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vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Definicija_i_primjer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija_i_primjer"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicija i primjer</span> </div> </a> <ul id="toc-Definicija_i_primjer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometrijska_interpretacija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometrijska_interpretacija"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geometrijska interpretacija</span> </div> </a> <ul id="toc-Geometrijska_interpretacija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vidjeti_također" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vidjeti_također"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vidjeti također</span> </div> </a> <ul id="toc-Vidjeti_također-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="Sadržaj" 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="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Skalarni umnožak</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="Idi na druge jezične varijante članka. Dostupan je na 70 jezika" > <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-70" 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">70 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%A5%E1%88%8B_%E1%89%A5%E1%8B%9C%E1%89%B5" title="ጥላ ብዜት – amharski" lang="am" hreflang="am" data-title="ጥላ ብዜት" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%AF%D8%A7%D8%A1_%D9%86%D9%82%D8%B7%D9%8A" title="جداء نقطي – arapski" lang="ar" hreflang="ar" data-title="جداء نقطي" data-language-autonym="العربية" data-language-local-name="arapski" 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/Productu_escalar" title="Productu escalar – asturijski" lang="ast" hreflang="ast" data-title="Productu escalar" data-language-autonym="Asturianu" data-language-local-name="asturijski" 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/Skalyar_hasil" title="Skalyar hasil – azerbajdžanski" lang="az" hreflang="az" data-title="Skalyar hasil" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1" title="Скаляр ҡабатландыҡ – baškirski" lang="ba" hreflang="ba" data-title="Скаляр ҡабатландыҡ" data-language-autonym="Башҡортса" data-language-local-name="baškirski" 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%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D1%8B_%D0%B7%D0%B4%D0%B0%D0%B1%D1%8B%D1%82%D0%B0%D0%BA" title="Скалярны здабытак – bjeloruski" lang="be" hreflang="be" data-title="Скалярны здабытак" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скаларно произведение – bugarski" lang="bg" hreflang="bg" data-title="Скаларно произведение" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%9F_%E0%A6%97%E0%A7%81%E0%A6%A3%E0%A6%A8" title="ডট গুণন – bangla" lang="bn" hreflang="bn" data-title="ডট গুণন" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Skalarni_proizvod" title="Skalarni proizvod – bosanski" lang="bs" hreflang="bs" data-title="Skalarni proizvod" data-language-autonym="Bosanski" data-language-local-name="bosanski" 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/Producte_escalar" title="Producte escalar – katalonski" lang="ca" hreflang="ca" data-title="Producte escalar" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%8E%DA%A9%D8%AF%D8%A7%D9%86%DB%8C_%D9%86%D8%A7%D9%88%DB%95%DA%A9%DB%8C" title="لێکدانی ناوەکی – soranski kurdski" lang="ckb" hreflang="ckb" data-title="لێکدانی ناوەکی" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din" title="Skalární součin – češki" lang="cs" hreflang="cs" data-title="Skalární součin" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BB%D0%B0_%D1%85%D1%83%D1%82%D0%BB%D0%B0%D0%B2" title="Скалярла хутлав – čuvaški" lang="cv" hreflang="cv" data-title="Скалярла хутлав" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Skalarprodukt" title="Skalarprodukt – danski" lang="da" hreflang="da" data-title="Skalarprodukt" data-language-autonym="Dansk" data-language-local-name="danski" 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/Skalarprodukt" title="Skalarprodukt – njemački" lang="de" hreflang="de" data-title="Skalarprodukt" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CF%8C_%CE%B3%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF" title="Εσωτερικό γινόμενο – grčki" lang="el" hreflang="el" data-title="Εσωτερικό γινόμενο" data-language-autonym="Ελληνικά" data-language-local-name="grčki" 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/Dot_product" title="Dot product – engleski" lang="en" hreflang="en" data-title="Dot product" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Skalara_produto" title="Skalara produto – esperanto" lang="eo" hreflang="eo" data-title="Skalara produto" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Producto_escalar" title="Producto escalar – španjolski" lang="es" hreflang="es" data-title="Producto escalar" data-language-autonym="Español" data-language-local-name="španjolski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Skalaarkorrutis" title="Skalaarkorrutis – estonski" lang="et" hreflang="et" data-title="Skalaarkorrutis" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Biderketa_eskalar" title="Biderketa eskalar – baskijski" lang="eu" hreflang="eu" data-title="Biderketa eskalar" data-language-autonym="Euskara" data-language-local-name="baskijski" 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%B6%D8%B1%D8%A8_%D8%AF%D8%A7%D8%AE%D9%84%DB%8C" title="ضرب داخلی – perzijski" lang="fa" hreflang="fa" data-title="ضرب داخلی" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pistetulo" title="Pistetulo – finski" lang="fi" hreflang="fi" data-title="Pistetulo" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Produit_scalaire" title="Produit scalaire – francuski" lang="fr" hreflang="fr" data-title="Produit scalaire" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Produto_escalar" title="Produto escalar – galicijski" lang="gl" hreflang="gl" data-title="Produto escalar" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99%D7%AA" title="מכפלה סקלרית – hebrejski" lang="he" hreflang="he" data-title="מכפלה סקלרית" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="अदिश गुणनफल – hindski" lang="hi" hreflang="hi" data-title="अदिश गुणनफल" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Skal%C3%A1ris_szorzat" title="Skaláris szorzat – mađarski" lang="hu" hreflang="hu" data-title="Skaláris szorzat" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%AF%D5%A1%D5%AC%D5%B5%D5%A1%D6%80_%D5%A1%D6%80%D5%BF%D5%A1%D5%A4%D6%80%D5%B5%D5%A1%D5%AC" title="Սկալյար արտադրյալ – armenski" lang="hy" hreflang="hy" data-title="Սկալյար արտադրյալ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Produk_dot" title="Produk dot – indonezijski" lang="id" hreflang="id" data-title="Produk dot" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Prodotto_scalare" title="Prodotto scalare – talijanski" lang="it" hreflang="it" data-title="Prodotto scalare" data-language-autonym="Italiano" data-language-local-name="talijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%89%E3%83%83%E3%83%88%E7%A9%8D" title="ドット積 – japanski" lang="ja" hreflang="ja" data-title="ドット積" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%99%E1%83%90%E1%83%9A%E1%83%90%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%9C%E1%83%90%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%9A%E1%83%98" title="სკალარული ნამრავლი – gruzijski" lang="ka" hreflang="ka" data-title="სკალარული ნამრავლი" data-language-autonym="ქართული" data-language-local-name="gruzijski" 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%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D0%BA%D3%A9%D0%B1%D0%B5%D0%B9%D1%82%D1%96%D0%BD%D0%B4%D1%96" title="Скаляр көбейтінді – kazaški" lang="kk" hreflang="kk" data-title="Скаляр көбейтінді" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="스칼라곱 – korejski" lang="ko" hreflang="ko" data-title="스칼라곱" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Productum_interius" title="Productum interius – latinski" lang="la" hreflang="la" data-title="Productum interius" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skaliarin%C4%97_sandauga" title="Skaliarinė sandauga – litavski" lang="lt" hreflang="lt" data-title="Skaliarinė sandauga" data-language-autonym="Lietuvių" data-language-local-name="litavski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skal%C4%81rais_reizin%C4%81jums" title="Skalārais reizinājums – latvijski" lang="lv" hreflang="lv" data-title="Skalārais reizinājums" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4" title="Скаларен производ – makedonski" lang="mk" hreflang="mk" data-title="Скаларен производ" data-language-autonym="Македонски" data-language-local-name="makedonski" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A4%BF%E0%A4%82%E0%A4%A6%E0%A5%82_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BE%E0%A4%95%E0%A4%BE%E0%A4%B0" title="बिंदू गुणाकार – marathski" lang="mr" hreflang="mr" data-title="बिंदू गुणाकार" data-language-autonym="मराठी" data-language-local-name="marathski" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hasil_darab_bintik" title="Hasil darab bintik – malajski" lang="ms" hreflang="ms" data-title="Hasil darab bintik" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inwendig_product" title="Inwendig product – nizozemski" lang="nl" hreflang="nl" data-title="Inwendig product" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Indreprodukt" title="Indreprodukt – norveški nynorsk" lang="nn" hreflang="nn" data-title="Indreprodukt" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Indreprodukt" title="Indreprodukt – norveški bokmål" lang="nb" hreflang="nb" data-title="Indreprodukt" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Iloczyn_skalarny" title="Iloczyn skalarny – poljski" lang="pl" hreflang="pl" data-title="Iloczyn skalarny" data-language-autonym="Polski" data-language-local-name="poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Prodot_%C3%ABscalar" title="Prodot ëscalar – Piedmontese" lang="pms" hreflang="pms" data-title="Prodot ëscalar" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Produto_escalar" title="Produto escalar – portugalski" lang="pt" hreflang="pt" data-title="Produto escalar" data-language-autonym="Português" data-language-local-name="portugalski" 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/Produs_scalar" title="Produs scalar – rumunjski" lang="ro" hreflang="ro" data-title="Produs scalar" data-language-autonym="Română" data-language-local-name="rumunjski" 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%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скалярное произведение – ruski" lang="ru" hreflang="ru" data-title="Скалярное произведение" data-language-autonym="Русский" data-language-local-name="ruski" 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%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скалярное произведение – jakutski" lang="sah" hreflang="sah" data-title="Скалярное произведение" data-language-autonym="Саха тыла" data-language-local-name="jakutski" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Dot_product" title="Dot product – škotski" lang="sco" hreflang="sco" data-title="Dot product" data-language-autonym="Scots" data-language-local-name="škotski" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Skalarni_proizvod_vektora" title="Skalarni proizvod vektora – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Skalarni proizvod vektora" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dot_product" title="Dot product – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dot product" 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/Skal%C3%A1rny_s%C3%BA%C4%8Din" title="Skalárny súčin – slovački" lang="sk" hreflang="sk" data-title="Skalárny súčin" data-language-autonym="Slovenčina" data-language-local-name="slovački" 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/Skalarni_produkt" title="Skalarni produkt – slovenski" lang="sl" hreflang="sl" data-title="Skalarni produkt" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Prodhimi_skalar" title="Prodhimi skalar – albanski" lang="sq" hreflang="sq" data-title="Prodhimi skalar" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%BD%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B0" title="Скаларни производ вектора – srpski" lang="sr" hreflang="sr" data-title="Скаларни производ вектора" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt – švedski" lang="sv" hreflang="sv" data-title="Skalärprodukt" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%8D" title="புள்ளிப் பெருக்கல் – tamilski" lang="ta" hreflang="ta" data-title="புள்ளிப் பெருக்கல்" data-language-autonym="தமிழ்" data-language-local-name="tamilski" 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%B8%9C%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%93%E0%B8%88%E0%B8%B8%E0%B8%94" title="ผลคูณจุด – tajlandski" lang="th" hreflang="th" data-title="ผลคูณจุด" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Produktong_tuldok" title="Produktong tuldok – tagalog" lang="tl" hreflang="tl" data-title="Produktong tuldok" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Nokta_%C3%A7arp%C4%B1m" title="Nokta çarpım – turski" lang="tr" hreflang="tr" data-title="Nokta çarpım" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D1%82%D0%B0%D0%BF%D0%BA%D1%8B%D1%80%D1%87%D1%8B%D0%B3%D1%8B%D1%88" title="Скаляр тапкырчыгыш – tatarski" lang="tt" hreflang="tt" data-title="Скаляр тапкырчыгыш" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarski" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%B8%D0%B9_%D0%B4%D0%BE%D0%B1%D1%83%D1%82%D0%BE%D0%BA" title="Скалярний добуток – ukrajinski" lang="uk" hreflang="uk" data-title="Скалярний добуток" data-language-autonym="Українська" data-language-local-name="ukrajinski" 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/%DA%88%D9%88%D9%B9_%D9%BE%D8%B1%D9%88%DA%88%DA%A9%D9%B9" title="ڈوٹ پروڈکٹ – urdski" lang="ur" hreflang="ur" data-title="ڈوٹ پروڈکٹ" data-language-autonym="اردو" data-language-local-name="urdski" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Skalyar_ko%CA%BBpaytmasi" title="Skalyar koʻpaytmasi – uzbečki" lang="uz" hreflang="uz" data-title="Skalyar koʻpaytmasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADch_v%C3%B4_h%C6%B0%E1%BB%9Bng" title="Tích vô hướng – vijetnamski" lang="vi" hreflang="vi" data-title="Tích vô hướng" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" 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/%E6%95%B0%E9%87%8F%E7%A7%AF" title="数量积 – wu kineski" lang="wuu" hreflang="wuu" data-title="数量积" data-language-autonym="吴语" data-language-local-name="wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%82%B9%E7%A7%AF" title="点积 – kineski" lang="zh" hreflang="zh" data-title="点积" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%BB%9E%E7%A9%8D" title="點積 – kantonski" lang="yue" hreflang="yue" data-title="點積" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> 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typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Scalarproduct.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Scalarproduct.gif/300px-Scalarproduct.gif" decoding="async" width="300" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Scalarproduct.gif/450px-Scalarproduct.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/7/72/Scalarproduct.gif 2x" data-file-width="562" data-file-height="474" /></a><figcaption>U geometriji, skalarni produkt dvaju vektora jest umnožak duljine jednoga vektora i duljine ortogonalne projekcije drugoga vektora na nj. Kada su vektori okomiti njihov je skalarni produkt nula.</figcaption></figure> <p><b>Skalarni umnožak</b> ili <b>skalarni produkt</b> dvaju <a href="/wiki/Vektor" title="Vektor">vektora</a> u <a href="/wiki/Geometrija" title="Geometrija">geometriji</a> se definira kao umnožak iznosa (modula, duljine, intenziteta) jednog i drugog vektora i <a href="/wiki/Kosinus" title="Kosinus">kosinusa</a> kuta između njih. Dobiveni je rezultat <a href="/wiki/Skalar" title="Skalar">skalar</a>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f801d00097189424c4a53060e0b6ecfc62c46928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:24.772ex; height:4.009ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}=\left|{\vec {a}}\right|\left|{\vec {b}}\right|\cos \phi }"></span></dd></dl> <p>Skalarni umnožak vektora sa samim sobom daje kvadrat njegova iznosa, jer je u tom slučaju kosinus <i>0°</i> jednak <i>1</i>. Skalarni umnožak vektora koji su pod pravim kutom (<i>90°</i>) jednak je <i>0</i>, jer je kosinus pravoga kuta <i>0</i>. </p><p>Skalarni umnožak je <a href="/wiki/Komutativnost" title="Komutativnost">komutativan</a>, <a href="/w/index.php?title=Distributivnost&action=edit&redlink=1" class="new" title="Distributivnost (stranica ne postoji)">distributivan</a> i <a href="/w/index.php?title=Linearnost&action=edit&redlink=1" class="new" title="Linearnost (stranica ne postoji)">linearan</a>. </p><p>Po uzoru na skalarni produkt u geometriji mogu se definirati skalarni produkti vektora u drugim, apstraktnim <a href="/wiki/Vektorski_prostor" title="Vektorski prostor">vektorskim prostorima</a>. Vektore čiji je skalarni produkt <a href="/wiki/Nula" title="Nula">nula</a> i tamo se naziva okomitima ili ortogonalnima. </p> <div class="mw-heading mw-heading2"><h2 id="Definicija_i_primjer">Definicija i primjer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&veaction=edit&section=1" title="Uredi odlomak: Definicija i primjer" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&action=edit&section=1" title="Uredi kôd odjeljka Definicija i primjer"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definicija skalarnog umnoška vektora <b>a</b> = [<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, … , <i>a</i><sub><i>n</i></sub>] i vektora <b>b</b> = [<i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, … , <i>b</i><sub><i>n</i></sub>] : </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}+\cdots +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}+\cdots +a_{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f8ac1d2b7ffb9ef70bb6b151a4b931f20087a5" 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}+\cdots +a_{n}b_{n}}"></span></dd></dl> <ul><li>gdje <i>Σ</i> označuje zbrajanje po komponentama.</li></ul> <p>Primjer skalarnog množenja na trodimenzionalnom vektoru <i>[1, 3, −5]</i> i <i>[4, −2, −1]</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}\cdot {\begin{bmatrix}4&-2&-1\end{bmatrix}}=(1)(4)+(3)(-2)+(-5)(-1)=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}\cdot {\begin{bmatrix}4&-2&-1\end{bmatrix}}=(1)(4)+(3)(-2)+(-5)(-1)=3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4111a43eca4706bdad9586ffc0810b691f2b6c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.401ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}\cdot {\begin{bmatrix}4&-2&-1\end{bmatrix}}=(1)(4)+(3)(-2)+(-5)(-1)=3.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Geometrijska_interpretacija">Geometrijska interpretacija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&veaction=edit&section=2" title="Uredi odlomak: Geometrijska interpretacija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&action=edit&section=2" title="Uredi kôd odjeljka Geometrijska interpretacija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>S obzirom na to da znamo da je skalarni umnožak i umnožak s kutom između dva vektora, možemo inverznom operacijom izračunati i kut. </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} =|\mathbf {a} |\,|\mathbf {b} |\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 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> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \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/49d5f609204676905eec218364aa9883427e1532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.684ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =|\mathbf {a} |\,|\mathbf {b} |\cos \theta \,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⟹</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Longrightarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052ac938cf46158e09d7a9528162518802cabd3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.806ex; height:1.843ex;" alt="{\displaystyle \Longrightarrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arccos \left({\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} ||\mathbf {b} |}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <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">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arccos \left({\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} ||\mathbf {b} |}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38ffb87b7a31695b67075fb3ff497900ecb5726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.683ex; height:6.343ex;" alt="{\displaystyle \theta =\arccos \left({\frac {\mathbf {a} \cdot \mathbf {b} }{|\mathbf {a} ||\mathbf {b} |}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vidjeti_također"><span id="Vidjeti_tako.C4.91er"></span>Vidjeti također</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&veaction=edit&section=3" title="Uredi odlomak: Vidjeti također" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Skalarni_umno%C5%BEak&action=edit&section=3" title="Uredi kôd odjeljka Vidjeti također"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Vektorski_umno%C5%BEak" class="mw-redirect" title="Vektorski umnožak">Vektorski umnožak</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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