px-Right_hand_rule_cross_product.svg.png" decoding="async" width="250" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Right_hand_rule_cross_product.svg/375px-Right_hand_rule_cross_product.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Right_hand_rule_cross_product.svg/500px-Right_hand_rule_cross_product.svg.png 2x" data-file-width="507" data-file-height="459" /></a><figcaption>Högerhandsregeln för en kryssprodukt.</figcaption></figure> <p><i><a href="/wiki/Kryssprodukt" title="Kryssprodukt">Kryssprodukten</a></i> (också kallad <i>vektorprodukt</i> eller <i>yttre produkt</i>) är bara meningsfull i tre eller sju dimensioner. Kryssprodukten skiljer sig från skalärprodukten genom att resultatet är en vektor. Kryssprodukten, betecknad <b>a</b> × <b>b</b>, är en vektor vinkelrät mot både <b>a</b> och <b>b</b> och definieras som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f960bc321540bc12366effa5a23ef5b5839c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.56ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} }"></span></dd></dl> <p>där <i>θ</i> är mätetalet för vinkeln mellan <b>a</b> och <b>b</b>, och <b>n</b> är en <a href="/wiki/Enhetsvektor" title="Enhetsvektor">enhetsvektor</a> vinkelrät mot både <b>a</b> och <b>b</b> som tillsammans med dessa bildar ett högerorienterat system. </p><p>Längden av <b>a</b> × <b>b</b> kan tolkas som arean av en parallellogram som har <b>a</b> och <b>b</b> som sidor. </p><p>Kryssprodukten kan i ett ortonormerat koordinatsystem också skrivas som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccc2e91be1f9539ff92525a2d428e5240148f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.137ex; height:2.843ex;" alt="{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1})}"></span></dd></dl> <p>Kryssprodukten är <a href="/wiki/Antikommutativ" class="mw-redirect" title="Antikommutativ">antikommutativ</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 \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870aac9633f39027b378ad2b4dcb8028ccf51b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.157ex; height:2.343ex;" alt="{\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }"></span></dd></dl> <p>Den är <a href="/wiki/Distributivitet" title="Distributivitet">distributiv</a> för addition: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=(\mathbf {a} \times \mathbf {b} )+(\mathbf {a} \times \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=(\mathbf {a} \times \mathbf {b} )+(\mathbf {a} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709f5bd9f07276d0fea92e83510efedbdc41572b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.973ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=(\mathbf {a} \times \mathbf {b} )+(\mathbf {a} \times \mathbf {c} )}"></span></dd></dl> <p>Kryssprodukten är relaterad till skalärprodukten enligt </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\|\mathbf {a} \times \mathbf {b} \right\|^{2}=\left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {a} \times \mathbf {b} \right\|^{2}=\left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e68dd7f607d60301968f6d2fee6a3e2289c3ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.814ex; height:3.343ex;" alt="{\displaystyle \left\|\mathbf {a} \times \mathbf {b} \right\|^{2}=\left\|\mathbf {a} \right\|^{2}\left\|\mathbf {b} \right\|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Vektoriell_trippelprodukt">Vektoriell trippelprodukt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=12" title="Redigera avsnitt: Vektoriell trippelprodukt" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=12" title="Redigera avsnitts källkod: Vektoriell trippelprodukt"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Trippelprodukt" title="Trippelprodukt">Trippelprodukt</a></dd></dl></div> <p>Den <i>vektoriella trippelprodukten</i> är kryssprodukten av en vektor och kryssprodukten av två andra vektorer: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \,(\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} \,(\mathbf {a} \cdot \mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \,(\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} \,(\mathbf {a} \cdot \mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1001ec1814d07e89367a5f9ae1f9d37b65556da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.098ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \,(\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} \,(\mathbf {a} \cdot \mathbf {b} )}"></span>.</dd></dl> <p>Då kryssprodukten är <a href="/wiki/Kommutativitet" title="Kommutativitet">antikommutativ</a> kan detta också skrivas </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\,\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\,\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\,\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\,\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/305360a124e6589a6c65c09013cd105f3a4ee37f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.276ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\,\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\,\mathbf {b} }"></span></dd></dl> <p>En annan användbar formulering är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\;-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\;-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa620ea72b99ce9cc0bdac8a794a9ad0876bc94a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.973ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\;-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vektorprojektion">Vektorprojektion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=13" title="Redigera avsnitt: Vektorprojektion" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=13" title="Redigera avsnitts källkod: Vektorprojektion"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Projektion_(algebra)" title="Projektion (algebra)">Projektion (algebra)</a></dd></dl></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Projections-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Projections-2.png/280px-Projections-2.png" decoding="async" width="280" height="307" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Projections-2.png/420px-Projections-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Projections-2.png/560px-Projections-2.png 2x" data-file-width="729" data-file-height="800" /></a><figcaption>Projektion av <b>a</b> på <b>b</b> (<b>a</b><sub>1</sub>). När 90° < <i>θ</i> ≤ 180°, har <b>a</b><sub>1</sub> en motsatt riktning med avseende på <b>b</b></figcaption></figure> <p>Projektionen av en vektor <b>a</b> på en vektor <b>b</b> (en vektorkomponent i <b>b</b>:s riktning) är den ortogonala projektionen av <b>a</b> på en rät linje parallell med <b>b</b> och definieras som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">b</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abc99128ca45844296477b0faabf1ef0c0c2670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.222ex; height:3.176ex;" alt="{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></span> är en skalär, kallad den skalära projektionen av <b>a</b> på <b>b</b> och <b>b̂</b> är enhetsvektorn i <b>b</b>:s riktning. Den skalära projektionen definieras i sin tur som </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} =\mathbf {a} \cdot {\frac {\mathbf {b} }{|\mathbf {b} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">b</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} =\mathbf {a} \cdot {\frac {\mathbf {b} }{|\mathbf {b} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ce5efc0e0104a62a0cad0b856fdaa00bb38ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.206ex; height:6.176ex;" alt="{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} =\mathbf {a} \cdot {\frac {\mathbf {b} }{|\mathbf {b} |}}}"></span></dd></dl> <p>där operatorn <b>·</b> betecknar <a href="/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt">skalärprodukt</a>, |<b>a</b>| är den <a href="/wiki/Euklidisk_norm" class="mw-redirect" title="Euklidisk norm">euklidiska normen</a> av <b>a</b> och <i>θ</i> är vinkeln mellan <b>a</b> och <b>b</b>. Den skalära projektionen har samma längd som vektorprojektionen. </p><p>Vektorkomponenten <b>a</b><sub>2</sub> av <b>a</b> vinkelrät mot <b>b</b> är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5f2b68b7c4ed888ea71b7ef2a00d416cc6ec19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.593ex; height:2.343ex;" alt="{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}.}"></span></dd></dl> <p>När vinkeln <i>θ</i> är okänd kan cosinus <i>θ</i> beräknas med hjälp av <b>a</b> och <b>b</b> och definitionen av skalärprodukt: </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 {\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}=\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}=\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a7b399aa9082990ac761d4253f514b3bc48507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.283ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}=\cos \theta }"></span></dd></dl> <p>Med hjälp av denna egenskap blir definitionen av den skalära projektionen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =|\mathbf {a} |{\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =|\mathbf {a} |{\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf120284f7869832fac9a44ddf2da032624f3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.024ex; height:6.176ex;" alt="{\displaystyle a_{1}=|\mathbf {a} |\cos \theta =|\mathbf {a} |{\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} |\,|\mathbf {b} |}}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}\,}"></span></dd></dl> <p>På liknande sätt blir definitionen av <b>a</b>:s vektorprojektion på <b>b</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}{\frac {\mathbf {b} }{|\mathbf {b} |}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">b</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}{\frac {\mathbf {b} }{|\mathbf {b} |}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc3068d1fec6d449c9b407578868f0e4f3c648f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.882ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |}}{\frac {\mathbf {b} }{|\mathbf {b} |}},}"></span></dd></dl> <p>vilket är ekvivalent med endera </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{1}=(\mathbf {a} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">b</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">b</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{1}=(\mathbf {a} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3453ab63e1328fb8f47f230a877a7dc4b399f67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.858ex; height:3.343ex;" alt="{\displaystyle \mathbf {a} _{1}=(\mathbf {a} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} ,}"></span></dd></dl> <p>eller<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-reference-link-bracket">[</span>3<span class="cite-reference-link-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ecdc8a9d5fb85f9323b55458ee87190618eebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.307ex; height:6.509ex;" alt="{\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {b} |^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Exempel">Exempel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=14" title="Redigera avsnitt: Exempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=14" title="Redigera avsnitts källkod: Exempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Linedistance-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Linedistance-2.png/300px-Linedistance-2.png" decoding="async" width="300" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Linedistance-2.png/450px-Linedistance-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Linedistance-2.png/600px-Linedistance-2.png 2x" data-file-width="1055" data-file-height="734" /></a><figcaption>Avståndet <i>PQ</i> mellan två linjer kan bestämmas genom en projektion av <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> på linjernas normalvektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span>. Linjerna ligger i två parallella plan som är bestämda av normalvektorn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span> och av punkterna <i>P</i> och <i>Q</i> som ligger i respektive plan</figcaption></figure> <p>Bestäm avståndet mellan två linjer i <b>R</b><sup>3</sup> givna i parameterformerna </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 (1,\ 1,\ 1)+t(1,\ 2,\ 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,\ 1,\ 1)+t(1,\ 2,\ 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73de9c20ae02abce0e5db77b2b2d0f9aca39d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.732ex; height:2.843ex;" alt="{\displaystyle (1,\ 1,\ 1)+t(1,\ 2,\ 1)}"></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 (1,\ 3,\ 4)+t(2,\ 2,\ 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,\ 3,\ 4)+t(2,\ 2,\ 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6ce9a996a0ea991e60590393bf131044e0c30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.732ex; height:2.843ex;" alt="{\displaystyle (1,\ 3,\ 4)+t(2,\ 2,\ 0)}"></span></dd></dl> <p>där riktningsvektorerna för linjerna är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}=(1,\ 2,\ 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{1}=(1,\ 2,\ 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381595d229cac6f605ba2b81e4d2c29d31a110f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.164ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} _{1}=(1,\ 2,\ 1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{2}=(2,\ 2,\ 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{2}=(2,\ 2,\ 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc37db858465cda5df972f8e6d1596afd9a75e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.164ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} _{2}=(2,\ 2,\ 0)}"></span></dd></dl> <p>Kortaste avståndet representeras av en sträcka <i>d</i> = <i>PQ</i> som är ortogonal mot linjerna. Vektorn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} =\mathbf {u} _{1}\times \mathbf {u} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} =\mathbf {u} _{1}\times \mathbf {u} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2abbc2e5a65f0c0a60a95156d3b6c87e440a2b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.503ex; height:2.009ex;" alt="{\displaystyle \mathbf {n} =\mathbf {u} _{1}\times \mathbf {u} _{2}}"></span> är linjernas normalvektor med samma riktning som sträckan <i>PQ</i>. En vektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> för linjen mellan linjernas fixa punkter är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =(1,\ 3,\ 4)-(1,\ 1,\ 1)=(0,\ 2,\ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =(1,\ 3,\ 4)-(1,\ 1,\ 1)=(0,\ 2,\ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf70d6d017c32ba6e441aa242dedfeb6f588ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.026ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} =(1,\ 3,\ 4)-(1,\ 1,\ 1)=(0,\ 2,\ 3)}"></span></dd></dl> <p>Avståndet <i>d</i> mellan linjerna är projektionen av <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> på <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"></span>: </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 d=|\mathbf {v} \cdot {\cfrac {\mathbf {n} }{\|\mathbf {n} \|}}|={\cfrac {1}{\sqrt {3}}}={\cfrac {\sqrt {3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=|\mathbf {v} \cdot {\cfrac {\mathbf {n} }{\|\mathbf {n} \|}}|={\cfrac {1}{\sqrt {3}}}={\cfrac {\sqrt {3}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8436ea739f4606ca2cfdfde358ff15f72b878f05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.41ex; height:7.176ex;" alt="{\displaystyle d=|\mathbf {v} \cdot {\cfrac {\mathbf {n} }{\|\mathbf {n} \|}}|={\cfrac {1}{\sqrt {3}}}={\cfrac {\sqrt {3}}{3}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vektorer_i_ℝ2_och_komplexa_tal"><span id="Vektorer_i_.E2.84.9D2_och_komplexa_tal"></span>Vektorer i ℝ<sup>2</sup> och komplexa tal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=15" title="Redigera avsnitt: Vektorer i ℝ2 och komplexa tal" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=15" title="Redigera avsnitts källkod: Vektorer i ℝ2 och komplexa tal"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Komplexa_tal" title="Komplexa tal">Komplexa tal</a> kan ses som ett fall av vektorer i ℝ<sup>2</sup>. Ett komplext tal har en realdel och en imaginärdel som kan representeras som komponenter i en vektor och som även kan ritas som en vektorpil i det komplexa talplanet. Addition, subtraktion, skalning och längdberäkning utförs som för rumsliga vektorer i ℝ<sup>2</sup>. Komplexa tal medger dessutom vanlig multiplikation och division. </p><p>En annan förbindelse mellan komplexa tal och vektorer är vektorer vars komponenter är komplexa tal (komplexvärda vektorer). </p> <div class="mw-heading mw-heading2"><h2 id="Vektorfält"><span id="Vektorf.C3.A4lt"></span>Vektorfält</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=16" title="Redigera avsnitt: Vektorfält" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=16" title="Redigera avsnitts källkod: Vektorfält"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint huvudartikel" style="font-style:italic;"> <dl><dd>Huvudartikel: <a href="/wiki/Vektorf%C3%A4lt" title="Vektorfält">Vektorfält</a></dd></dl></div> <p>Ett <a href="/wiki/Vektorf%C3%A4lt" title="Vektorfält">vektorfält</a> är en tilldelning av en vektor till varje punkt i en <a href="/wiki/Delm%C3%A4ngd" title="Delmängd">delmängd</a> av rummet. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 152px"> <div class="thumb" style="width: 150px;"><span typeof="mw:File"><a href="/wiki/Fil:VectorField.svg" class="mw-file-description" title="En del av vektorfältet (sin y, sin x)"><img alt="En del av vektorfältet (sin y, sin x)" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/225px-VectorField.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/338px-VectorField.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/VectorField.svg/450px-VectorField.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div> <div class="gallerytext">En del av vektorfältet (sin <i>y</i>, sin <i>x</i>)</div> </li> <li class="gallerybox" style="width: 152px"> <div class="thumb" style="width: 150px;"><span typeof="mw:File"><a href="/wiki/Fil:Vector_Field_on_a_Sphere.png" class="mw-file-description" title="Ett vektorfält på en sfär"><img alt="Ett vektorfält på en sfär" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Vector_Field_on_a_Sphere.png/225px-Vector_Field_on_a_Sphere.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Vector_Field_on_a_Sphere.png/338px-Vector_Field_on_a_Sphere.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Vector_Field_on_a_Sphere.png/450px-Vector_Field_on_a_Sphere.png 2x" data-file-width="1200" data-file-height="1200" /></a></span></div> <div class="gallerytext">Ett vektorfält på en sfär</div> </li> <li class="gallerybox" style="width: 262px"> <div class="thumb" style="width: 260px;"><span typeof="mw:File"><a href="/wiki/Fil:Cessna_182_model-wingtip-vortex.jpg" class="mw-file-description" title="Vektorfält runt ett flygplan, visualiserat med bubblor"><img alt="Vektorfält runt ett flygplan, visualiserat med bubblor" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/390px-Cessna_182_model-wingtip-vortex.jpg" decoding="async" width="260" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/585px-Cessna_182_model-wingtip-vortex.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Cessna_182_model-wingtip-vortex.jpg/779px-Cessna_182_model-wingtip-vortex.jpg 2x" data-file-width="2271" data-file-height="1313" /></a></span></div> <div class="gallerytext">Vektorfält runt ett flygplan, visualiserat med bubblor</div> </li> </ul> <p>Exempel på vektorfält: </p> <ul><li><a href="/wiki/Gradient" class="mw-redirect" title="Gradient">Gradientfält</a></li> <li><a href="/wiki/Divergens_(vektoranalys)" title="Divergens (vektoranalys)">Divergensfält</a></li> <li><a href="/wiki/Rotation_(vektoranalys)" title="Rotation (vektoranalys)">Rotationsfält</a></li> <li><a href="/wiki/Skal%C3%A4rpotential" title="Skalärpotential">Skalärpotential</a>, gradientens invers</li></ul> <div class="mw-heading mw-heading2"><h2 id="Se_även"><span id="Se_.C3.A4ven"></span>Se även</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=17" title="Redigera avsnitt: Se även" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=17" title="Redigera avsnitts källkod: Se även"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Enhetsvektor" title="Enhetsvektor">Enhetsvektor</a></li> <li><a href="/wiki/Kontravariant_vektor" class="mw-redirect" title="Kontravariant vektor">Kontravariant vektor</a></li> <li><a href="/wiki/Kovariant_vektor" title="Kovariant vektor">Kovariant vektor</a></li> <li><a href="/wiki/Kovarians_och_kontravarians_(vektorer)" title="Kovarians och kontravarians (vektorer)">Kovarians och kontravarians (vektorer)</a></li> <li><a href="/wiki/Linj%C3%A4rt_rum" title="Linjärt rum">Linjärt rum</a> (vektorrum)</li> <li><a href="/wiki/Normalvektor" title="Normalvektor">Normalvektor</a></li> <li><a href="/wiki/Ortsvektor" title="Ortsvektor">Ortsvektor</a></li> <li><a href="/wiki/Poyntings_vektor" title="Poyntings vektor">Poyntings vektor</a></li> <li><a href="/wiki/Pseudovektor" title="Pseudovektor">Pseudovektor</a> (axiell vektor)</li> <li><a href="/wiki/Rumsvektor" title="Rumsvektor">Rumsvektor</a></li> <li><a href="/wiki/Talf%C3%B6ljd" title="Talföljd">Talföljd</a></li> <li><a href="/wiki/Vektoranalys" title="Vektoranalys">Vektoranalys</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referenser">Referenser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=18" title="Redigera avsnitt: Referenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=18" title="Redigera avsnitts källkod: Referenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Folke Eriksson, Flerdimensionell Analys, Studentlitteratur Lund 1971</li></ul> <div class="mw-heading mw-heading3"><h3 id="Noter">Noter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=19" title="Redigera avsnitt: Noter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=19" title="Redigera avsnitts källkod: Noter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Crowe-1">^ [<a href="#cite_ref-Crowe_1-0"><small>a</small></a> <a href="#cite_ref-Crowe_1-1"><small>b</small></a> <a href="#cite_ref-Crowe_1-2"><small>c</small></a> <a href="#cite_ref-Crowe_1-3"><small>d</small></a>] <span class="reference-text">Michael J. Crowe, <i>A History of Vector Analysis</i>; se även hans föreläsning med samma titel <cite style="font-style:normal" class="web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040126161844/http://www.nku.edu/~curtin/crowe_oresme.pdf">”A History of Vector Analysis”</a>. Arkiverad från <a rel="nofollow" class="external text" href="http://www.nku.edu/~curtin/crowe_oresme.pdf">originalet</a> den 26 januari 2004<span class="printonly">. <a rel="nofollow" class="external free" href="https://web.archive.org/web/20040126161844/http://www.nku.edu/~curtin/crowe_oresme.pdf">https://web.archive.org/web/20040126161844/http://www.nku.edu/~curtin/crowe_oresme.pdf</a></span><span class="reference-accessdate">. Läst 4 september 2010</span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=A+History+of+Vector+Analysis&rft.atitle=&rft_id=https%3A%2F%2Fweb.archive.org%2Fweb%2F20040126161844%2Fhttp%3A%2F%2Fwww.nku.edu%2F%7Ecurtin%2Fcrowe_oresme.pdf&rfr_id=info:sid/en.wikipedia.org:Vektor"><span style="display: none;"> </span></span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <span class="reference-text">W. R. Hamilton (1846) <i>London, Edinburgh & Dublin Philosophical Magazine</i> 3rd series 29 27</span> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <span class="reference-text"><cite style="font-style:normal" class="web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160531080405/http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html">”Dot Products and Projections”</a>. Arkiverad från <a rel="nofollow" class="external text" href="http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html">originalet</a> den 31 maj 2016<span class="printonly">. <a rel="nofollow" class="external free" href="https://web.archive.org/web/20160531080405/http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html">https://web.archive.org/web/20160531080405/http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html</a></span><span class="reference-accessdate">. Läst 3 november 2017</span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Dot+Products+and+Projections&rft.atitle=&rft_id=https%3A%2F%2Fweb.archive.org%2Fweb%2F20160531080405%2Fhttp%3A%2F%2Fmath.oregonstate.edu%2Fhome%2Fprograms%2Fundergrad%2FCalculusQuestStudyGuides%2Fvcalc%2Fdotprod%2Fdotprod.html&rfr_id=info:sid/en.wikipedia.org:Vektor"><span style="display: none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Vidare_läsning"><span id="Vidare_l.C3.A4sning"></span>Vidare läsning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=20" title="Redigera avsnitt: Vidare läsning" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=20" title="Redigera avsnitts källkod: Vidare läsning"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="book" id="CITEREFGavel2011">Gavel, Hillevi (2011). <i><span>Grundläggande linjär algebra</span></i>. Studentlitteratur. <a href="/wiki/Libris_(bibliotekskatalog)" title="Libris (bibliotekskatalog)">Libris</a> <a rel="nofollow" class="external text" href="http://libris.kb.se/bib/12242337">12242337</a>. <a href="/wiki/Special:Bokk%C3%A4llor/9789144076058" title="Special:Bokkällor/9789144076058">ISBN 9789144076058</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundl%C3%A4ggande+linj%C3%A4r+algebra&rft.aulast=Gavel&rft.aufirst=Hillevi&rft.au=Gavel%2C+Hillevi&rft.date=2011&rft.pub=Studentlitteratur&rft.isbn=9789144076058&rfr_id=info:sid/en.wikipedia.org:Vektor"><span style="display: none;"> </span></span></li> <li><cite style="font-style:normal" class="book" id="CITEREFSparr,_Gunnar,_1942-">Sparr, Gunnar, 1942- (1995 ;). <i><a rel="nofollow" class="external text" href="http://worldcat.org/oclc/187001658">Linjär algebra</a></i>. Studentlitteratur. <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/187001658">187001658</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/oclc/187001658">http://worldcat.org/oclc/187001658</a></span><span class="reference-accessdate">. Läst 15 juni 2019</span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linj%C3%A4r+algebra&rft.aulast=Sparr%2C+Gunnar%2C+1942-&rft.au=Sparr%2C+Gunnar%2C+1942-&rft.date=1995+%3B&rft.pub=Studentlitteratur&rft_id=info:oclcnum/187001658&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F187001658&rfr_id=info:sid/en.wikipedia.org:Vektor"><span style="display: none;"> </span></span></li> <li><cite style="font-style:normal" class="book" id="CITEREFMånsson,_Jonas.">Månsson, Jonas.. <i><a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1097632984">Linjär algebra.</a></i>. <a href="/wiki/Special:Bokk%C3%A4llor/9789144127408" title="Special:Bokkällor/9789144127408">ISBN 9789144127408</a>. <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1097632984">1097632984</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://worldcat.org/oclc/1097632984">http://worldcat.org/oclc/1097632984</a></span><span class="reference-accessdate">. Läst 15 juni 2019</span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linj%C3%A4r+algebra.&rft.aulast=M%C3%A5nsson%2C+Jonas.&rft.au=M%C3%A5nsson%2C+Jonas.&rft_id=info:oclcnum/1097632984&rft.isbn=9789144127408&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F1097632984&rfr_id=info:sid/en.wikipedia.org:Vektor"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externa_länkar"><span id="Externa_l.C3.A4nkar"></span>Externa länkar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vektor&veaction=edit&section=21" title="Redigera avsnitt: Externa länkar" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Vektor&action=edit&section=21" title="Redigera avsnitts källkod: Externa länkar"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> Wikimedia Commons har media som rör <a href="https://commons.wikimedia.org/wiki/Category:Vectors" class="extiw" title="commons:Category:Vectors">Vektor</a>.<div class="interProject commons" style="display:none;"><a href="https://commons.wikimedia.org/wiki/Category:Vectors" class="extiw" title="commons:Category:Vectors">Bilder & media</a></div></li></ul> <style data-mw-deduplicate="TemplateStyles:r56287950">.mw-parser-output table.navbox{border:#aaa 1px 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.navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background:#d0e0f5}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background:#deeafa}.mw-parser-output .navbox-even{background:#f7f7f7}.mw-parser-output .navbox-odd{background:transparent}</style><table class="navbox" style="border-spacing:0; ;"><tbody><tr><td style="padding:2px;"><table class="collapsible collapsed" style="width:100%;border-spacing:0;background:transparent;color:inherit;;"><tbody><tr><th style=";" colspan="3" class="navbox-title"><div style="float:left; width:3em;text-align:left;"><div class="noprint plainlinks" style="background-color:transparent; padding:0; white-space:nowrap; font-weight:normal; font-size:80%; border:none;; color: inherit;"><a href="/wiki/Mall:Linj%C3%A4r-algebra" title="Mall:Linjär-algebra"><span title="Visa denna mall" style="border:none;;">v</span></a> <span style="font-size:80%;">•</span> <a class="external text" href="https://sv.wikipedia.org/w/index.php?title=Mall:Linj%C3%A4r-algebra&action=edit"><span style="border:none;;" title="Redigera den här mallen">r</span></a></div></div><span style="font-size:110%;"><a href="/wiki/Linj%C3%A4r_algebra" title="Linjär algebra">Linjär algebra</a></span></th></tr><tr style="height:2px;"><td></td></tr><tr><td class="navbox-group" style=";;">Grundläggande begrepp</td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Skal%C3%A4r" title="Skalär">Skalär</a><span style="font-weight:bold;"> · </span><a class="mw-selflink selflink">Vektor</a><span style="font-weight:bold;"> · </span><a href="/wiki/Nollvektor" title="Nollvektor">Noll</a><span style="font-weight:bold;"> · </span><a href="/wiki/Ortogonalitet" title="Ortogonalitet">Ortogonalitet</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4rt_ekvationssystem" title="Linjärt ekvationssystem">Ekvationssystem</a><span style="font-weight:bold;"> · </span><a href="/wiki/Vektorrum" class="mw-redirect" title="Vektorrum">Rum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4rkombination" title="Linjärkombination">Linjärkombination</a><span style="font-weight:bold;"> · </span><a href="/wiki/Inre_produktrum" title="Inre produktrum">Inre produkt</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4rt_oberoende" title="Linjärt oberoende">Oberoende</a><span style="font-weight:bold;"> · </span><a href="/wiki/Bas_(linj%C3%A4r_algebra)" title="Bas (linjär algebra)">Bas</a><span style="font-weight:bold;"> · </span><a href="/wiki/Radrum" title="Radrum">Radrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Kolonnrum" title="Kolonnrum">Kolonnrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Nollrum" title="Nollrum">Nollrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Gram%E2%80%93Schmidts_ortogonaliseringsprocess" title="Gram–Schmidts ortogonaliseringsprocess">Gram-Schimdt</a><span style="font-weight:bold;"> · </span><a href="/wiki/Egenv%C3%A4rde_(matematik)" class="mw-redirect" title="Egenvärde (matematik)">Egenvärde</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4rt_h%C3%B6lje" title="Linjärt hölje">Hölje</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4ritet" title="Linjäritet">Linjäritet</a></div></td><td style="width:0%;padding:0px 0px 0px 2px;" rowspan="11"><span typeof="mw:File"><a href="/wiki/Fil:Linear_subspaces_with_shading.svg" class="mw-file-description"><img alt="Bild på euklidiska rummet" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></td></tr><tr style="height:2px"><td></td></tr><tr><td class="navbox-group" style=";;"><a href="/wiki/Linj%C3%A4r_algebra" title="Linjär algebra">Linjär algebra</a></td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Kryssprodukt" title="Kryssprodukt">Kryssprodukt</a><span style="font-weight:bold;"> · </span><a href="/wiki/Trippelprodukt" title="Trippelprodukt">Trippelprodukt</a></div></td></tr><tr style="height:2px"><td></td></tr><tr><td class="navbox-group" style=";;"><a href="/wiki/Matris" title="Matris">Matriser</a></td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Element%C3%A4r_matris" title="Elementär matris">Elementär</a><span style="font-weight:bold;"> · </span><a href="/wiki/Blockmatris" title="Blockmatris">Block</a><span style="font-weight:bold;"> · </span><a href="/wiki/Enhetsmatris" title="Enhetsmatris">Enhet</a><span style="font-weight:bold;"> · </span><a href="/wiki/Determinant" title="Determinant">Determinant</a><span style="font-weight:bold;"> · </span><a href="/wiki/Matrisnorm" title="Matrisnorm">Norm</a><span style="font-weight:bold;"> · </span><a href="/wiki/Matrisrang" title="Matrisrang">Rang</a><span style="font-weight:bold;"> · </span><a href="/wiki/Linj%C3%A4r_avbildning" title="Linjär avbildning">Transformation</a><span style="font-weight:bold;"> · </span><a href="/wiki/Rotation_(avbildning)" title="Rotation (avbildning)">Rotation</a><span style="font-weight:bold;"> · </span><a href="/wiki/Inverterbar_matris" title="Inverterbar matris">Invers</a><span style="font-weight:bold;"> · </span><a href="/wiki/Cramers_regel" title="Cramers regel">Cramers regel</a><span style="font-weight:bold;"> · </span><a href="/wiki/Trappstegsform" title="Trappstegsform">Trappstegsform</a><span style="font-weight:bold;"> · </span><a href="/wiki/Sp%C3%A5r_(matematik)" title="Spår (matematik)">Spår</a><span style="font-weight:bold;"> · </span><a href="/wiki/Transponat" title="Transponat">Transponat</a><span style="font-weight:bold;"> · </span><a href="/wiki/Gausselimination" title="Gausselimination">Gausselimination</a><span style="font-weight:bold;"> · </span><a href="/wiki/Symmetrisk_matris" title="Symmetrisk matris">Symmetri</a><span style="font-weight:bold;"> · </span><a href="/wiki/Matrisaddition" title="Matrisaddition">Addition</a></div></td></tr><tr style="height:2px"><td></td></tr><tr><td class="navbox-group" style=";;"><a href="/w/index.php?title=Multilinj%C3%A4r_algebra&action=edit&redlink=1" class="new" title="Multilinjär algebra [inte skriven än]">Multilinjär algebra</a></td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Geometrisk_algebra" class="mw-redirect" title="Geometrisk algebra">Geometrisk algebra</a><span style="font-weight:bold;"> · </span><a href="/wiki/Yttre_algebra" title="Yttre algebra">Yttre algebra</a><span style="font-weight:bold;"> · </span><a href="/wiki/Bivektor" title="Bivektor">Bivektor</a><span style="font-weight:bold;"> · </span><a href="/w/index.php?title=Multivektor&action=edit&redlink=1" class="new" title="Multivektor [inte skriven än]">Multivektor</a><span style="font-weight:bold;"> · </span><a href="/wiki/Tensor" title="Tensor">Tensor</a></div></td></tr><tr style="height:2px"><td></td></tr><tr><td class="navbox-group" style=";;"><a href="/wiki/Abstrakt_algebra" title="Abstrakt algebra">Konstruktioner</a></td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Delrum" title="Delrum">Delrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Dualrum" title="Dualrum">Dualrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Funktionsrum" title="Funktionsrum">Funktionsrum</a><span style="font-weight:bold;"> · </span><a href="/w/index.php?title=Kvotrum&action=edit&redlink=1" class="new" title="Kvotrum [inte skriven än]">Kvotrum</a><span style="font-weight:bold;"> · </span><a href="/wiki/Tensorprodukt" title="Tensorprodukt">Tensorprodukt</a></div></td></tr><tr style="height:2px"><td></td></tr><tr><td class="navbox-group" style=";;"><a href="/wiki/Numerisk_analys" title="Numerisk analys">Numerik</a></td><td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-"><div style="padding:0em 0.25em"><a href="/wiki/Flyttal" title="Flyttal">Flyttal</a><span style="font-weight:bold;"> · </span><a href="/wiki/Gles_matris" title="Gles matris">Gles matris</a></div></td></tr><tr style="height:2px;"><td></td></tr><tr><td class="navbox-abovebelow" style=";font-weight:bold;" colspan="3"><span typeof="mw:File"><span title="Kategori"><img alt="Kategori" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/16px-Folder_Hexagonal_Icon.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/24px-Folder_Hexagonal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/32px-Folder_Hexagonal_Icon.svg.png 2x" data-file-width="36" data-file-height="31" /></span></span> <a href="/wiki/Kategori:Linj%C3%A4r_algebra" title="Kategori:Linjär 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