삼중곱 - 위키백과, 우리 모두의 백과사전

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[o]" accesskey="o" class=""><span>로그인</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="더 많은 옵션" > <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="개인 도구" > <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">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <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_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%82%BC%EC%A4%91%EA%B3%B1" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%82%BC%EC%A4%91%EA%B3%B1" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-스칼라_삼중곱" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#스칼라_삼중곱"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>스칼라 삼중곱</span> </div> </a> <button aria-controls="toc-스칼라_삼중곱-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>스칼라 삼중곱 하위섹션 토글하기</span> </button> <ul id="toc-스칼라_삼중곱-sublist" class="vector-toc-list"> <li id="toc-기하학적_의미" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#기하학적_의미"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>기하학적 의미</span> </div> </a> <ul id="toc-기하학적_의미-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>성질</span> </div> </a> <ul id="toc-성질-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-스칼라_삼중곱과_행렬식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#스칼라_삼중곱과_행렬식"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>스칼라 삼중곱과 행렬식</span> </div> </a> <ul id="toc-스칼라_삼중곱과_행렬식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-스칼라_또는_유사_스칼라" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#스칼라_또는_유사_스칼라"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>스칼라 또는 유사 스칼라</span> </div> </a> <ul id="toc-스칼라_또는_유사_스칼라-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-스칼라_삼중곱과_쐐기곱" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#스칼라_삼중곱과_쐐기곱"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>스칼라 삼중곱과 쐐기곱</span> </div> </a> <ul id="toc-스칼라_삼중곱과_쐐기곱-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-그라스만_기호" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#그라스만_기호"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>그라스만 기호</span> </div> </a> <ul id="toc-그라스만_기호-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-벡터_삼중곱" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#벡터_삼중곱"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>벡터 삼중곱</span> </div> </a> <button aria-controls="toc-벡터_삼중곱-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>벡터 삼중곱 하위섹션 토글하기</span> </button> <ul id="toc-벡터_삼중곱-sublist" class="vector-toc-list"> <li id="toc-벡터_삼중곱의_전개" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#벡터_삼중곱의_전개"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>벡터 삼중곱의 전개</span> </div> </a> <ul id="toc-벡터_삼중곱의_전개-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-정의_불가능한_삼중곱들" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의_불가능한_삼중곱들"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>정의 불가능한 삼중곱들</span> </div> </a> <ul id="toc-정의_불가능한_삼중곱들-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>각주</span> </div> </a> <ul id="toc-각주-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="목차" 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="목차 토글" > <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">목차 토글</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">삼중곱</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="다른 언어로 문서를 방문합니다. 28개 언어로 읽을 수 있습니다" > <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">28개 언어</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 – 아랍어" lang="ar" hreflang="ar" data-title="جداء ثلاثي" data-language-autonym="العربية" data-language-local-name="아랍어" 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 – 카탈로니아어" lang="ca" hreflang="ca" data-title="Producte mixt" data-language-autonym="Català" data-language-local-name="카탈로니아어" 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 – 체코어" lang="cs" hreflang="cs" data-title="Smíšený součin" data-language-autonym="Čeština" data-language-local-name="체코어" 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 – 추바시어" lang="cv" hreflang="cv" data-title="Хутăш хутлав" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" 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 – 독일어" lang="de" hreflang="de" data-title="Spatprodukt" data-language-autonym="Deutsch" data-language-local-name="독일어" 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 – 영어" lang="en" hreflang="en" data-title="Triple product" data-language-autonym="English" data-language-local-name="영어" 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 – 스페인어" lang="es" hreflang="es" data-title="Producto mixto" data-language-autonym="Español" data-language-local-name="스페인어" 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 – 에스토니아어" lang="et" hreflang="et" data-title="Segakorrutis" data-language-autonym="Eesti" data-language-local-name="에스토니아어" 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 – 프랑스어" lang="fr" hreflang="fr" data-title="Produit mixte" data-language-autonym="Français" data-language-local-name="프랑스어" 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 – 히브리어" lang="he" hreflang="he" data-title="מכפלה מעורבת" data-language-autonym="עברית" data-language-local-name="히브리어" 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%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="त्रिक गुणनफल – 힌디어" lang="hi" hreflang="hi" data-title="त्रिक गुणनफल" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vegyes_szorzat" title="Vegyes szorzat – 헝가리어" lang="hu" hreflang="hu" data-title="Vegyes szorzat" data-language-autonym="Magyar" data-language-local-name="헝가리어" 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 – 아르메니아어" lang="hy" hreflang="hy" data-title="Խառն արտադրյալ" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" 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 – 이탈리아어" lang="it" hreflang="it" data-title="Prodotto misto" data-language-autonym="Italiano" data-language-local-name="이탈리아어" 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) – 일본어" 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="일본어" 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 – 카자흐어" lang="kk" hreflang="kk" data-title="Аралас көбейтінді" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</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 – 키르기스어" lang="ky" hreflang="ky" data-title="Аралаш көбөйтүндү" data-language-autonym="Кыргызча" data-language-local-name="키르기스어" 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 – 리투아니아어" lang="lt" hreflang="lt" data-title="Mišrioji sandauga" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" 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 – 폴란드어" lang="pl" hreflang="pl" data-title="Iloczyn mieszany" data-language-autonym="Polski" data-language-local-name="폴란드어" 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 – 포르투갈어" lang="pt" hreflang="pt" data-title="Produto triplo" data-language-autonym="Português" data-language-local-name="포르투갈어" 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 – 러시아어" lang="ru" hreflang="ru" data-title="Смешанное произведение" data-language-autonym="Русский" data-language-local-name="러시아어" 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 – 슬로베니아어" lang="sl" hreflang="sl" data-title="Mešani produkt" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" 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 – 스웨덴어" lang="sv" hreflang="sv" data-title="Trippelprodukt" data-language-autonym="Svenska" data-language-local-name="스웨덴어" 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 – 타밀어" lang="ta" hreflang="ta" data-title="திசையிலி முப்பெருக்கம்" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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 – 터키어" lang="tr" hreflang="tr" data-title="Üçlü çarpım" data-language-autonym="Türkçe" data-language-local-name="터키어" 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 – 우크라이나어" lang="uk" hreflang="uk" data-title="Мішаний добуток" data-language-autonym="Українська" data-language-local-name="우크라이나어" 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 – 우즈베크어" lang="uz" hreflang="uz" data-title="Aralash koʻpaytma" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" 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 – 중국어" lang="zh" hreflang="zh" data-title="三重积" data-language-autonym="中文" data-language-local-name="중국어" 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="언어 간 링크 편집" class="wbc-editpage">링크 편집</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="이름공간"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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항목</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> <b>삼중곱</b>(<span lang="en">triple product</span>) 또는 <b>삼중 벡터곱</b>(<span lang="en">triple vector product</span>)는 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="벡터 미적분학">벡터 미적분학</a>에서 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EB%B2%A1%ED%84%B0" title="유클리드 벡터">벡터</a> 3개를 곱하는 방법을 말하는 것으로 스칼라 삼중곱과 벡터 삼중곱 2가지가 있다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="스칼라_삼중곱"><span id=".EC.8A.A4.EC.B9.BC.EB.9D.BC_.EC.82.BC.EC.A4.91.EA.B3.B1"></span>스칼라 삼중곱</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=1" title="부분 편집: 스칼라 삼중곱"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Parallelepiped_volume.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/240px-Parallelepiped_volume.svg.png" decoding="async" width="240" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/360px-Parallelepiped_volume.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Parallelepiped_volume.svg/480px-Parallelepiped_volume.svg.png 2x" data-file-width="1002" data-file-height="770" /></a><figcaption>세개의 벡터로 정의된 평행 육면체</figcaption></figure> <p><b>스칼라 삼중곱</b>(<span lang="en">scalar triple product</span>)은 두개의 벡터의 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a>을 나머지 벡터와 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="스칼라곱">스칼라곱</a>한 것으로 정의된다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c3eb0e68741b1979a8b6a210462615e383049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.302ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}"></span></dd></dl> <p>보통 괄호 없이 이를 표기하기도 하는데, 점곱을 먼저 계산하면 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a>이 불가능하기 때문에 <a href="/wiki/%EC%A4%91%EC%9D%98%EC%84%B1" title="중의성">중의적</a>이지 않기 때문이다. </p> <div class="mw-heading mw-heading3"><h3 id="기하학적_의미"><span id=".EA.B8.B0.ED.95.98.ED.95.99.EC.A0.81_.EC.9D.98.EB.AF.B8"></span>기하학적 의미</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=2" title="부분 편집: 기하학적 의미"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라 삼중곱의 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값">절댓값</a>은 <a href="/wiki/%EA%B8%B0%ED%95%98%ED%95%99" title="기하학">기하학</a>적으로 스칼라 삼중곱의 3개의 벡터로 정의되는 <a href="/wiki/%ED%8F%89%ED%96%89%EC%9C%A1%EB%A9%B4%EC%B2%B4" title="평행육면체">평행육면체</a>의 <a href="/wiki/%EB%B6%80%ED%94%BC" title="부피">부피</a>로 정의된다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\left|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <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> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\left|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc2a76de7471b9be9768a7f12fda86bd432db62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.481ex; height:2.843ex;" alt="{\displaystyle V=\left|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\right|}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=3" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라 삼중곱은 다음과 같이 벡터의 순서를 <a href="/wiki/%EC%A7%9D%EC%88%9C%EC%97%B4" class="mw-redirect" title="짝순열">짝순열</a>이 되도록 바꾸면 값이 변하지 않는다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1d6095438e953d75300658518683d98f37c19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.102ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}"></span></dd></dl> <p>또한, 만약 스칼라 삼중곱의 값이 0이면 세 벡터 <b>a</b>, <b>b</b>, <b>c</b>는 모두 동일평면상의 벡터라는 성질이 있다. </p> <div class="mw-heading mw-heading3"><h3 id="스칼라_삼중곱과_행렬식"><span id=".EC.8A.A4.EC.B9.BC.EB.9D.BC_.EC.82.BC.EC.A4.91.EA.B3.B1.EA.B3.BC_.ED.96.89.EB.A0.AC.EC.8B.9D"></span>스칼라 삼중곱과 행렬식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=4" title="부분 편집: 스칼라 삼중곱과 행렬식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>세 벡터의 스칼라 삼중곱은 그 세 벡터들을 <a href="/wiki/%ED%96%89%EB%B2%A1%ED%84%B0" class="mw-redirect" title="행벡터">행벡터</a> 또는 <a href="/wiki/%EC%97%B4%EB%B2%A1%ED%84%B0" class="mw-redirect" title="열벡터">열벡터</a>로 갖는 3 x 3 <a href="/wiki/%ED%96%89%EB%A0%AC" title="행렬">행렬</a>의 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>이다. 이를 <a href="/wiki/%EB%8D%B0%EC%B9%B4%EB%A5%B4%ED%8A%B8_%EC%A2%8C%ED%91%9C%EA%B3%84" title="데카르트 좌표계">데카르트 좌표계</a>의 성분으로 써보면 (<a href="/wiki/%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8_%ED%91%9C%EA%B8%B0%EB%B2%95" title="아인슈타인 표기법">아인슈타인 표기법</a> 사용) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=a^{i}(\mathbf {b} \times \mathbf {c} )_{i}\\&=a^{i}(\epsilon _{ijk}b^{j}c^{k})\\&=\epsilon _{ijk}a^{i}b^{j}c^{k}\\&={\begin{vmatrix}a^{1}&b^{1}&c^{1}\\a^{2}&b^{2}&c^{2}\\a^{3}&b^{3}&c^{3}\end{vmatrix}}\\&={\begin{vmatrix}a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\c^{1}&c^{2}&c^{3}\end{vmatrix}}\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"> <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> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <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> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=a^{i}(\mathbf {b} \times \mathbf {c} )_{i}\\&=a^{i}(\epsilon _{ijk}b^{j}c^{k})\\&=\epsilon _{ijk}a^{i}b^{j}c^{k}\\&={\begin{vmatrix}a^{1}&b^{1}&c^{1}\\a^{2}&b^{2}&c^{2}\\a^{3}&b^{3}&c^{3}\end{vmatrix}}\\&={\begin{vmatrix}a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\c^{1}&c^{2}&c^{3}\end{vmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628fffb3b104bf20f577ea498013a23a4edad11d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:27.694ex; height:29.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=a^{i}(\mathbf {b} \times \mathbf {c} )_{i}\\&=a^{i}(\epsilon _{ijk}b^{j}c^{k})\\&=\epsilon _{ijk}a^{i}b^{j}c^{k}\\&={\begin{vmatrix}a^{1}&b^{1}&c^{1}\\a^{2}&b^{2}&c^{2}\\a^{3}&b^{3}&c^{3}\end{vmatrix}}\\&={\begin{vmatrix}a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\c^{1}&c^{2}&c^{3}\end{vmatrix}}\end{aligned}}}"></span></dd></dl> <p>이 되어 쉽게 이를 확인할 수 있다. (여기서 ε<sub>ijk</sub>는 <a href="/wiki/%EB%A0%88%EB%B9%84%EC%B9%98%EB%B9%84%ED%83%80_%EA%B8%B0%ED%98%B8" title="레비치비타 기호">레비치비타 기호</a>이다.) </p><p>또한, <a href="/w/index.php?title=%ED%9A%8C%EC%A0%84%EB%B3%80%ED%99%98&action=edit&redlink=1" class="new" title="회전변환 (없는 문서)">회전변환</a> 행렬의 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>의 값이 1이기 때문에, 스칼라 삼중곱의 값은 좌표의 회전에 대해 값이 변하지 않음을 쉽게 확인할 수 있다. </p> <div class="mw-heading mw-heading3"><h3 id="스칼라_또는_유사_스칼라"><span id=".EC.8A.A4.EC.B9.BC.EB.9D.BC_.EB.98.90.EB.8A.94_.EC.9C.A0.EC.82.AC_.EC.8A.A4.EC.B9.BC.EB.9D.BC"></span>스칼라 또는 유사 스칼라</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=5" title="부분 편집: 스칼라 또는 유사 스칼라"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라 삼중곱의 결과는 보통 <a href="/w/index.php?title=%EC%9C%A0%EC%82%AC%EC%8A%A4%EC%B9%BC%EB%9D%BC&action=edit&redlink=1" class="new" title="유사스칼라 (없는 문서)">유사스칼라</a>이다. 만약 좌표계의 <a href="/wiki/%EB%B0%A9%ED%96%A5" title="방향">방향</a>이 미리 주어지고 고정되면 유사스칼라는 (진짜) <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC" class="mw-disambig" title="스칼라">스칼라</a>와 같아진다. </p><p>좀 더 정확히 말하면, <b>a</b> · (<b>b</b> × <b>c</b>) 는 </p> <ul><li><b>a</b>, <b>b</b> × <b>c</b>가 모두 (진짜) 벡터이거나,</li> <li>둘 모두 <a href="/wiki/%EC%9C%A0%EC%82%AC%EB%B2%A1%ED%84%B0" title="유사벡터">유사벡터</a></li></ul> <p>일 때만 (진짜) 스칼라이다. 다른 경우, 스칼라 삼중곱의 결과는 <a href="/w/index.php?title=%EC%9C%A0%EC%82%AC%EC%8A%A4%EC%B9%BC%EB%9D%BC&action=edit&redlink=1" class="new" title="유사스칼라 (없는 문서)">유사스칼라</a>이다. </p> <div class="mw-heading mw-heading3"><h3 id="스칼라_삼중곱과_쐐기곱"><span id=".EC.8A.A4.EC.B9.BC.EB.9D.BC_.EC.82.BC.EC.A4.91.EA.B3.B1.EA.B3.BC_.EC.90.90.EA.B8.B0.EA.B3.B1"></span>스칼라 삼중곱과 쐐기곱</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=6" title="부분 편집: 스칼라 삼중곱과 쐐기곱"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Exterior_calc_triple_product.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Exterior_calc_triple_product.png/220px-Exterior_calc_triple_product.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Exterior_calc_triple_product.png/330px-Exterior_calc_triple_product.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Exterior_calc_triple_product.png/440px-Exterior_calc_triple_product.png 2x" data-file-width="1068" data-file-height="836" /></a><figcaption><a href="/wiki/%EC%B0%A8%EC%9B%90" title="차원">3차원</a> 공간에서의 <a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="삼중벡터 (없는 문서)">삼중벡터</a>는 방향이 있는 <a href="/w/index.php?title=%EB%B6%80%ED%94%BC%EC%9A%94%EC%86%8C&action=edit&redlink=1" class="new" title="부피요소 (없는 문서)">부피요소</a>이다. 이것의 <a href="/wiki/%ED%98%B8%EC%A7%80_%EC%8C%8D%EB%8C%80" title="호지 쌍대">호지 쌍대</a>로 얻어지는 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC" class="mw-disambig" title="스칼라">스칼라</a>의 크기는 삼중벡터의 <a href="/wiki/%EB%B6%80%ED%94%BC" title="부피">부피</a>와 같다.</figcaption></figure> <p>스칼라 삼중곱은 <a href="/wiki/%EC%99%B8%EB%8C%80%EC%88%98" title="외대수">외대수</a>에서의 <a href="/wiki/%EC%90%90%EA%B8%B0%EA%B3%B1" class="mw-redirect" title="쐐기곱">쐐기곱</a>을 사용해 표현할 수 있다. </p> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%99%B8%EB%8C%80%EC%88%98" title="외대수">외대수</a>입니다.</div> <p>먼저, 외대수의 요소들과 쐐기곱에 대해 간단히 알아보자. <a href="/w/index.php?title=%EC%99%B8_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99&action=edit&redlink=1" class="new" title="외 미적분학 (없는 문서)">외 미적분학</a>에서 두 벡터를 쐐기곱하면 <a href="/w/index.php?title=%EC%9D%B4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="이중벡터 (없는 문서)">이중벡터</a>를 얻고, 세 벡터를 쐐기곱하면 <a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="삼중벡터 (없는 문서)">삼중벡터</a>를 얻는다. 간단히 설명하면, 외 미적분학의 이중벡터란, 일종의 방향이 있는 <a href="/w/index.php?title=%ED%8F%89%EB%A9%B4%EC%9A%94%EC%86%8C&action=edit&redlink=1" class="new" title="평면요소 (없는 문서)">평면요소</a>이고, 삼중벡터는 일종의 방향이 있는 <a href="/w/index.php?title=%EB%B6%80%ED%94%BC%EC%9A%94%EC%86%8C&action=edit&redlink=1" class="new" title="부피요소 (없는 문서)">부피요소</a>이다. 비슷하게 벡터는 방향이 있는 <a href="/w/index.php?title=%EC%84%A0%EC%9A%94%EC%86%8C&action=edit&redlink=1" class="new" title="선요소 (없는 문서)">선요소</a>이다. 여기서 삼중벡터 <b>a</b>∧<b>b</b>∧<b>c</b>는 세 백터 <b>a</b>, <b>b</b>, and <b>c</b>로 정의된 평행육면체로 볼 수 있는데 각각의 면은 이중벡터 <b>a</b>∧<b>b</b>, <b>a</b>∧<b>c</b>, <b>b</b>∧<b>c</b>에 해당한다. </p><p>이를 이용해 스칼라 삼중곱과 쐐기곱의 관계를 표현하면, 임의의 주어진 벡터 <b>a</b>, <b>b</b>, <b>c</b>의 스칼라 삼중곱은 삼중벡터의 <a href="/wiki/%ED%98%B8%EC%A7%80_%EC%8C%8D%EB%8C%80" title="호지 쌍대">호지 쌍대</a>로 얻어지는 스칼라와 같다. (비슷하게, 이중벡터의 삼중곱은 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a>과 같다.). </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=*(\mathbf {a} \wedge \mathbf {b} \wedge \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>∗</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>∧</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} \wedge \mathbf {b} \wedge \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b484e34c96310ae288cc429ba26a818921eb3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.51ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=*(\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} )}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="그라스만_기호"><span id=".EA.B7.B8.EB.9D.BC.EC.8A.A4.EB.A7.8C_.EA.B8.B0.ED.98.B8"></span>그라스만 기호</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=7" title="부분 편집: 그라스만 기호"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라 삼중곱을 다음과 같이 쓰기도 한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv [\mathbf {a} \mathbf {b} \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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <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} )\equiv [\mathbf {a} \mathbf {b} \mathbf {c} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7ce96946060325022aba20697c45a5bd21d4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.667ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\equiv [\mathbf {a} \mathbf {b} \mathbf {c} ]}"></span>.</dd></dl> <p>이와 같은 기호를 <b>그라스만 기호</b>라 한다.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 이는 독일의 수학자 <a href="/wiki/%ED%97%A4%EB%A5%B4%EB%A7%8C_%EA%B7%B8%EB%9D%BC%EC%8A%A4%EB%A7%8C" title="헤르만 그라스만">헤르만 그라스만</a>(<span lang="de">Hermann Graßmann</span>)의 이름을 딴 것이다. </p> <div class="mw-heading mw-heading2"><h2 id="벡터_삼중곱"><span id=".EB.B2.A1.ED.84.B0_.EC.82.BC.EC.A4.91.EA.B3.B1"></span>벡터 삼중곱</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=8" title="부분 편집: 벡터 삼중곱"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>벡터 삼중곱</b>(<span lang="en">vector triple product</span>)은 두 벡터의 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a>에 다시 다른 벡터와 벡터곱을 한 것을 말한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )}"> <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} \times (\mathbf {b} \times \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b52ea9cf5c396cfce1dec618898d66bb8df06a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.463ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="벡터_삼중곱의_전개"><span id=".EB.B2.A1.ED.84.B0_.EC.82.BC.EC.A4.91.EA.B3.B1.EC.9D.98_.EC.A0.84.EA.B0.9C"></span>벡터 삼중곱의 전개</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=9" title="부분 편집: 벡터 삼중곱의 전개"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f5ab653b62f34e9e02e8addb76e3572c5032dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.324ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {b} \cdot \mathbf {c} )\mathbf {a} +(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <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">c</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {b} \cdot \mathbf {c} )\mathbf {a} +(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606c00c10e67552a5f9574f4f45c7b574709cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.502ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {b} \cdot \mathbf {c} )\mathbf {a} +(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} }"></span></dd></dl> <p>위의 첫 번째 공식은 흔히 <b>삼중곱 전개</b> 또는 <b>라그랑주 공식</b> <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 또는 <b>백캡 규칙</b>(<span lang="en">BAC-CAB rule</span>) <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> 이라고 불린다. </p><p>또한 <a href="/wiki/%EA%B7%B8%EB%9E%98%EB%94%94%EC%96%B8%ED%8A%B8" class="mw-redirect" title="그래디언트">그래디언트</a>가 들어간 삼중곱과 관계된 항등식은 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="벡터 미적분학">벡터 미적분학</a>과 여러 <a href="/wiki/%EB%AC%BC%EB%A6%AC%ED%95%99" title="물리학">물리학</a>의 분야에서 유용하게 쓰인다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \times (\nabla \times \mathbf {f} )&{}=\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} \\&{}={\mbox{grad }}({\mbox{div }}\mathbf {f} )-{\mbox{laplacian }}\mathbf {f} .\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> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>grad </mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>div </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>laplacian </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \times (\nabla \times \mathbf {f} )&{}=\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} \\&{}={\mbox{grad }}({\mbox{div }}\mathbf {f} )-{\mbox{laplacian }}\mathbf {f} .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df25911ae1c3b4c5ffabc52e59fb723b47922c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.151ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\nabla \times (\nabla \times \mathbf {f} )&{}=\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} \\&{}={\mbox{grad }}({\mbox{div }}\mathbf {f} )-{\mbox{laplacian }}\mathbf {f} .\end{aligned}}}"></span></dd></dl> <p>이 식은 <a href="/w/index.php?title=%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4-%EB%93%9C_%EB%9E%8C_%EC%97%B0%EC%82%B0%EC%9E%90&action=edit&redlink=1" class="new" title="라플라스-드 람 연산자 (없는 문서)">라플라스-드 람 연산자</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =d\delta +\delta d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mi>d</mi> <mi>δ</mi> <mo>+</mo> <mi>δ</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =d\delta +\delta d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92a393f3c3b5c6102a33f0f0b924476ba09e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.404ex; height:2.509ex;" alt="{\displaystyle \Delta =d\delta +\delta d}"></span> 의 특별한 경우로 볼 수도 있다. </p><p>중 하나가 <a href="/wiki/%EC%9C%A0%EC%82%AC%EB%B2%A1%ED%84%B0" title="유사벡터">유사벡터</a>라면 삼중곱 <b>a</b> × (<b>b</b> × <b>c</b>)의 결과는 벡터이다. 하지만 다른경우엔 모두 <a href="/wiki/%EC%9C%A0%EC%82%AC%EB%B2%A1%ED%84%B0" title="유사벡터">유사벡터</a>이다. 예를 들어, 만약 <b>a</b>, <b>b</b>, <b>c</b>가 모두 벡터라면, <b>b</b> × <b>c</b>는 유사벡터이고, <b>a</b> × (<b>b</b> × <b>c</b>)는 벡터가 된다. </p> <div class="mw-heading mw-heading2"><h2 id="정의_불가능한_삼중곱들"><span id=".EC.A0.95.EC.9D.98_.EB.B6.88.EA.B0.80.EB.8A.A5.ED.95.9C_.EC.82.BC.EC.A4.91.EA.B3.B1.EB.93.A4"></span>정의 불가능한 삼중곱들</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=10" title="부분 편집: 정의 불가능한 삼중곱들"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>위의 두 삼중곱과 마찬가지로 다음과 같은 삼중곱들을 생각해 볼 수도 있다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \times \left(\mathbf {b} \cdot \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> <mrow> <mo>(</mo> <mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times \left(\mathbf {b} \cdot \mathbf {c} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9891e73c74573d4babbfdac2085cb90286b3a903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.302ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \times \left(\mathbf {b} \cdot \mathbf {c} \right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \left(\mathbf {b} \cdot \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> <mrow> <mo>(</mo> <mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \left(\mathbf {b} \cdot \mathbf {c} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5549a231acdcef036af50a0b4b33d1084d7761" 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 \left(\mathbf {b} \cdot \mathbf {c} \right)}"></span></dd></dl> <p>하지만 위 두 곱은 <a href="/wiki/%EC%A0%90%EA%B3%B1" class="mw-redirect" title="점곱">점곱</a>이 주는 값이 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC" class="mw-disambig" title="스칼라">스칼라</a>이기 때문에, 괄호를 계산한 뒤에 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a>과 <a href="/wiki/%EC%A0%90%EA%B3%B1" class="mw-redirect" title="점곱">점곱</a>을 하는 것이 불가능하다. 따라서, 위 두 삼중곱은 정의되지 않는다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%82%BC%EC%A4%91%EA%B3%B1&action=edit&section=11" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Martin Lipschutz, 전재복 역, 《미분기하학개론》, 경문사, 2008, 17쪽.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="/wiki/%EC%A1%B0%EC%A0%9C%ED%94%84%EB%A3%A8%EC%9D%B4_%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC" title="조제프루이 라그랑주">조제프루이 라그랑주</a>는 벡터곱을 벡터에 대한 대수적 곱으로 전개하진 않았다. 하지만 그는 성분으로 구성된 동등한 형태를 사용했다. Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires", Oeuvres <b>vol 3</b>. 참조. 또한 그는 벡터 삼중곱 전개의 성분으로 된 형태를 사용했었다. <a href="/w/index.php?title=%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC%EC%9D%98_%ED%95%AD%EB%93%B1%EC%8B%9D&action=edit&redlink=1" class="new" title="라그랑주의 항등식 (없는 문서)">라그랑주의 항등식</a> 또는 Kiyoshi Ito (1987). <i>Encyclopedic Dictionary of Mathematics</i>. MIT Press, p. 1679. <style data-mw-deduplicate="TemplateStyles:r38117996">.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.id-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.id-lock-limited a,.mw-parser-output .id-lock-registration.id-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.id-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}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.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:#085;margin-left:0.3em}.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}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-262-59020-4" title="특수:책찾기/0-262-59020-4">0-262-59020-4</a>. 참조.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Reitz, Milford, Christy(2006). <i>Foundations of Electromagnetic Theory</i>. Pearson Education, Inc, Benjamin Cummings. p. 5.</span> </li> </ol></div></div> <ul><li>Lass, Harry (1950). <i>Vector and Tensor Analysis</i>. McGraw-Hill Book Company, Inc., pp. 23–25.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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href="/wiki/%EC%84%A0%ED%98%95%EC%83%9D%EC%84%B1" class="mw-redirect" title="선형생성">선형생성</a></li> <li><a href="/wiki/%EC%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환">선형 변환</a></li> <li><a href="/wiki/%EC%82%AC%EC%98%81%EC%9E%91%EC%9A%A9%EC%86%8C" title="사영작용소">사영작용소</a></li> <li><a href="/wiki/%EC%9D%BC%EC%B0%A8_%EB%8F%85%EB%A6%BD_%EC%A7%91%ED%95%A9" title="일차 독립 집합">일차 독립 집합</a></li> <li><a href="/wiki/%EC%84%A0%ED%98%95_%EA%B2%B0%ED%95%A9" title="선형 결합">선형 결합</a></li> <li><a href="/wiki/%EA%B8%B0%EC%A0%80_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="기저 (선형대수학)">기저</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%EB%B3%80%EA%B2%BD&action=edit&redlink=1" class="new" title="기저 변경 (없는 문서)">기저 변경</a></li> <li><a href="/w/index.php?title=%ED%96%89_%EB%B2%A1%ED%84%B0%EC%99%80_%EC%97%B4_%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="행 벡터와 열 벡터 (없는 문서)">행 벡터와 열 벡터</a></li> <li><a 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class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B6%80%EB%8F%99%EC%86%8C%EC%88%98%EC%A0%90" title="부동소수점">부동소수점</a></li> <li><a href="/w/index.php?title=%EC%88%98%EC%B9%98%EC%A0%81_%EC%95%88%EC%A0%95%EC%84%B1&action=edit&redlink=1" class="new" title="수치적 안정성 (없는 문서)">수치적 안정성</a></li> <li><a href="/w/index.php?title=BLAS&action=edit&redlink=1" class="new" title="BLAS (없는 문서)">BLAS</a>(Basic Linear Algebra Subprogram)</li> <li><a href="/wiki/%ED%9D%AC%EC%86%8C%ED%96%89%EB%A0%AC" class="mw-redirect" title="희소행렬">희소행렬</a></li> <li><a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99_%EB%9D%BC%EC%9D%B4%EB%B8%8C%EB%9F%AC%EB%A6%AC_%EB%B9%84%EA%B5%90&action=edit&redlink=1" class="new" title="선형대수학 라이브러리 비교 (없는 문서)">선형대수학 라이브러리 비교</a></li> <li><a href="/w/index.php?title=%EC%88%98%EC%B9%98_%EB%B6%84%EC%84%9D_%EC%86%8C%ED%94%84%ED%8A%B8%EC%9B%A8%EC%96%B4_%EB%B9%84%EA%B5%90&action=edit&redlink=1" class="new" title="수치 분석 소프트웨어 비교 (없는 문서)">수치 분석 소프트웨어 비교</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span typeof="mw:File"><span title="분류"><img alt="분류" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/16px-Folder_Hexagonal_Icon.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/24px-Folder_Hexagonal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/32px-Folder_Hexagonal_Icon.svg.png 2x" data-file-width="36" data-file-height="31" /></span></span> <a href="/wiki/%EB%B6%84%EB%A5%98:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="분류:선형대수학">분류</a></li> <li><span 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