त्रिक गुणनफल - विकिपीडिया

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src="/static/images/mobile/copyright/wikipedia-tagline-hi.svg" width="108" height="18" style="width: 6.75em; height: 1.125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A5%8B%E0%A4%9C" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="इस विकि में खोजें [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>खोजें</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> 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<div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="दिखावट"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="पृष्ठ के फ़ॉन्ट का आकार, चौड़ाई और रंग बदलें" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="दिखावट" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">दिखावट</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi" class=""><span>दान करें</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95+%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है" class=""><span>खाता बनाएँ</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95+%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o" class=""><span>लॉग-इन करें</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="अन्य विकल्प" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="व्यक्तिगत उपकरण" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">व्यक्तिगत उपकरण</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hi.wikipedia.org&uselang=hi"><span>दान करें</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%96%E0%A4%BE%E0%A4%A4%E0%A4%BE_%E0%A4%AC%E0%A4%A8%E0%A4%BE%E0%A4%8F%E0%A4%81&returnto=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95+%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="हमारा सुझाव है की आप खाता बनाएँ और लॉगिन करें, परन्तु यह अनिवार्य नहीं है"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>खाता बनाएँ</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A4%BF%E0%A4%A8&returnto=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95+%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="आपको सत्रारम्भ करने के लिए प्रोत्साहित किया जाता है; लेकिन यह अनिवार्य नहीं है [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>लॉग-इन करें</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> लॉग-आउट किए गए संपादकों के लिए पृष्ठ <a href="/wiki/%E0%A4%B8%E0%A4%B9%E0%A4%BE%E0%A4%AF%E0%A4%A4%E0%A4%BE:%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%9A%E0%A4%AF" aria-label="संपादन के बारे में अधिक जानें"><span>अधिक जानें</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%87_%E0%A4%AF%E0%A5%8B%E0%A4%97%E0%A4%A6%E0%A4%BE%E0%A4%A8" title="इस आईपी पते से संपादन की सूची [y]" accesskey="y"><span>योगदान</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AE%E0%A5%87%E0%A4%B0%E0%A5%80_%E0%A4%B5%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%BE" title="इस आईपी एड्रेससे हुए बदलावों के बारे में वार्ता [n]" accesskey="n"><span>वार्ता</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="साइट"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="सामग्री" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">सामग्री</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">साइडबार पर जाएँ</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">छुपाएँ</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">शुरुआत</div> </a> </li> <li id="toc-अदिश_त्रिक_गुणनफल" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#अदिश_त्रिक_गुणनफल"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>अदिश त्रिक गुणनफल</span> </div> </a> <button aria-controls="toc-अदिश_त्रिक_गुणनफल-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>अदिश त्रिक गुणनफल उप-अनुभाग को टॉगल करें</span> </button> <ul id="toc-अदिश_त्रिक_गुणनफल-sublist" class="vector-toc-list"> <li id="toc-गुण" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#गुण"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>गुण</span> </div> </a> <ul id="toc-गुण-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-सदिश_त्रिक_गुणनफल" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#सदिश_त्रिक_गुणनफल"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>सदिश त्रिक गुणनफल</span> </div> </a> <ul id="toc-सदिश_त्रिक_गुणनफल-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-सन्दर्भ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#सन्दर्भ"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>सन्दर्भ</span> </div> </a> <ul id="toc-सन्दर्भ-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="सामग्री" 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="विषयसूची को टॉगल करें" > <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">विषयसूची को टॉगल करें</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">त्रिक गुणनफल</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="दूसरे भाषा में लेख पर जाएँ। 28 भाषाओं में उपलब्ध" > <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-28" 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">28 भाषाएँ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%AF%D8%A7%D8%A1_%D8%AB%D9%84%D8%A7%D8%AB%D9%8A" title="جداء ثلاثي – अरबी" lang="ar" hreflang="ar" data-title="جداء ثلاثي" data-language-autonym="العربية" data-language-local-name="अरबी" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Producte_mixt" title="Producte mixt – कातालान" lang="ca" hreflang="ca" data-title="Producte mixt" data-language-autonym="Català" data-language-local-name="कातालान" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sm%C3%AD%C5%A1en%C3%BD_sou%C4%8Din" title="Smíšený součin – चेक" lang="cs" hreflang="cs" data-title="Smíšený součin" data-language-autonym="Čeština" data-language-local-name="चेक" 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%A5%D1%83%D1%82%C4%83%D1%88_%D1%85%D1%83%D1%82%D0%BB%D0%B0%D0%B2" title="Хутăш хутлав – चूवाश" lang="cv" hreflang="cv" data-title="Хутăш хутлав" data-language-autonym="Чӑвашла" data-language-local-name="चूवाश" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spatprodukt" title="Spatprodukt – जर्मन" lang="de" hreflang="de" data-title="Spatprodukt" data-language-autonym="Deutsch" data-language-local-name="जर्मन" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Triple_product" title="Triple product – अंग्रेज़ी" lang="en" hreflang="en" data-title="Triple product" data-language-autonym="English" data-language-local-name="अंग्रेज़ी" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Producto_mixto" title="Producto mixto – स्पेनिश" lang="es" hreflang="es" data-title="Producto mixto" data-language-autonym="Español" data-language-local-name="स्पेनिश" 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/Segakorrutis" title="Segakorrutis – एस्टोनियाई" lang="et" hreflang="et" data-title="Segakorrutis" data-language-autonym="Eesti" data-language-local-name="एस्टोनियाई" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Produit_mixte" title="Produit mixte – फ़्रेंच" lang="fr" hreflang="fr" data-title="Produit mixte" data-language-autonym="Français" data-language-local-name="फ़्रेंच" class="interlanguage-link-target"><span>Français</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%9E%D7%A2%D7%95%D7%A8%D7%91%D7%AA" title="מכפלה מעורבת – हिब्रू" lang="he" hreflang="he" data-title="מכפלה מעורבת" data-language-autonym="עברית" data-language-local-name="हिब्रू" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vegyes_szorzat" title="Vegyes szorzat – हंगेरियाई" lang="hu" hreflang="hu" data-title="Vegyes szorzat" data-language-autonym="Magyar" data-language-local-name="हंगेरियाई" 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/%D4%BD%D5%A1%D5%BC%D5%B6_%D5%A1%D6%80%D5%BF%D5%A1%D5%A4%D6%80%D5%B5%D5%A1%D5%AC" title="Խառն արտադրյալ – आर्मेनियाई" lang="hy" hreflang="hy" data-title="Խառն արտադրյալ" data-language-autonym="Հայերեն" data-language-local-name="आर्मेनियाई" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Prodotto_misto" title="Prodotto misto – इतालवी" lang="it" hreflang="it" data-title="Prodotto misto" data-language-autonym="Italiano" data-language-local-name="इतालवी" 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/%E4%B8%89%E9%87%8D%E7%A9%8D_(%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E8%A7%A3%E6%9E%90)" title="三重積 (ベクトル解析) – जापानी" lang="ja" hreflang="ja" data-title="三重積 (ベクトル解析)" data-language-autonym="日本語" data-language-local-name="जापानी" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D0%BB%D0%B0%D1%81_%D0%BA%D3%A9%D0%B1%D0%B5%D0%B9%D1%82%D1%96%D0%BD%D0%B4%D1%96" title="Аралас көбейтінді – कज़ाख़" lang="kk" hreflang="kk" data-title="Аралас көбейтінді" data-language-autonym="Қазақша" data-language-local-name="कज़ाख़" 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%82%BC%EC%A4%91%EA%B3%B1" title="삼중곱 – कोरियाई" lang="ko" hreflang="ko" data-title="삼중곱" data-language-autonym="한국어" data-language-local-name="कोरियाई" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D0%BB%D0%B0%D1%88_%D0%BA%D3%A9%D0%B1%D3%A9%D0%B9%D1%82%D2%AF%D0%BD%D0%B4%D2%AF" title="Аралаш көбөйтүндү – किर्गीज़" lang="ky" hreflang="ky" data-title="Аралаш көбөйтүндү" data-language-autonym="Кыргызча" data-language-local-name="किर्गीज़" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mi%C5%A1rioji_sandauga" title="Mišrioji sandauga – लिथुआनियाई" lang="lt" hreflang="lt" data-title="Mišrioji sandauga" data-language-autonym="Lietuvių" data-language-local-name="लिथुआनियाई" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Iloczyn_mieszany" title="Iloczyn mieszany – पोलिश" lang="pl" hreflang="pl" data-title="Iloczyn mieszany" data-language-autonym="Polski" data-language-local-name="पोलिश" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Produto_triplo" title="Produto triplo – पुर्तगाली" lang="pt" hreflang="pt" data-title="Produto triplo" data-language-autonym="Português" data-language-local-name="पुर्तगाली" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BC%D0%B5%D1%88%D0%B0%D0%BD%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="Смешанное произведение – रूसी" lang="ru" hreflang="ru" data-title="Смешанное произведение" data-language-autonym="Русский" data-language-local-name="रूसी" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Me%C5%A1ani_produkt" title="Mešani produkt – स्लोवेनियाई" lang="sl" hreflang="sl" data-title="Mešani produkt" data-language-autonym="Slovenščina" data-language-local-name="स्लोवेनियाई" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trippelprodukt" title="Trippelprodukt – स्वीडिश" lang="sv" hreflang="sv" data-title="Trippelprodukt" data-language-autonym="Svenska" data-language-local-name="स्वीडिश" 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%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%B2%E0%AE%BF_%E0%AE%AE%E0%AF%81%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%AE%E0%AF%8D" title="திசையிலி முப்பெருக்கம் – तमिल" lang="ta" hreflang="ta" data-title="திசையிலி முப்பெருக்கம்" data-language-autonym="தமிழ்" data-language-local-name="तमिल" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7l%C3%BC_%C3%A7arp%C4%B1m" title="Üçlü çarpım – तुर्की" lang="tr" hreflang="tr" data-title="Üçlü çarpım" data-language-autonym="Türkçe" data-language-local-name="तुर्की" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D1%96%D1%88%D0%B0%D0%BD%D0%B8%D0%B9_%D0%B4%D0%BE%D0%B1%D1%83%D1%82%D0%BE%D0%BA" title="Мішаний добуток – यूक्रेनियाई" lang="uk" hreflang="uk" data-title="Мішаний добуток" data-language-autonym="Українська" data-language-local-name="यूक्रेनियाई" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Aralash_ko%CA%BBpaytma" title="Aralash koʻpaytma – उज़्बेक" lang="uz" hreflang="uz" data-title="Aralash koʻpaytma" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="उज़्बेक" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E9%87%8D%E7%A7%AF" title="三重积 – चीनी" lang="zh" hreflang="zh" data-title="三重积" data-language-autonym="中文" data-language-local-name="चीनी" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q36248#sitelinks-wikipedia" title="अंतरभाषा कड़ियाँ संपादित करें" class="wbc-editpage">कड़ियाँ संपादित करें</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="नामस्थान"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="सामग्री वाला पृष्ठ देखें [c]" accesskey="c"><span>लेख</span></a></li><li id="ca-talk" class="new vector-tab-noicon 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सम्पादित करें: अदिश त्रिक गुणनफल"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%B0:Parallelepiped_volume.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/220px-Parallelepiped_volume.svg.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/330px-Parallelepiped_volume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/440px-Parallelepiped_volume.svg.png 2x" data-file-width="1002" data-file-height="770" /></a><figcaption></figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a}\cdot ({b}\times {c})={c}\cdot ({a}\times {b})={b}\cdot ({c}\times {a})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a}\cdot ({b}\times {c})={c}\cdot ({a}\times {b})={b}\cdot ({c}\times {a})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3202b98a16b4e1ad62a9a60bb1f615eb8c370a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.886ex; height:2.843ex;" alt="{\displaystyle {a}\cdot ({b}\times {c})={c}\cdot ({a}\times {b})={b}\cdot ({c}\times {a})}"></span></dd></dl> <p>इस गुणन्फल का मान तीनों सदिशों <span class="mwe-math-element"><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> और <span class="mwe-math-element"><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> से बने हुए <a href="/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0%E0%A4%B7%E0%A4%9F%E0%A4%AB%E0%A4%B2%E0%A4%95" title="समान्तरषटफलक">समान्तरषटफलक</a> के <a href="/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8" title="आयतन">आयतन</a> के बराबर होता है। </p> <div class="mw-heading mw-heading3"><h3 id="गुण"><span id=".E0.A4.97.E0.A5.81.E0.A4.A3"></span>गुण</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2&action=edit&section=2" title="अनुभाग सम्पादित करें: गुण"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%E0%A4%9A%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%AF_%E0%A4%B6%E0%A4%BF%E0%A4%AB%E0%A5%8D%E0%A4%9F&action=edit&redlink=1" class="new" title="चक्रीय शिफ्ट (पृष्ठ मौजूद नहीं है)">चक्रीय शिफ्ट</a> करने पर अदिश त्रिक गुणनफल (<b>a</b>, <b>b</b>, <b>c</b>) का मान नहीं बदलता। : <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1d6095438e953d75300658518683d98f37c19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.102ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"></span></dd></dl></li> <li>Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product. <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv (\mathbf {a} \times \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> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv (\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fde9d679c6fa996266381c7439a6f4d215c258e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.702ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv (\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} }"></span></dd></dl></li> <li>Swapping any two of the three operands <a href="/w/index.php?title=Additive_inverse&action=edit&redlink=1" class="new" title="Additive inverse (पृष्ठ मौजूद नहीं है)">negates</a> the triple product. This follows from the circular-shift property and the <a href="/w/index.php?title=Anticommutativity&action=edit&redlink=1" class="new" title="Anticommutativity (पृष्ठ मौजूद नहीं है)">anticommutativity</a> of the cross product. <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\\\equiv -&\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\\equiv -&\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\\equiv -&\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\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 /> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <mo>≡</mo> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>≡</mo> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>≡</mo> <mo>−</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\\\equiv -&\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\\equiv -&\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\\equiv -&\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea225f242944137f63124f970a86138d922450b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:15.314ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}&\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\\\equiv -&\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\\equiv -&\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\\equiv -&\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}}"></span></dd></dl></li> <li>The scalar triple product can also be understood as the <a href="/w/index.php?title=Determinant&action=edit&redlink=1" class="new" title="Determinant (पृष्ठ मौजूद नहीं है)">determinant</a> of the <span style="white-space: nowrap">3<span style="margin-left: 0.25em">×</span><span style="margin-left: 0.25em">3</span></span> matrix (thus also its <a href="/w/index.php?title=Invertible_matrix&action=edit&redlink=1" class="new" title="Invertible matrix (पृष्ठ मौजूद नहीं है)">inverse</a>) having the three vectors either as its rows or its columns (a matrix has the same determinant as its <a href="/w/index.php?title=Transpose&action=edit&redlink=1" class="new" title="Transpose (पृष्ठ मौजूद नहीं है)">transpose</a>): <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv \det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \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> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv \det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3efb64854ece413bec38a7d213263c1c533c0406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:47.965ex; height:9.176ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv \det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \right).}"></span></dd></dl></li> <li>If the scalar triple product is equal to zero, then the three vectors <b>a</b>, <b>b</b>, and <b>c</b> are <a href="/w/index.php?title=Coplanar&action=edit&redlink=1" class="new" title="Coplanar (पृष्ठ मौजूद नहीं है)">coplanar</a>, since the parallelepiped defined by them would be flat and have no volume.</li> <li>If any two vectors in the triple scalar product are equal, then its value is zero: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {a} \times \mathbf {a} )\equiv 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">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">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">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {a} \times \mathbf {a} )\equiv 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f46750bb7fd7791f4b7dddcd518e8efc2526240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.209ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )\equiv \mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )\equiv \mathbf {a} \cdot (\mathbf {a} \times \mathbf {a} )\equiv 0}"></span></dd></dl></li> <li>Moreover, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )]\mathbf {a} \equiv (\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )]\mathbf {a} \equiv (\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9c573c7edc157fee3c81b56ccc4b7ff390d77dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.405ex; height:2.843ex;" alt="{\displaystyle [\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )]\mathbf {a} \equiv (\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}"></span></dd></dl></li> <li>The <a href="/w/index.php?title=Multiplication&action=edit&redlink=1" class="new" title="Multiplication (पृष्ठ मौजूद नहीं है)">simple product</a> of two triple products (or the square of a triple product), may be expanded in terms of dot products:<sup id="cite_ref-Wong_1-0" class="reference"><a href="#cite_note-Wong-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><div class="paragraphbreak" style="margin-top:0.5em"></div><dl class="glossary"><dd class="glossary"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ([\mathbf {a} \times \mathbf {b} ]\cdot \mathbf {c} )\;([\mathbf {d} \times \mathbf {e} ]\cdot \mathbf {f} )\equiv \det \left[{\begin{pmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {d} &\mathbf {e} &\mathbf {f} \end{pmatrix}}\right]\equiv \det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mo>≡</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi 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</mrow> <annotation encoding="application/x-tex">{\displaystyle ([\mathbf {a} \times \mathbf {b} ]\cdot \mathbf {c} )\;([\mathbf {d} \times \mathbf {e} ]\cdot \mathbf {f} )\equiv \det \left[{\begin{pmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {d} &\mathbf {e} &\mathbf {f} \end{pmatrix}}\right]\equiv \det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc04f06c43ed2f1a2c26f053df2d1e5084da5b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:80.864ex; height:9.176ex;" alt="{\displaystyle ([\mathbf {a} \times \mathbf {b} ]\cdot \mathbf {c} )\;([\mathbf {d} \times \mathbf {e} ]\cdot \mathbf {f} )\equiv \det \left[{\begin{pmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {d} &\mathbf {e} &\mathbf {f} \end{pmatrix}}\right]\equiv \det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}}"></span></dd></dl><div class="paragraphbreak" style="margin-top:0.5em"></div>This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.</li></ul> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="सदिश_त्रिक_गुणनफल"><span id=".E0.A4.B8.E0.A4.A6.E0.A4.BF.E0.A4.B6_.E0.A4.A4.E0.A5.8D.E0.A4.B0.E0.A4.BF.E0.A4.95_.E0.A4.97.E0.A5.81.E0.A4.A3.E0.A4.A8.E0.A4.AB.E0.A4.B2"></span>सदिश त्रिक गुणनफल</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2&action=edit&section=3" title="अनुभाग सम्पादित करें: सदिश त्रिक गुणनफल"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>यह गुणनफल एक सदिश राशि होती है। </p> <dl><dd>a*(b*c)=(a.c)b-(a.b)c</dd></dl> <div class="mw-heading mw-heading2"><h2 id="सन्दर्भ"><span id=".E0.A4.B8.E0.A4.A8.E0.A5.8D.E0.A4.A6.E0.A4.B0.E0.A5.8D.E0.A4.AD"></span>सन्दर्भ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2&action=edit&section=4" title="अनुभाग सम्पादित करें: सन्दर्भ"><span>संपादित करें</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-Wong-1"><span class="mw-cite-backlink"><a href="#cite_ref-Wong_1-0">↑</a></span> <span class="reference-text"><cite class="citation book">Wong, Chun Wa (2013). <a rel="nofollow" class="external text" href="https://books.google.co.uk/books?id=JePI32FCqBYC&pg=PA215#v=onepage&f=true"><i>Introduction to Mathematical Physics: Methods & Concepts</i></a>. Oxford University Press. पृ॰ 215. <a href="/wiki/%E0%A4%86%E0%A4%88%E0%A5%B0%E0%A4%8D%E0%A4%B8%E0%A5%B0%E0%A4%AC%E0%A5%80%E0%A5%B0%E0%A4%8D%E0%A4%A8%E0%A5%B0" class="mw-redirect" title="आई॰ऍस॰बी॰ऍन॰">आई॰ऍस॰बी॰ऍन॰</a> <a href="/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%87%E0%A4%B7:%E0%A4%AA%E0%A5%81%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%95_%E0%A4%B8%E0%A5%8D%E0%A4%B0%E0%A5%8B%E0%A4%A4/9780199641390" title="विशेष:पुस्तक स्रोत/9780199641390"><bdi>9780199641390</bdi></a>. मूल से 4 जनवरी 2017 को <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170104000218/https://books.google.co.uk/books?id=JePI32FCqBYC&pg=PA215#v=onepage&f=true">पुरालेखित</a><span class="reference-accessdate">. अभिगमन तिथि 3 जनवरी 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Physics%3A+Methods+%26+Concepts&rft.pages=215&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=9780199641390&rft.aulast=Wong&rft.aufirst=Chun+Wa&rft_id=https%3A%2F%2Fbooks.google.co.uk%2Fbooks%3Fid%3DJePI32FCqBYC%26pg%3DPA215%23v%3Donepage%26f%3Dtrue&rfr_id=info%3Asid%2Fhi.wikipedia.org%3A%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95+%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" class="Z3988"></span><style data-mw-deduplicate="TemplateStyles:r5547232">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.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-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg");background-repeat:no-repeat;background-size:12px;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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