Prodhimi skalar - Wikipedia

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subsection</span> </button> <ul id="toc-E_ç'është_prodhimi_skalar?-sublist" class="vector-toc-list"> <li id="toc-Përkufizimi_koordinativ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Përkufizimi_koordinativ"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Përkufizimi koordinativ</span> </div> </a> <ul id="toc-Përkufizimi_koordinativ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Përkufizimi_gjeometrik" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Përkufizimi_gjeometrik"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Përkufizimi gjeometrik</span> </div> </a> <ul id="toc-Përkufizimi_gjeometrik-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vetitë" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vetitë"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vetitë</span> </div> </a> <button aria-controls="toc-Vetitë-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>Toggle Vetitë subsection</span> </button> <ul id="toc-Vetitë-sublist" class="vector-toc-list"> <li id="toc-Zbatimi_në_ligjin_e_kosinusit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zbatimi_në_ligjin_e_kosinusit"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Zbatimi në ligjin e kosinusit</span> </div> </a> <ul id="toc-Zbatimi_në_ligjin_e_kosinusit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fizika" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fizika"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Fizika</span> </div> </a> <ul id="toc-Fizika-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Prodhimi skalar</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="Go to an article in another language. Available in 70 languages" > <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 gjuhë</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 – amarisht" lang="am" hreflang="am" data-title="ጥላ ብዜት" data-language-autonym="አማርኛ" data-language-local-name="amarisht" 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 – arabisht" lang="ar" hreflang="ar" data-title="جداء نقطي" data-language-autonym="العربية" data-language-local-name="arabisht" 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 – asturisht" lang="ast" hreflang="ast" data-title="Productu escalar" data-language-autonym="Asturianu" data-language-local-name="asturisht" 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 – azerbajxhanisht" lang="az" hreflang="az" data-title="Skalyar hasil" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajxhanisht" 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 – bashkirisht" lang="ba" hreflang="ba" data-title="Скаляр ҡабатландыҡ" data-language-autonym="Башҡортса" data-language-local-name="bashkirisht" 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 – bjellorusisht" lang="be" hreflang="be" data-title="Скалярны здабытак" data-language-autonym="Беларуская" data-language-local-name="bjellorusisht" 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 – bullgarisht" lang="bg" hreflang="bg" data-title="Скаларно произведение" data-language-autonym="Български" data-language-local-name="bullgarisht" 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 – bengalisht" lang="bn" hreflang="bn" data-title="ডট গুণন" data-language-autonym="বাংলা" data-language-local-name="bengalisht" 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 – boshnjakisht" lang="bs" hreflang="bs" data-title="Skalarni proizvod" data-language-autonym="Bosanski" data-language-local-name="boshnjakisht" 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 – katalonisht" lang="ca" hreflang="ca" data-title="Producte escalar" data-language-autonym="Català" data-language-local-name="katalonisht" 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 – kurdishte qendrore" lang="ckb" hreflang="ckb" data-title="لێکدانی ناوەکی" data-language-autonym="کوردی" data-language-local-name="kurdishte qendrore" 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 – çekisht" lang="cs" hreflang="cs" data-title="Skalární součin" data-language-autonym="Čeština" data-language-local-name="çekisht" 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 – çuvashisht" lang="cv" hreflang="cv" data-title="Скалярла хутлав" data-language-autonym="Чӑвашла" data-language-local-name="çuvashisht" 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 – danisht" lang="da" hreflang="da" data-title="Skalarprodukt" data-language-autonym="Dansk" data-language-local-name="danisht" 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 – gjermanisht" lang="de" hreflang="de" data-title="Skalarprodukt" data-language-autonym="Deutsch" data-language-local-name="gjermanisht" 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 – greqisht" lang="el" hreflang="el" data-title="Εσωτερικό γινόμενο" data-language-autonym="Ελληνικά" data-language-local-name="greqisht" 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 – anglisht" lang="en" hreflang="en" data-title="Dot product" data-language-autonym="English" data-language-local-name="anglisht" 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 – spanjisht" lang="es" hreflang="es" data-title="Producto escalar" data-language-autonym="Español" data-language-local-name="spanjisht" 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 – estonisht" lang="et" hreflang="et" data-title="Skalaarkorrutis" data-language-autonym="Eesti" data-language-local-name="estonisht" 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 – baskisht" lang="eu" hreflang="eu" data-title="Biderketa eskalar" data-language-autonym="Euskara" data-language-local-name="baskisht" 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 – persisht" lang="fa" hreflang="fa" data-title="ضرب داخلی" data-language-autonym="فارسی" data-language-local-name="persisht" 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 – finlandisht" lang="fi" hreflang="fi" data-title="Pistetulo" data-language-autonym="Suomi" data-language-local-name="finlandisht" 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 – frëngjisht" lang="fr" hreflang="fr" data-title="Produit scalaire" data-language-autonym="Français" data-language-local-name="frëngjisht" 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 – galicisht" lang="gl" hreflang="gl" data-title="Produto escalar" data-language-autonym="Galego" data-language-local-name="galicisht" 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 – hebraisht" lang="he" hreflang="he" data-title="מכפלה סקלרית" data-language-autonym="עברית" data-language-local-name="hebraisht" 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 – indisht" lang="hi" hreflang="hi" data-title="अदिश गुणनफल" data-language-autonym="हिन्दी" data-language-local-name="indisht" 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/Skalarni_umno%C5%BEak" title="Skalarni umnožak – kroatisht" lang="hr" hreflang="hr" data-title="Skalarni umnožak" data-language-autonym="Hrvatski" data-language-local-name="kroatisht" class="interlanguage-link-target"><span>Hrvatski</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 – hungarisht" lang="hu" hreflang="hu" data-title="Skaláris szorzat" data-language-autonym="Magyar" data-language-local-name="hungarisht" 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 – armenisht" lang="hy" hreflang="hy" data-title="Սկալյար արտադրյալ" data-language-autonym="Հայերեն" data-language-local-name="armenisht" 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 – indonezisht" lang="id" hreflang="id" data-title="Produk dot" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezisht" 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 – italisht" lang="it" hreflang="it" data-title="Prodotto scalare" data-language-autonym="Italiano" data-language-local-name="italisht" 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 – japonisht" lang="ja" hreflang="ja" data-title="ドット積" data-language-autonym="日本語" data-language-local-name="japonisht" 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 – gjeorgjisht" lang="ka" hreflang="ka" data-title="სკალარული ნამრავლი" data-language-autonym="ქართული" data-language-local-name="gjeorgjisht" 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 – kazakisht" lang="kk" hreflang="kk" data-title="Скаляр көбейтінді" data-language-autonym="Қазақша" data-language-local-name="kazakisht" 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 – koreanisht" lang="ko" hreflang="ko" data-title="스칼라곱" data-language-autonym="한국어" data-language-local-name="koreanisht" 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 – latinisht" lang="la" hreflang="la" data-title="Productum interius" data-language-autonym="Latina" data-language-local-name="latinisht" 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 – lituanisht" lang="lt" hreflang="lt" data-title="Skaliarinė sandauga" data-language-autonym="Lietuvių" data-language-local-name="lituanisht" 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 – letonisht" lang="lv" hreflang="lv" data-title="Skalārais reizinājums" data-language-autonym="Latviešu" data-language-local-name="letonisht" 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 – maqedonisht" lang="mk" hreflang="mk" data-title="Скаларен производ" data-language-autonym="Македонски" data-language-local-name="maqedonisht" 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 – maratisht" lang="mr" hreflang="mr" data-title="बिंदू गुणाकार" data-language-autonym="मराठी" data-language-local-name="maratisht" 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 – malajisht" lang="ms" hreflang="ms" data-title="Hasil darab bintik" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisht" 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 – holandisht" lang="nl" hreflang="nl" data-title="Inwendig product" data-language-autonym="Nederlands" data-language-local-name="holandisht" 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 – norvegjishte nynorsk" lang="nn" hreflang="nn" data-title="Indreprodukt" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegjishte 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 – norvegjishte letrare" lang="nb" hreflang="nb" data-title="Indreprodukt" data-language-autonym="Norsk bokmål" data-language-local-name="norvegjishte letrare" 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 – polonisht" lang="pl" hreflang="pl" data-title="Iloczyn skalarny" data-language-autonym="Polski" data-language-local-name="polonisht" 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 – portugalisht" lang="pt" hreflang="pt" data-title="Produto escalar" data-language-autonym="Português" data-language-local-name="portugalisht" 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 – rumanisht" lang="ro" hreflang="ro" data-title="Produs scalar" data-language-autonym="Română" data-language-local-name="rumanisht" 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 – rusisht" lang="ru" hreflang="ru" data-title="Скалярное произведение" data-language-autonym="Русский" data-language-local-name="rusisht" 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 – sakaisht" lang="sah" hreflang="sah" data-title="Скалярное произведение" data-language-autonym="Саха тыла" data-language-local-name="sakaisht" 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 – skotisht" lang="sco" hreflang="sco" data-title="Dot product" data-language-autonym="Scots" data-language-local-name="skotisht" 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 – serbo-kroatisht" lang="sh" hreflang="sh" data-title="Skalarni proizvod vektora" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-kroatisht" 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 – sllovakisht" lang="sk" hreflang="sk" data-title="Skalárny súčin" data-language-autonym="Slovenčina" data-language-local-name="sllovakisht" 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 – sllovenisht" lang="sl" hreflang="sl" data-title="Skalarni produkt" data-language-autonym="Slovenščina" data-language-local-name="sllovenisht" class="interlanguage-link-target"><span>Slovenščina</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 – serbisht" lang="sr" hreflang="sr" data-title="Скаларни производ вектора" data-language-autonym="Српски / srpski" data-language-local-name="serbisht" 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 – suedisht" lang="sv" hreflang="sv" data-title="Skalärprodukt" data-language-autonym="Svenska" data-language-local-name="suedisht" 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 – tamilisht" lang="ta" hreflang="ta" data-title="புள்ளிப் பெருக்கல்" data-language-autonym="தமிழ்" data-language-local-name="tamilisht" 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 – tajlandisht" lang="th" hreflang="th" data-title="ผลคูณจุด" data-language-autonym="ไทย" data-language-local-name="tajlandisht" 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 – turqisht" lang="tr" hreflang="tr" data-title="Nokta çarpım" data-language-autonym="Türkçe" data-language-local-name="turqisht" 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 – tatarisht" lang="tt" hreflang="tt" data-title="Скаляр тапкырчыгыш" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarisht" 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 – ukrainisht" lang="uk" hreflang="uk" data-title="Скалярний добуток" data-language-autonym="Українська" data-language-local-name="ukrainisht" 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 – urduisht" lang="ur" hreflang="ur" data-title="ڈوٹ پروڈکٹ" data-language-autonym="اردو" data-language-local-name="urduisht" 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 – uzbekisht" lang="uz" hreflang="uz" data-title="Skalyar koʻpaytmasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekisht" 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 – vietnamisht" lang="vi" hreflang="vi" data-title="Tích vô hướng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamisht" 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 – kinezishte vu" lang="wuu" hreflang="wuu" data-title="数量积" data-language-autonym="吴语" data-language-local-name="kinezishte vu" 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 – kinezisht" lang="zh" hreflang="zh" data-title="点积" data-language-autonym="中文" data-language-local-name="kinezisht" 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 – kantonezisht" lang="yue" hreflang="yue" data-title="點積" data-language-autonym="粵語" data-language-local-name="kantonezisht" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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Në <a href="/wiki/Gjeometria_Euklidiane" class="mw-redirect" title="Gjeometria Euklidiane">gjeometrinë Euklidiane</a>, prodhimi skalar i <a href="/wiki/Sistemi_koordinativ_kartezian" title="Sistemi koordinativ kartezian">koordinatave karteziane</a> të dy <a href="/wiki/Vektori" title="Vektori">vektorëve</a> përdoret gjerësisht. Shpesh quhet <b>produkt i brendshëm</b> (ose rrallë <b>produkt i projeksionit</b> ) i hapësirës Euklidiane, edhe pse nuk është i vetmi prodhim i brendshëm që mund të përcaktohet në hapësirën Euklidiane (shih hapësirën e brendshme të prodhimit për më shumë). </p><p>Nga ana algjebrike, prodhimi me pikë është shuma e produkteve të hyrjeve përkatëse të dy vargjeve të numrave. Gjeometrikisht, është prodhim i <a href="/wiki/Vektori" title="Vektori">madhësive Euklidiane</a> të dy vektorëve dhe <a href="/wiki/Funksionet_trigonometrike" title="Funksionet trigonometrike">kosinusit</a> të këndit ndërmjet tyre. Këto përkufizime janë të njëvlershme kur përdoren koordinatat karteziane. Në <a href="/wiki/Gjeometria" title="Gjeometria">gjeometrinë</a> moderne, hapësirat Euklidiane shpesh përcaktohen duke përdorur <a href="/wiki/Hap%C3%ABsira_vektoriale" title="Hapësira vektoriale">hapësira vektoriale</a> . Në këtë rast, prodhimi me pikë përdoret për përcaktimin e gjatësive (gjatësia e një vektori është <a href="/w/index.php?title=Rrenja_katrore&action=edit&redlink=1" class="new" title="Rrenja katrore (nuk është shkruar akoma)">rrënja katrore</a> e prodhimit me pikë të vektorit me veten) dhe këndet (kosinusi i këndit midis dy vektorëve është herësi i prodhimit të tyre me pikë me nga prodhimin e gjatësisë së tyre). </p><p>Emri "prodhim me pikë" rrjedh nga pika e përqendruar " <b>·</b> " që përdoret shpesh për të përcaktuar këtë veprim; <sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> emri alternativ "prodhim skalar" thekson se rezultati është një <a href="/wiki/Skalar_(matematik%C3%AB)" title="Skalar (matematikë)">skalar</a>, në vend të një vektor (si me produktin vektorial në hapësirën tre-dimensionale). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="E_ç'është_prodhimi_skalar?"><span id="E_.C3.A7.27.C3.ABsht.C3.AB_prodhimi_skalar.3F"></span>E ç'është prodhimi skalar?</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=1" title="Redakto pjesën: E ç'është prodhimi skalar?" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=1" title="Edit section's source code: E ç'është prodhimi skalar?"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prodhimi me pikë mund të përcaktohet në mënyrë algjebrike ose gjeometrike. Përkufizimi gjeometrik bazohet në nocionet e këndit dhe largësisë (madhësia) mes vektorëve. Njëvlershmëria e këtyre dy përkufizimeve mbështetet në të paturit e një <a href="/wiki/Sistemi_koordinativ_kartezian" title="Sistemi koordinativ kartezian">sistemi koordinativ kartezian</a> për hapësirën Euklidiane. </p> <div class="mw-heading mw-heading3"><h3 id="Përkufizimi_koordinativ"><span id="P.C3.ABrkufizimi_koordinativ"></span>Përkufizimi koordinativ</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=2" title="Redakto pjesën: Përkufizimi koordinativ" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=2" title="Edit section's source code: Përkufizimi koordinativ"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prodhimi skalar i dy vektorëve <span class="mwe-math-element"><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},\cdots ,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> <msub> <mi>a</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> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =[a_{1},a_{2},\cdots ,a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5284f6fd0c1181f08a22db25e7a51668b0621db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.92ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =[a_{1},a_{2},\cdots ,a_{n}]}"></span> dhe <span class="nowrap"><span class="mwe-math-element"><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},b_{2},\cdots ,b_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <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> <mo>⋯</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =[b_{1},b_{2},\cdots ,b_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6049394efd0b0a5bedeafa4fcbd0fd35a4d8846f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.409ex; height:2.843ex;" alt="{\displaystyle \mathbf {b} =[b_{1},b_{2},\cdots ,b_{n}]}"></span>,</span> i specifikuar në lidhje me një <a href="/w/index.php?title=Baza_ortonormale&action=edit&redlink=1" class="new" title="Baza ortonormale (nuk është shkruar akoma)">bazë ortonormale</a>, përkufizohet si: <sup id="cite_ref-Lipschutz2009_3-0" class="reference"><a href="#cite_note-Lipschutz2009-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {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-display 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>ku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> tregon shumën dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> është dimensioni i hapësirës vektoriale . Për shembull, në hapësirën tre-dimensionale, produkti me pika i vektorëve <span class="nowrap"><span class="mwe-math-element"><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,-5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,3,-5]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34361be3217025716bc493edaf428109cdde996a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.657ex; height:2.843ex;" alt="{\displaystyle [1,3,-5]}"></span></span> dhe <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [4,-2,-1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [4,-2,-1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa56bf9b7ea1fc8fdb00b036c4c246b5f653a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.465ex; height:2.843ex;" alt="{\displaystyle [4,-2,-1]}"></span></span> është:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mtext> </mtext> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>×</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f1f0d7669d35eb1220c3256ea458319c80f713" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:57.261ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}"></span>Po kështu, prodhimi skalar i vektorit <span class="nowrap"><span class="mwe-math-element"><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,-5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,3,-5]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34361be3217025716bc493edaf428109cdde996a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.657ex; height:2.843ex;" alt="{\displaystyle [1,3,-5]}"></span></span> me vetveten është:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mtext> </mtext> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>×</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>9</mn> <mo>+</mo> <mn>25</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>35</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1e6ff09018948273e2f6375b7d0c6196ee1c23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:53.645ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}"></span>Nëse vektorët identifikohen me vektorët kolonë, prodhimi me pikë mund të shkruhet gjithashtu si prodhim matricor<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730b0c5d8ee397842e852cc1526c840b5cadebf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.345ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathsf {T}}\mathbf {b} ,}"></span>ku <span class="mwe-math-element"><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{^{\mathsf {T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{^{\mathsf {T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e64e782571e10a0e7cfa9082af85f58f3455a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle a{^{\mathsf {T}}}}"></span> tregon transpozimin e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span> . </p> <div class="mw-heading mw-heading3"><h3 id="Përkufizimi_gjeometrik"><span id="P.C3.ABrkufizimi_gjeometrik"></span>Përkufizimi gjeometrik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=3" title="Redakto pjesën: Përkufizimi gjeometrik" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=3" title="Edit section's source code: Përkufizimi gjeometrik"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Skeda:Inner-product-angle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Inner-product-angle.svg/220px-Inner-product-angle.svg.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Inner-product-angle.svg/330px-Inner-product-angle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Inner-product-angle.svg/440px-Inner-product-angle.svg.png 2x" data-file-width="385" data-file-height="287" /></a><figcaption> Ilustrim që tregon se si të gjendet këndi midis vektorëve duke përdorur prodjhimin skalar</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Skeda:Tetrahedral_angle_calculation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Tetrahedral_angle_calculation.svg/216px-Tetrahedral_angle_calculation.svg.png" decoding="async" width="216" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Tetrahedral_angle_calculation.svg/324px-Tetrahedral_angle_calculation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Tetrahedral_angle_calculation.svg/432px-Tetrahedral_angle_calculation.svg.png 2x" data-file-width="512" data-file-height="304" /></a><figcaption> Llogaritja e këndeve të lidhjes së një gjeometrie molekulare simetrike tetraedrale duke përdorur një prodhim skalar</figcaption></figure> <p>Në hapësirën Euklidiane, një <a href="/wiki/Vektori" title="Vektori">vektor Euklidian</a> është një objekt gjeometrik që zotëron një madhësi dhe një drejtim. Një vektor mund të paraqitet si një shigjetë. Madhësia e tij është gjatësia e tij, dhe drejtimi i tij është drejtimi në të cilin tregon shigjeta. Madhësia e një vektori <span class="mwe-math-element"><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> shënohet me <span class="mwe-math-element"><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\|}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {a} \right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec49df2fa8265066b59f02d363bbc490ba023c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.624ex; height:2.843ex;" alt="{\displaystyle \left\|\mathbf {a} \right\|}"></span> . Prodhimi skalar i dy vektorëve Euklidianë <span class="mwe-math-element"><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> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> është përcaktuar nga <sup id="cite_ref-Spiegel2009_4-0" class="reference"><a href="#cite_note-Spiegel2009-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {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> <mo>,</mo> </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/2ed1f590c477f4f86793ed25a3f20c3633f742ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.007ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,}"></span>ku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> është <a href="/wiki/K%C3%ABndi" title="Këndi">këndi</a> ndërmjet <span class="mwe-math-element"><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> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> . </p><p>Në veçanti, nëse vektorët <span class="mwe-math-element"><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> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> janë <a href="/w/index.php?title=Ortogonaliteti&action=edit&redlink=1" class="new" title="Ortogonaliteti (nuk është shkruar akoma)">ortogonalë</a> (d.m.th., këndi i tyre është <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"></span> ose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span> ), pastaj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\pi }{2}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\pi }{2}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a42c21d362dc99b3986486f963a3cce908269d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.927ex; height:4.676ex;" alt="{\displaystyle \cos {\frac {\pi }{2}}=0}"></span>, që nënkupton se<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe01527f19f9dee4c44eb76180ed04cfae7b02a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.372ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0.}"></span>Në skajin tjetër, nëse ata janë të njëanshëm, atëherë këndi ndërmjet tyre është zero me <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 0=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9946942d786042c531c821bec01e8bf09f8bab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.922ex; height:2.176ex;" alt="{\displaystyle \cos 0=1}"></span> dhe<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}"> <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> <mspace width="thinmathspace" /> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d59f442438a6f8c8af9a0882b57350d991b6f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.771ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|}"></span>Kjo nënkupton që produkti me pikë i një vektori <span class="mwe-math-element"><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> me vetveten është<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},}"> <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">a</mi> </mrow> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d670295f64b1f8de1a98aa2782590aa20a64415" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.702ex; height:3.343ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {a} =\left\|\mathbf {a} \right\|^{2},}"></span>që jep<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mo>,</mo> </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/3dbaf4a9a824c63de38a24d569fd58d8286c9c29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.584ex; height:3.009ex;" alt="{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }},}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Vetitë"><span id="Vetit.C3.AB"></span>Vetitë</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=4" title="Redakto pjesën: Vetitë" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=4" title="Edit section's source code: Vetitë"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prodhimi skalar plotëson vetitë e mëposhtme nëse <span class="mwe-math-element"><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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span>, dhe <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {c} }"></span> janë <a href="/wiki/Vektori" title="Vektori">vektorë</a> realë dhe <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></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 c_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7dc6d279091d354e0b90889b463bfa7eb7247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}"></span> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b30ba1b247fb8d334580cec68561e749d24aff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{2}}"></span> janë skalarë . <sup id="cite_ref-Lipschutz2009_3-1" class="reference"><a href="#cite_note-Lipschutz2009-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-Spiegel2009_4-1" class="reference"><a href="#cite_note-Spiegel2009-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dt><a href="/wiki/Vetia_e_nd%C3%ABrrimit" title="Vetia e ndërrimit">Ndërruese</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23da9e9ff4be4e3c6abcc7b7678f63383423a47d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.673ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} ,}"></span></dd> <dt>Shpërndarëse në lidhje me mbledhjen e vektorëve</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <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">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f483a3722bee6d25aaee76359d3c80a15898086" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.518ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}"></span></dd> <dt>Bilineare</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (r\mathbf {b} +\mathbf {c} )=r(\mathbf {a} \cdot \mathbf {b} )+(\mathbf {a} \cdot \mathbf {c} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>r</mi> <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> <mi>r</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (r\mathbf {b} +\mathbf {c} )=r(\mathbf {a} \cdot \mathbf {b} )+(\mathbf {a} \cdot \mathbf {c} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f53cbee56949f939497c8b5ed0ceb9a2f25912" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.234ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (r\mathbf {b} +\mathbf {c} )=r(\mathbf {a} \cdot \mathbf {b} )+(\mathbf {a} \cdot \mathbf {c} ).}"></span></dd> <dt>Shumëzimin skalar</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{1}\mathbf {a} )\cdot (c_{2}\mathbf {b} )=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{1}\mathbf {a} )\cdot (c_{2}\mathbf {b} )=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d167dd8c90f29a915bf474aa78c34fd29ab75f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.345ex; height:2.843ex;" alt="{\displaystyle (c_{1}\mathbf {a} )\cdot (c_{2}\mathbf {b} )=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} ).}"></span></dd> <dt>Jo <a href="/wiki/Vetia_e_shoq%C3%ABrimit" title="Vetia e shoqërimit">shoqëruese</a></dt> <dd>sepse prodhimi me pikë ndërmjet një skalari <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494aed3b5e94f1c0ee071debc707d2700c0e0390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.464ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} }"></span> dhe një vektori <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {c} }"></span> nuk është i përcaktuar, që do të thotë se shprehjet e përfshira në vetinë e shoqërimit, <span class="mwe-math-element"><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} )\cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8d72072d1f2a19495a141c6ce1fd560722977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.14ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }"></span> ose <span class="mwe-math-element"><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} \cdot \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <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} \cdot \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1baea8c615db765ff89591b667ae571aee5aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.14ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}"></span> janë të dyja të keqpërcaktuara. <sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Sidoqoftë, vini re se vetia e shumëzimit skalar e përmendur më parë ndonjëherë quhet "ligji shoqërues për prodhimin skalar dhe atë me pikë" <sup id="cite_ref-BanchoffWermer1983_7-0" class="reference"><a href="#cite_note-BanchoffWermer1983-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> ose mund të thuhet se "produkti me pikë është shoqërues në lidhje me shumëzimin skalar" sepse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\mathbf {a} \cdot \mathbf {b} )=(c\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (c\mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\mathbf {a} \cdot \mathbf {b} )=(c\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (c\mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a98791f396522497bd10bc4661af5d881f62fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.037ex; height:2.843ex;" alt="{\displaystyle c(\mathbf {a} \cdot \mathbf {b} )=(c\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (c\mathbf {b} )}"></span> . <sup id="cite_ref-BedfordFowler2008_8-0" class="reference"><a href="#cite_note-BedfordFowler2008-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd> <dt>Ortogonale</dt> <dd>Dy vektorë jo zero <span class="mwe-math-element"><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> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> janë <i>ortogonalë</i> atëherë dhe vetëm atëherë kur <span class="mwe-math-element"><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} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c416b33910828e0941fec78eec1170c79e7ca146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.725ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}"></span> .</dd> <dt>Asnjë anulim</dt> <dd>Ndryshe nga shumëzimi i numrave të zakonshëm, ku nëse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab=ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66bc658be9d643119e9b7fc0df7f0bbe37b99d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.562ex; height:2.176ex;" alt="{\displaystyle ab=ac}"></span>, pastaj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> gjithmonë të barabartë <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> përveç nëse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> është zero, produkti me pikë nuk i bindet ligjit të anulimit :<div class="paragraphbreak" style="margin-top:0.5em"></div> Nëse <span class="mwe-math-element"><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} \cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ef6fab228b0e35c52d1c80ed024fdfe4bc25f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.729ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }"></span> dhe <span class="mwe-math-element"><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} \neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbd86f21598eb5c1c24152737ed1c22835368e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.735ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \neq \mathbf {0} }"></span>, atëherë mund të shkruajmë: <span class="mwe-math-element"><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 {c} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} -\mathbf {c} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc4c343455359751e77d16ff1efa7737c8915d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.563ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} -\mathbf {c} )=0}"></span> sipas ligjit shpërndarës ; rezultati i mësipërm thotë se kjo do të thotë vetëm se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> është pingul me <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {b} -\mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {b} -\mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2438332cf0cef7a424101c79a241b8d797f3de9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.323ex; height:2.843ex;" alt="{\displaystyle (\mathbf {b} -\mathbf {c} )}"></span>, e cila ende lejon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {b} -\mathbf {c} )\neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {b} -\mathbf {c} )\neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6043c901bf44ab6e0078dcd8119e5432629b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.758ex; height:2.843ex;" alt="{\displaystyle (\mathbf {b} -\mathbf {c} )\neq \mathbf {0} }"></span>, dhe për këtë arsye lejon <span class="mwe-math-element"><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} \neq \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \neq \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd7fe932f086de06c1b134c3a43790c215aaa2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.772ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} \neq \mathbf {c} }"></span> .</dd> <dt>Rregulli i prodhimit</dt> <dd>Nëse <span class="mwe-math-element"><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> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ebf4628a1adf07133a6009e4a78bdd990c6eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }"></span> janë funksione të diferencueshme me vlerë vektoriale, pastaj derivati ( i shënuar me një të thjeshtë <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea6544b21c7df37230227c33a42eda9aa3f078e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.685ex; height:2.343ex;" alt="{\displaystyle {}'}"></span> ) të <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494aed3b5e94f1c0ee071debc707d2700c0e0390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.464ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} }"></span> jepet nga rregulli <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a} \cdot \mathbf {b} )'=\mathbf {a} '\cdot \mathbf {b} +\mathbf {a} \cdot \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> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <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> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \cdot \mathbf {b} )'=\mathbf {a} '\cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {b} '.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef7a8a0489f4844895df1ade3e9f02810600070" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.841ex; height:3.009ex;" alt="{\displaystyle (\mathbf {a} \cdot \mathbf {b} )'=\mathbf {a} '\cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {b} '.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Zbatimi_në_ligjin_e_kosinusit"><span id="Zbatimi_n.C3.AB_ligjin_e_kosinusit"></span>Zbatimi në ligjin e kosinusit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=5" title="Redakto pjesën: Zbatimi në ligjin e kosinusit" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=5" title="Edit section's source code: Zbatimi në ligjin e kosinusit"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Skeda:Dot_product_cosine_rule.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/100px-Dot_product_cosine_rule.svg.png" decoding="async" width="100" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/150px-Dot_product_cosine_rule.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/199px-Dot_product_cosine_rule.svg.png 2x" data-file-width="106" data-file-height="135" /></a><figcaption> Trekëndësh me skajet vektoriale <b>a</b> dhe <b>b</b>, të ndara me kënd <i>θ</i> .</figcaption></figure> <p>Jepen dy vektorë <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {red}\mathbf {a} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {red}\mathbf {a} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5791a6b67777df8db7d6f2508967d7ebb6ed07" 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 {\color {red}\mathbf {a} }}"></span> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {blue}\mathbf {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {blue}\mathbf {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b392ac93e227c2eae68b47eb927547fa6c4812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\color {blue}\mathbf {b} }}"></span> të ndara sipas këndit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> (shih imazhin djathtas), ato formojnë një trekëndësh me një anë të tretë <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="orange"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcd6288734559d216bf776b403d47dd87b5cc8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.912ex; height:2.343ex;" alt="{\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }}"></span> . Le <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> tregojnë gjatësitë e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {red}\mathbf {a} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {red}\mathbf {a} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5791a6b67777df8db7d6f2508967d7ebb6ed07" 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 {\color {red}\mathbf {a} }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {blue}\mathbf {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {blue}\mathbf {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b392ac93e227c2eae68b47eb927547fa6c4812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\color {blue}\mathbf {b} }}"></span>, dhe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {orange}\mathbf {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="orange"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {orange}\mathbf {c} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d928cc0bbfa8ee90f5f5fa49debc12705e5581c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle {\color {orange}\mathbf {c} }}"></span>, respektivisht. Produkti me pika i kësaj me vetveten është:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {\color {orange}c} \cdot \mathbf {\color {orange}c} &=(\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\cdot (\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\\&=\mathbf {\color {red}a} \cdot \mathbf {\color {red}a} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {blue}b} \cdot \mathbf {\color {red}a} +\mathbf {\color {blue}b} \cdot \mathbf {\color {blue}b} \\&={\color {red}a}^{2}-\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\&={\color {red}a}^{2}-2\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\{\color {orange}c}^{2}&={\color {red}a}^{2}+{\color {blue}b}^{2}-2{\color {red}a}{\color {blue}b}\cos \mathbf {\color {purple}\theta } \\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="orange"> <mi mathvariant="bold">c</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="orange"> <mi mathvariant="bold">c</mi> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>a</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>b</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>a</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi mathvariant="bold">a</mi> </mstyle> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi mathvariant="bold">b</mi> </mstyle> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>b</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="orange"> <mi>c</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>a</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>b</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mi>a</mi> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#0000ff"> <mi>b</mi> </mstyle> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="purple"> <mi>θ</mi> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {\color {orange}c} \cdot \mathbf {\color {orange}c} &=(\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\cdot (\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\\&=\mathbf {\color {red}a} \cdot \mathbf {\color {red}a} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {blue}b} \cdot \mathbf {\color {red}a} +\mathbf {\color {blue}b} \cdot \mathbf {\color {blue}b} \\&={\color {red}a}^{2}-\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\&={\color {red}a}^{2}-2\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\{\color {orange}c}^{2}&={\color {red}a}^{2}+{\color {blue}b}^{2}-2{\color {red}a}{\color {blue}b}\cos \mathbf {\color {purple}\theta } \\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c729be2c3b2d9881787b32492412a0322b217ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:34.282ex; height:15.676ex;" alt="{\displaystyle {\begin{aligned}\mathbf {\color {orange}c} \cdot \mathbf {\color {orange}c} &=(\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\cdot (\mathbf {\color {red}a} -\mathbf {\color {blue}b} )\\&=\mathbf {\color {red}a} \cdot \mathbf {\color {red}a} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {blue}b} \cdot \mathbf {\color {red}a} +\mathbf {\color {blue}b} \cdot \mathbf {\color {blue}b} \\&={\color {red}a}^{2}-\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} -\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\&={\color {red}a}^{2}-2\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} +{\color {blue}b}^{2}\\{\color {orange}c}^{2}&={\color {red}a}^{2}+{\color {blue}b}^{2}-2{\color {red}a}{\color {blue}b}\cos \mathbf {\color {purple}\theta } \\\end{aligned}}}"></span>i cili është <a href="/wiki/Teorema_e_kosinusit" title="Teorema e kosinusit">ligji i kosinusit</a> . </p> <div class="mw-heading mw-heading2"><h2 id="Fizika">Fizika</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prodhimi_skalar&veaction=edit&section=6" title="Redakto pjesën: Fizika" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Prodhimi_skalar&action=edit&section=6" title="Edit section's source code: Fizika"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Në <a href="/wiki/Fizika" title="Fizika">fizikë</a>, madhësia vektoriale është një skalar në kuptimin fizik (dmth., një madhësi fizike e pavarur nga sistemi i koordinatave), e shprehur si prodhim i një <a href="/wiki/Numri" title="Numri">vlere numerike</a> dhe një <a href="/wiki/Nj%C3%ABsia_mat%C3%ABse" title="Njësia matëse">njësie fizike</a>, jo thjesht një numër. Prodhimi me pikë është gjithashtu një skalar në këtë kuptim, i dhënë nga formula, i pavarur nga sistemi i koordinatave. Për shembull: <sup id="cite_ref-Riley2010_9-0" class="reference"><a href="#cite_note-Riley2010-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Puna_(fizik%C3%AB)" title="Puna (fizikë)">Puna mekanike</a> është prodhimi me pikë i vektorëve të <a href="/wiki/Forca" title="Forca">forcës</a> dhe <a href="/wiki/Zhvendosja_(fizik%C3%AB)" title="Zhvendosja (fizikë)">zhvendosjes</a> ,</li> <li><a href="/wiki/Fuqia_(fizik%C3%AB)" title="Fuqia (fizikë)">Fuqia</a> është prodhimi me pikë i <a href="/wiki/Forca" title="Forca">forcës</a> dhe <a href="/wiki/Shpejt%C3%ABsia_(fizik%C3%AB)" title="Shpejtësia (fizikë)">shpejtësisë</a> .</li></ul> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <span class="reference-text">The term <i>scalar product</i> means literally "product with a <a href="/w/index.php?title=Scalar_(mathematics)&action=edit&redlink=1" class="new" title="Scalar (mathematics) (nuk është shkruar akoma)">scalar</a> as a result". It is also used sometimes for other <a href="/w/index.php?title=Symmetric_bilinear_form&action=edit&redlink=1" class="new" title="Symmetric bilinear form (nuk është shkruar akoma)">symmetric bilinear forms</a>, for example in a <a href="/w/index.php?title=Pseudo-Euclidean_space&action=edit&redlink=1" class="new" title="Pseudo-Euclidean space (nuk është shkruar akoma)">pseudo-Euclidean space</a>.</span> </li> </ol> <ol class="references"> <li id="cite_note-:1-2">^ <a href="#cite_ref-:1_2-0"><sup>a</sup></a> <a href="#cite_ref-:1_2-1"><sup>b</sup></a> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r2706204">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/vectors-dot-product.html">"Dot Product"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Marrë më <span class="nowrap">2020-09-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Dot+Product&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Falgebra%2Fvectors-dot-product.html&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_web" title="Stampa:Cite web">cite web</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span> <span class="error mw-ext-cite-error" lang="sq" dir="ltr">Gabim referencash: Invalid <code><ref></code> tag; name ":1" defined multiple times with different content</span></span> </li> <li id="cite_note-Lipschutz2009-3">^ <a href="#cite_ref-Lipschutz2009_3-0"><sup>a</sup></a> <a href="#cite_ref-Lipschutz2009_3-1"><sup>b</sup></a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFS._LipschutzM._Lipson2009" class="citation book cs1">S. Lipschutz; M. Lipson (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebra0000lips_a2h3"><i>Linear Algebra (Schaum's Outlines)</i></a> (bot. 4th). McGraw Hill. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-0-07-154352-1" title="Speciale:BurimetELibrave/978-0-07-154352-1"><bdi>978-0-07-154352-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+%28Schaum%27s+Outlines%29&rft.edition=4th&rft.pub=McGraw+Hill&rft.date=2009&rft.isbn=978-0-07-154352-1&rft.au=S.+Lipschutz&rft.au=M.+Lipson&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearalgebra0000lips_a2h3&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span> <span class="error mw-ext-cite-error" lang="sq" dir="ltr">Gabim referencash: Invalid <code><ref></code> tag; name "Lipschutz2009" defined multiple times with different content</span></span> </li> <li id="cite_note-Spiegel2009-4">^ <a href="#cite_ref-Spiegel2009_4-0"><sup>a</sup></a> <a href="#cite_ref-Spiegel2009_4-1"><sup>b</sup></a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFM.R._SpiegelS._LipschutzD._Spellman2009" class="citation book cs1">M.R. Spiegel; S. Lipschutz; D. Spellman (2009). <i>Vector Analysis (Schaum's Outlines)</i> (bot. 2nd). McGraw Hill. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-0-07-161545-7" title="Speciale:BurimetELibrave/978-0-07-161545-7"><bdi>978-0-07-161545-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Analysis+%28Schaum%27s+Outlines%29&rft.edition=2nd&rft.pub=McGraw+Hill&rft.date=2009&rft.isbn=978-0-07-161545-7&rft.au=M.R.+Spiegel&rft.au=S.+Lipschutz&rft.au=D.+Spellman&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span> <span class="error mw-ext-cite-error" lang="sq" dir="ltr">Gabim referencash: Invalid <code><ref></code> tag; name "Spiegel2009" defined multiple times with different content</span></span> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFA_I_BorisenkoI_E_Taparov1968" class="citation book cs1">A I Borisenko; I E Taparov (1968). <i>Vector and tensor analysis with applications</i>. Përkthyer nga Richard Silverman. Dover. fq. 14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+and+tensor+analysis+with+applications&rft.pages=14&rft.pub=Dover&rft.date=1968&rft.au=A+I+Borisenko&rft.au=I+E+Taparov&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span></span> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <span class="reference-text">Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/DotProduct.html">http://mathworld.wolfram.com/DotProduct.html</a></span> </li> <li id="cite_note-BanchoffWermer1983-7"><a href="#cite_ref-BanchoffWermer1983_7-0">^</a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFT._BanchoffJ._Wermer1983" class="citation book cs1">T. Banchoff; J. Wermer (1983). <a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrathr00banc_0/page/12/mode/2up"><i>Linear Algebra Through Geometry</i></a>. Springer Science & Business Media. fq. 12. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-1-4684-0161-5" title="Speciale:BurimetELibrave/978-1-4684-0161-5"><bdi>978-1-4684-0161-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Through+Geometry&rft.pages=12&rft.pub=Springer+Science+%26+Business+Media&rft.date=1983&rft.isbn=978-1-4684-0161-5&rft.au=T.+Banchoff&rft.au=J.+Wermer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearalgebrathr00banc_0%2Fpage%2F12%2Fmode%2F2up&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span></span> </li> <li id="cite_note-BedfordFowler2008-8"><a href="#cite_ref-BedfordFowler2008_8-0">^</a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFA._BedfordWallace_L._Fowler2008" class="citation book cs1">A. Bedford; Wallace L. Fowler (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/engineeringmecha0000bedf_h4p8"><i>Engineering Mechanics: Statics</i></a> (bot. 5th). Prentice Hall. fq. <a rel="nofollow" class="external text" href="https://archive.org/details/engineeringmecha0000bedf_h4p8/page/60">60</a>. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-0-13-612915-8" title="Speciale:BurimetELibrave/978-0-13-612915-8"><bdi>978-0-13-612915-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Statics&rft.pages=60&rft.edition=5th&rft.pub=Prentice+Hall&rft.date=2008&rft.isbn=978-0-13-612915-8&rft.au=A.+Bedford&rft.au=Wallace+L.+Fowler&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fengineeringmecha0000bedf_h4p8&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span></span> </li> <li id="cite_note-Riley2010-9"><a href="#cite_ref-Riley2010_9-0">^</a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFK.F._RileyM.P._HobsonS.J._Bence2010" class="citation book cs1">K.F. Riley; M.P. Hobson; S.J. Bence (2010). <span class="cs1-lock-registration" title="Kërkohet regjistrim"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth00rile"><i>Mathematical methods for physics and engineering</i></a></span> (bot. 3rd). Cambridge University Press. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-0-521-86153-3" title="Speciale:BurimetELibrave/978-0-521-86153-3"><bdi>978-0-521-86153-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+methods+for+physics+and+engineering&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-86153-3&rft.au=K.F.+Riley&rft.au=M.P.+Hobson&rft.au=S.J.+Bence&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalmeth00rile&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span></span> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2706204"><cite id="CITEREFM._MansfieldC._O'Sullivan2011" class="citation book cs1">M. Mansfield; C. O'Sullivan (2011). <i>Understanding Physics</i> (bot. 4th). John Wiley & Sons. <a href="https://en.wikipedia.org/wiki/en:ISBN_(identifier)" class="extiw" title="w:en:ISBN (identifier)">ISBN</a> <a href="/wiki/Speciale:BurimetELibrave/978-0-47-0746370" title="Speciale:BurimetELibrave/978-0-47-0746370"><bdi>978-0-47-0746370</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Physics&rft.edition=4th&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-0-47-0746370&rft.au=M.+Mansfield&rft.au=C.+O%27Sullivan&rfr_id=info%3Asid%2Fsq.wikipedia.org%3AProdhimi+skalar" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Stampa:Cite_book" title="Stampa:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Mungon ose është bosh parametri <code class="cs1-code">|language=</code> (<a href="/wiki/Ndihm%C3%AB:Gabimet_CS1#language_missing" title="Ndihmë:Gabimet CS1">Ndihmë!</a>)</span></span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Marrë nga "<a dir="ltr" href="https://sq.wikipedia.org/w/index.php?title=Prodhimi_skalar&oldid=2603130">https://sq.wikipedia.org/w/index.php?title=Prodhimi_skalar&oldid=2603130</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Speciale:Kategorit%C3%AB" title="Speciale:Kategoritë">Kategoritë</a>: <ul><li><a href="/wiki/Kategoria:Gabime_CS1:_Mungon_parametri_i_gjuh%C3%ABs" title="Kategoria:Gabime CS1: Mungon parametri i gjuhës">Gabime CS1: Mungon parametri i gjuhës</a></li><li><a href="/wiki/Kategoria:Matematik%C3%AB" title="Kategoria:Matematikë">Matematikë</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kategori të fshehura: <ul><li><a href="/wiki/Kategoria:Pages_with_reference_errors" title="Kategoria:Pages with reference errors">Pages with reference errors</a></li><li><a href="/wiki/Kategoria:Faqe_me_p%C3%ABrkthime_t%C3%AB_pashqyrtuara" title="Kategoria:Faqe me përkthime të pashqyrtuara">Faqe me përkthime të pashqyrtuara</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Kjo faqe është redaktuar për herë te fundit më 27 tetor 2023, në orën 20:26.</li> <li id="footer-info-copyright">Të gjitha materialet që gjenden në këtë faqë janë të mbrojtura nga <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike License</a>;. 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