스칼라곱 - 위키백과, 우리 모두의 백과사전

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="ko" dir="ltr"> <head> <meta charset="UTF-8"> <title>스칼라곱 - 위키백과, 우리 모두의 백과사전</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )kowikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"ko", "wgMonthNames":["","1월","2월","3월","4월","5월","6월","7월","8월","9월","10월","11월","12월"],"wgRequestId":"3eae9766-fac7-4da2-ab00-1b9bb8cb9c6e","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"스칼라곱","wgTitle":"스칼라곱","wgCurRevisionId":37274980,"wgRevisionId":37274980,"wgArticleId":798,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["해결되지 않은 속성이 있는 문서","CS1 - 영어 인용 (en)","위키데이터 속성 P18을 사용하는 문서","위키데이터 속성 P373을 사용하는 문서","위키데이터 속성 P227을 사용하는 문서","위키데이터 속성 P7859를 사용하는 문서","영어 표기를 포함한 문서","GND 식별자를 포함한 위키백과 문서","구식 포맷의 math 태그를 사용하는 문서","선형대수학","벡터","해석기하학","텐서"],"wgPageViewLanguage":"ko", "wgPageContentLanguage":"ko","wgPageContentModel":"wikitext","wgRelevantPageName":"스칼라곱","wgRelevantArticleId":798,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"ko","pageLanguageDir":"ltr","pageVariantFallbacks":"ko"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q181365","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness", "fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.SectionFont":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP", "ext.centralNotice.startUp","ext.gadget.directcommons","ext.gadget.edittools","ext.gadget.refToolbar","ext.gadget.siteNotice","ext.gadget.scrollUpButton","ext.gadget.strikethroughTOC","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=ko&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=ko&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=ko&modules=ext.gadget.SectionFont&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=ko&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="스칼라곱 - 위키백과, 우리 모두의 백과사전"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//ko.m.wikipedia.org/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1"> <link rel="alternate" type="application/x-wiki" title="편집" href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="위키백과 (ko)"> <link rel="EditURI" type="application/rsd+xml" href="//ko.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://ko.wikipedia.org/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ko"> <link rel="alternate" type="application/atom+xml" title="위키백과 아톰 피드" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EC%B5%9C%EA%B7%BC%EB%B0%94%EB%80%9C&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-스칼라곱 rootpage-스칼라곱 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">본문으로 이동</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="주 메뉴" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">주 메뉴</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">주 메뉴</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">숨기기</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 둘러보기 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8C%80%EB%AC%B8" title="대문으로 가기 [z]" accesskey="z"><span>대문</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B5%9C%EA%B7%BC%EB%B0%94%EB%80%9C" title="위키의 최근 바뀐 목록 [r]" accesskey="r"><span>최근 바뀜</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%ED%8F%AC%ED%84%B8:%EC%9A%94%EC%A6%98_%ED%99%94%EC%A0%9C" title="최근의 소식 알아 보기"><span>요즘 화제</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EC%9E%84%EC%9D%98%EB%AC%B8%EC%84%9C" title="무작위로 선택된 문서 불러오기 [x]" accesskey="x"><span>임의의 문서로</span></a></li> </ul> </div> </div> <div id="p-사용자_모임" class="vector-menu mw-portlet mw-portlet-사용자_모임" > <div class="vector-menu-heading"> 사용자 모임 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-projectchat" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%82%AC%EB%9E%91%EB%B0%A9"><span>사랑방</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%82%AC%EC%9A%A9%EC%9E%90_%EB%AA%A8%EC%9E%84" title="위키백과 참여자를 위한 토론/대화 공간입니다."><span>사용자 모임</span></a></li><li id="n-request" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%9A%94%EC%B2%AD"><span>관리 요청</span></a></li> </ul> </div> </div> <div id="p-편집_안내" class="vector-menu mw-portlet mw-portlet-편집_안내" > <div class="vector-menu-heading"> 편집 안내 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-helpintro" class="mw-list-item"><a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C"><span>소개</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8F%84%EC%9B%80%EB%A7%90" title="도움말"><span>도움말</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%A0%95%EC%B1%85%EA%B3%BC_%EC%A7%80%EC%B9%A8"><span>정책과 지침</span></a></li><li id="n-qna" class="mw-list-item"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%A7%88%EB%AC%B8%EB%B0%A9"><span>질문방</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8C%80%EB%AC%B8" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="위키백과" src="/static/images/mobile/copyright/wikipedia-wordmark-ko.svg" style="width: 7.5em; height: 1.75em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-ko.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </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/%ED%8A%B9%EC%88%98:%EA%B2%80%EC%83%89" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="위키백과 검색 [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>검색</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"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="위키백과 검색" aria-label="위키백과 검색" autocapitalize="sentences" title="위키백과 검색 [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="특수:검색"> </div> <button class="cdx-button cdx-search-input__end-button">검색</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="개인 도구"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" 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-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="문서의 글꼴 크기, 폭, 색의 모습을 변경합니다" > <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="보이기" > <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">보이기</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_ko.wikipedia.org&uselang=ko" class=""><span>기부</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=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</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=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [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%8A%A4%EC%B9%BC%EB%9D%BC%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%8A%A4%EC%B9%BC%EB%9D%BC%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-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-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>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-대수적_정의" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#대수적_정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>대수적 정의</span> </div> </a> <ul id="toc-대수적_정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-기하학적_정의" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#기하학적_정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>기하학적 정의</span> </div> </a> <ul id="toc-기하학적_정의-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>성질</span> </div> </a> <ul id="toc-성질-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-응용" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#응용"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>응용</span> </div> </a> <button aria-controls="toc-응용-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>응용 하위섹션 토글하기</span> </button> <ul id="toc-응용-sublist" class="vector-toc-list"> <li id="toc-스칼라_사영" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#스칼라_사영"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>스칼라 사영</span> </div> </a> <ul id="toc-스칼라_사영-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-코사인_법칙" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#코사인_법칙"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>코사인 법칙</span> </div> </a> <ul id="toc-코사인_법칙-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-삼중곱" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#삼중곱"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>삼중곱</span> </div> </a> <ul id="toc-삼중곱-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-물리학" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#물리학"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>물리학</span> </div> </a> <ul id="toc-물리학-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-일반화" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#일반화"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>일반화</span> </div> </a> <button aria-controls="toc-일반화-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>일반화 하위섹션 토글하기</span> </button> <ul id="toc-일반화-sublist" class="vector-toc-list"> <li id="toc-복소수_벡터의_경우" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#복소수_벡터의_경우"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>복소수 벡터의 경우</span> </div> </a> <ul id="toc-복소수_벡터의_경우-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-내적" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#내적"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>내적</span> </div> </a> <ul id="toc-내적-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-함수의_경우" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#함수의_경우"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>함수의 경우</span> </div> </a> <ul id="toc-함수의_경우-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-행렬의_경우" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#행렬의_경우"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>행렬의 경우</span> </div> </a> <ul id="toc-행렬의_경우-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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">6</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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">스칼라곱</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. 70개 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-70" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">70개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8C%A5%E1%88%8B_%E1%89%A5%E1%8B%9C%E1%89%B5" title="ጥላ ብዜት – 암하라어" lang="am" hreflang="am" data-title="ጥላ ብዜት" data-language-autonym="አማርኛ" data-language-local-name="암하라어" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%AF%D8%A7%D8%A1_%D9%86%D9%82%D8%B7%D9%8A" title="جداء نقطي – 아랍어" lang="ar" hreflang="ar" data-title="جداء نقطي" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Productu_escalar" title="Productu escalar – 아스투리아어" lang="ast" hreflang="ast" data-title="Productu escalar" data-language-autonym="Asturianu" data-language-local-name="아스투리아어" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Skalyar_hasil" title="Skalyar hasil – 아제르바이잔어" lang="az" hreflang="az" data-title="Skalyar hasil" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D2%A1%D0%B0%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D0%BD%D0%B4%D1%8B%D2%A1" title="Скаляр ҡабатландыҡ – 바슈키르어" lang="ba" hreflang="ba" data-title="Скаляр ҡабатландыҡ" data-language-autonym="Башҡортса" data-language-local-name="바슈키르어" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D1%8B_%D0%B7%D0%B4%D0%B0%D0%B1%D1%8B%D1%82%D0%B0%D0%BA" title="Скалярны здабытак – 벨라루스어" lang="be" hreflang="be" data-title="Скалярны здабытак" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скаларно произведение – 불가리아어" lang="bg" hreflang="bg" data-title="Скаларно произведение" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%9F_%E0%A6%97%E0%A7%81%E0%A6%A3%E0%A6%A8" title="ডট গুণন – 벵골어" lang="bn" hreflang="bn" data-title="ডট গুণন" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Skalarni_proizvod" title="Skalarni proizvod – 보스니아어" lang="bs" hreflang="bs" data-title="Skalarni proizvod" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Producte_escalar" title="Producte escalar – 카탈로니아어" lang="ca" hreflang="ca" data-title="Producte escalar" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%8E%DA%A9%D8%AF%D8%A7%D9%86%DB%8C_%D9%86%D8%A7%D9%88%DB%95%DA%A9%DB%8C" title="لێکدانی ناوەکی – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="لێکدانی ناوەکی" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Skal%C3%A1rn%C3%AD_sou%C4%8Din" title="Skalární součin – 체코어" lang="cs" hreflang="cs" data-title="Skalární 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%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BB%D0%B0_%D1%85%D1%83%D1%82%D0%BB%D0%B0%D0%B2" title="Скалярла хутлав – 추바시어" lang="cv" hreflang="cv" data-title="Скалярла хутлав" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Skalarprodukt" title="Skalarprodukt – 덴마크어" lang="da" hreflang="da" data-title="Skalarprodukt" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Skalarprodukt" title="Skalarprodukt – 독일어" lang="de" hreflang="de" data-title="Skalarprodukt" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CF%8C_%CE%B3%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF" title="Εσωτερικό γινόμενο – 그리스어" lang="el" hreflang="el" data-title="Εσωτερικό γινόμενο" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Dot_product" title="Dot product – 영어" lang="en" hreflang="en" data-title="Dot product" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Skalara_produto" title="Skalara produto – 에스페란토어" lang="eo" hreflang="eo" data-title="Skalara produto" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Producto_escalar" title="Producto escalar – 스페인어" lang="es" hreflang="es" data-title="Producto escalar" 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/Skalaarkorrutis" title="Skalaarkorrutis – 에스토니아어" lang="et" hreflang="et" data-title="Skalaarkorrutis" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Biderketa_eskalar" title="Biderketa eskalar – 바스크어" lang="eu" hreflang="eu" data-title="Biderketa eskalar" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B6%D8%B1%D8%A8_%D8%AF%D8%A7%D8%AE%D9%84%DB%8C" title="ضرب داخلی – 페르시아어" lang="fa" hreflang="fa" data-title="ضرب داخلی" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pistetulo" title="Pistetulo – 핀란드어" lang="fi" hreflang="fi" data-title="Pistetulo" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Produit_scalaire" title="Produit scalaire – 프랑스어" lang="fr" hreflang="fr" data-title="Produit scalaire" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Produto_escalar" title="Produto escalar – 갈리시아어" lang="gl" hreflang="gl" data-title="Produto escalar" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99%D7%AA" title="מכפלה סקלרית – 히브리어" lang="he" hreflang="he" data-title="מכפלה סקלרית" 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%85%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%AB%E0%A4%B2" title="अदिश गुणनफल – 힌디어" lang="hi" hreflang="hi" data-title="अदिश गुणनफल" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Skalarni_umno%C5%BEak" title="Skalarni umnožak – 크로아티아어" lang="hr" hreflang="hr" data-title="Skalarni umnožak" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Skal%C3%A1ris_szorzat" title="Skaláris szorzat – 헝가리어" lang="hu" hreflang="hu" data-title="Skaláris 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/%D5%8D%D5%AF%D5%A1%D5%AC%D5%B5%D5%A1%D6%80_%D5%A1%D6%80%D5%BF%D5%A1%D5%A4%D6%80%D5%B5%D5%A1%D5%AC" title="Սկալյար արտադրյալ – 아르메니아어" lang="hy" hreflang="hy" data-title="Սկալյար արտադրյալ" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Produk_dot" title="Produk dot – 인도네시아어" lang="id" hreflang="id" data-title="Produk dot" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Prodotto_scalare" title="Prodotto scalare – 이탈리아어" lang="it" hreflang="it" data-title="Prodotto scalare" 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/%E3%83%89%E3%83%83%E3%83%88%E7%A9%8D" title="ドット積 – 일본어" lang="ja" hreflang="ja" data-title="ドット積" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%99%E1%83%90%E1%83%9A%E1%83%90%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%9C%E1%83%90%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%9A%E1%83%98" title="სკალარული ნამრავლი – 조지아어" lang="ka" hreflang="ka" data-title="სკალარული ნამრავლი" data-language-autonym="ქართული" data-language-local-name="조지아어" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D0%BA%D3%A9%D0%B1%D0%B5%D0%B9%D1%82%D1%96%D0%BD%D0%B4%D1%96" title="Скаляр көбейтінді – 카자흐어" lang="kk" hreflang="kk" data-title="Скаляр көбейтінді" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Productum_interius" title="Productum interius – 라틴어" lang="la" hreflang="la" data-title="Productum interius" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Skaliarin%C4%97_sandauga" title="Skaliarinė sandauga – 리투아니아어" lang="lt" hreflang="lt" data-title="Skaliarinė sandauga" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Skal%C4%81rais_reizin%C4%81jums" title="Skalārais reizinājums – 라트비아어" lang="lv" hreflang="lv" data-title="Skalārais reizinājums" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4" title="Скаларен производ – 마케도니아어" lang="mk" hreflang="mk" data-title="Скаларен производ" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A4%BF%E0%A4%82%E0%A4%A6%E0%A5%82_%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BE%E0%A4%95%E0%A4%BE%E0%A4%B0" title="बिंदू गुणाकार – 마라티어" lang="mr" hreflang="mr" data-title="बिंदू गुणाकार" data-language-autonym="मराठी" data-language-local-name="마라티어" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hasil_darab_bintik" title="Hasil darab bintik – 말레이어" lang="ms" hreflang="ms" data-title="Hasil darab bintik" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inwendig_product" title="Inwendig product – 네덜란드어" lang="nl" hreflang="nl" data-title="Inwendig product" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Indreprodukt" title="Indreprodukt – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Indreprodukt" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Indreprodukt" title="Indreprodukt – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Indreprodukt" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Iloczyn_skalarny" title="Iloczyn skalarny – 폴란드어" lang="pl" hreflang="pl" data-title="Iloczyn skalarny" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Prodot_%C3%ABscalar" title="Prodot ëscalar – Piedmontese" lang="pms" hreflang="pms" data-title="Prodot ëscalar" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Produto_escalar" title="Produto escalar – 포르투갈어" lang="pt" hreflang="pt" data-title="Produto escalar" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Produs_scalar" title="Produs scalar – 루마니아어" lang="ro" hreflang="ro" data-title="Produs scalar" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скалярное произведение – 러시아어" lang="ru" hreflang="ru" data-title="Скалярное произведение" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Скалярное произведение – 야쿠트어" lang="sah" hreflang="sah" data-title="Скалярное произведение" data-language-autonym="Саха тыла" data-language-local-name="야쿠트어" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Dot_product" title="Dot product – 스코틀랜드어" lang="sco" hreflang="sco" data-title="Dot product" data-language-autonym="Scots" data-language-local-name="스코틀랜드어" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Skalarni_proizvod_vektora" title="Skalarni proizvod vektora – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Skalarni proizvod vektora" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dot_product" title="Dot product – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dot product" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Skal%C3%A1rny_s%C3%BA%C4%8Din" title="Skalárny súčin – 슬로바키아어" lang="sk" hreflang="sk" data-title="Skalárny súčin" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Skalarni_produkt" title="Skalarni produkt – 슬로베니아어" lang="sl" hreflang="sl" data-title="Skalarni produkt" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Prodhimi_skalar" title="Prodhimi skalar – 알바니아어" lang="sq" hreflang="sq" data-title="Prodhimi skalar" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%BD%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B0" title="Скаларни производ вектора – 세르비아어" lang="sr" hreflang="sr" data-title="Скаларни производ вектора" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt – 스웨덴어" lang="sv" hreflang="sv" data-title="Skalärprodukt" 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%AA%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B2%E0%AF%8D" title="புள்ளிப் பெருக்கல் – 타밀어" lang="ta" hreflang="ta" data-title="புள்ளிப் பெருக்கல்" data-language-autonym="தமிழ்" data-language-local-name="타밀어" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9C%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%93%E0%B8%88%E0%B8%B8%E0%B8%94" title="ผลคูณจุด – 태국어" lang="th" hreflang="th" data-title="ผลคูณจุด" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Produktong_tuldok" title="Produktong tuldok – 타갈로그어" lang="tl" hreflang="tl" data-title="Produktong tuldok" data-language-autonym="Tagalog" data-language-local-name="타갈로그어" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Nokta_%C3%A7arp%C4%B1m" title="Nokta çarpım – 터키어" lang="tr" hreflang="tr" data-title="Nokta çarpım" data-language-autonym="Türkçe" data-language-local-name="터키어" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80_%D1%82%D0%B0%D0%BF%D0%BA%D1%8B%D1%80%D1%87%D1%8B%D0%B3%D1%8B%D1%88" title="Скаляр тапкырчыгыш – 타타르어" lang="tt" hreflang="tt" data-title="Скаляр тапкырчыгыш" data-language-autonym="Татарча / tatarça" data-language-local-name="타타르어" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%B8%D0%B9_%D0%B4%D0%BE%D0%B1%D1%83%D1%82%D0%BE%D0%BA" title="Скалярний добуток – 우크라이나어" lang="uk" hreflang="uk" data-title="Скалярний добуток" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%88%D9%88%D9%B9_%D9%BE%D8%B1%D9%88%DA%88%DA%A9%D9%B9" title="ڈوٹ پروڈکٹ – 우르두어" lang="ur" hreflang="ur" data-title="ڈوٹ پروڈکٹ" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Skalyar_ko%CA%BBpaytmasi" title="Skalyar koʻpaytmasi – 우즈베크어" lang="uz" hreflang="uz" data-title="Skalyar koʻpaytmasi" 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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADch_v%C3%B4_h%C6%B0%E1%BB%9Bng" title="Tích vô hướng – 베트남어" lang="vi" hreflang="vi" data-title="Tích vô hướng" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E9%87%8F%E7%A7%AF" title="数量积 – 우어" lang="wuu" hreflang="wuu" data-title="数量积" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%82%B9%E7%A7%AF" title="点积 – 중국어" lang="zh" hreflang="zh" data-title="点积" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%BB%9E%E7%A9%8D" title="點積 – 광둥어" lang="yue" hreflang="yue" data-title="點積" data-language-autonym="粵語" data-language-local-name="광둥어" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q181365#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 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/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="본문 보기 [c]" accesskey="c"><span>문서</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%ED%86%A0%EB%A1%A0:%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" rel="discussion" title="문서의 내용에 대한 토론 문서 [t]" accesskey="t"><span>토론</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="언어 변종 바꾸기" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">한국어</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="보기"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1"><span>읽기</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit" title="이 문서의 원본 코드를 편집 [e]" accesskey="e"><span>편집</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=history" title="이 문서의 과거 편집 내역입니다. [h]" accesskey="h"><span>역사 보기</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="도구" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">도구</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">도구</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">숨기기</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="더 많은 옵션" > <div class="vector-menu-heading"> 동작 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1"><span>읽기</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit" title="이 문서의 원본 코드를 편집 [e]" accesskey="e"><span>편집</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=history"><span>역사 보기</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 일반 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EA%B0%80%EB%A6%AC%ED%82%A4%EB%8A%94%EB%AC%B8%EC%84%9C/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="여기를 가리키는 모든 위키 문서의 목록 [j]" accesskey="j"><span>여기를 가리키는 문서</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%A7%81%ED%81%AC%EC%B5%9C%EA%B7%BC%EB%B0%94%EB%80%9C/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" rel="nofollow" title="이 문서에서 링크한 문서의 최근 바뀜 [k]" accesskey="k"><span>가리키는 글의 최근 바뀜</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/위키백과:파일_올리기" title="파일 올리기 [u]" accesskey="u"><span>파일 올리기</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%ED%8A%B9%EC%88%98%EB%AC%B8%EC%84%9C" title="모든 특수 문서의 목록 [q]" accesskey="q"><span>특수 문서 목록</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&oldid=37274980" title="이 문서의 이 판에 대한 고유 링크"><span>고유 링크</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=info" title="이 문서에 대한 자세한 정보"><span>문서 정보</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EC%9D%B4%EB%AC%B8%EC%84%9C%EC%9D%B8%EC%9A%A9&page=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&id=37274980&wpFormIdentifier=titleform" title="이 문서를 인용하는 방법에 대한 정보"><span>이 문서 인용하기</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:UrlShortener&url=https%3A%2F%2Fko.wikipedia.org%2Fwiki%2F%25EC%258A%25A4%25EC%25B9%25BC%25EB%259D%25BC%25EA%25B3%25B1"><span>축약된 URL 얻기</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:QrCode&url=https%3A%2F%2Fko.wikipedia.org%2Fwiki%2F%25EC%258A%25A4%25EC%25B9%25BC%25EB%259D%25BC%25EA%25B3%25B1"><span>QR 코드 다운로드</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> 인쇄/내보내기 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EC%B1%85&bookcmd=book_creator&referer=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1"><span>책 만들기</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:DownloadAsPdf&page=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=show-download-screen"><span>PDF로 다운로드</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&printable=yes" title="이 문서의 인쇄용 판 [p]" accesskey="p"><span>인쇄용 판</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 다른 프로젝트 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Scalar_product" hreflang="en"><span>위키미디어 공용</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q181365" title="데이터 저장소에 연결된 항목을 가리키는 링크 [g]" accesskey="g"><span>위키데이터 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> </p> <div class="dablink hatnote"><span typeof="mw:File"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98%EC%96%B4_%EB%AC%B8%EC%84%9C" title="위키백과:동음이의어 문서"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/23px-Disambig_grey.svg.png" decoding="async" width="23" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/35px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/46px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span> 이 문서는 유클리드 공간 위의 내적에 관한 것입니다. 벡터 공간 위의 내적에 대해서는 <a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a> 문서를, 벡터와 스칼라의 곱셈에 대해서는 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EA%B3%B1%EC%85%88" title="스칼라 곱셈">스칼라 곱셈</a> 문서를 참고하십시오.</div> <p><a href="/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학">선형대수학</a>에서 <b>스칼라곱</b>(scalar곱, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">scalar product</span>) 또는 <b>점곱</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">dot product</span>)은 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</a>의 두 벡터로부터 <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a> <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC" class="mw-disambig" title="스칼라">스칼라</a>를 얻는 연산이다. 스칼라곱이 유클리드 공간의 <a href="/wiki/%EB%82%B4%EC%A0%81" class="mw-redirect" title="내적">내적</a>을 이루므로, 이를 단순히 '내적'이라고 부르기도 한다. 스칼라곱의 개념의 <a href="/wiki/%EB%AC%BC%EB%A6%AC%ED%95%99" title="물리학">물리학</a> 배경은 주어진 <a href="/wiki/%ED%9E%98_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="힘 (물리학)">힘</a>이 주어진 <a href="/wiki/%EB%B3%80%EC%9C%84" title="변위">변위</a>의 물체에 가한 <a href="/wiki/%EC%9D%BC_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일 (물리학)">일</a>을 구하는 문제이다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%B0%A8%EC%9B%90" title="차원">차원</a>이 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>인 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>의 두 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f47b03efec8aaa368a671bd60936c05d5385445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.556ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}"></span>의 <b>스칼라곱</b> <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} \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfd66541b4bc8611d7723a550abc03262f44aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.983ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} \in \mathbb {R} }"></span>은 두 가지로 정의할 수 있으며, 이 두 정의는 서로 동치이다. 스칼라곱의 기호에는 <a href="/wiki/%EA%B0%80%EC%9A%B4%EB%8E%83%EC%A0%90" title="가운뎃점">가운뎃점</a> '⋅'을 사용하며, 수의 곱셈 기호와는 다르게 생략할 수 없다. </p> <div class="mw-heading mw-heading3"><h3 id="대수적_정의"><span id=".EB.8C.80.EC.88.98.EC.A0.81_.EC.A0.95.EC.9D.98"></span>대수적 정의</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=2" title="부분 편집: 대수적 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>두 벡터의 좌표가 각각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003ebf2334646f14c8eee0c6ade2944917768ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.436ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}"></span>와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =(b_{1},b_{2},\dots ,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =(b_{1},b_{2},\dots ,b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f00bd63b03e63e1937e685f63b8c603b456b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.925ex; height:2.843ex;" alt="{\displaystyle \mathbf {b} =(b_{1},b_{2},\dots ,b_{n})}"></span>라면, 이 둘의 스칼라곱은 같은 위치의 성분을 곱한 뒤 모두 합하여 얻는 값이다. </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} =a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9c6b9882834ac8015329755c8dc1858fe5fc1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.143ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}"></span></dd></dl> <p>예를 들어, 두 3차원 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,3,-2),(4,2,1)\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,3,-2),(4,2,1)\in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a093d231f2eeb7bccca2b8483628af2d618c5d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.144ex; height:3.176ex;" alt="{\displaystyle (1,3,-2),(4,2,1)\in \mathbb {R} ^{3}}"></span>의 스칼라곱은 다음과 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 (1,3,-2)\cdot (4,2,1)=1\times 4+3\times 2+(-2)\times 1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,3,-2)\cdot (4,2,1)=1\times 4+3\times 2+(-2)\times 1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4732cc101de7b0a053ada6b6229102068e9e51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.37ex; height:2.843ex;" alt="{\displaystyle (1,3,-2)\cdot (4,2,1)=1\times 4+3\times 2+(-2)\times 1=8}"></span></dd></dl> <p>이 경우 스칼라곱의 정의는 벡터의 좌표에 의존하여 정의하지만, <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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>에 기존의 좌표계가 아닌 새로운 좌표계를 주더라도, 이 좌표계가 <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">정규 직교 좌표계</a>라면, 스칼라곱을 나타내는 공식은 바뀌지 않는다. 즉, 임의의 정규 직교 좌표계 아래 스칼라곱은 위치가 같은 두 좌표의 곱을 합한 것과 같다. </p><p>유클리드 공간의 벡터는 종종 <a href="/wiki/%EC%97%B4%EB%B2%A1%ED%84%B0" class="mw-redirect" title="열벡터">열벡터</a>로 간주되며, 이 경우 두 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f1488994015f56ee267a2dfadf01a1d067e7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.819ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} }"></span>의 스칼라곱은 다음과 같이 나타낼 수 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} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8badd8257cf1a0807b592671e7c9f9c715cf25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.766ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} }"></span></dd></dl> <p>여기서 우변의 <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} ^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad623d9c620003152572f77bcb708fc360a3a799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.718ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} ^{\operatorname {T} }}"></span>는 <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>의 <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a>이며, 곱셈 기호가 생략된 곱셈은 <a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a>이다. </p><p>이 경우 앞선 예시에서의 내적은 다음과 같이 나타낼 수 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 {\begin{pmatrix}1&3&-2\end{pmatrix}}{\begin{pmatrix}4\\2\\1\end{pmatrix}}=1\times 4+3\times 2+(-2)\times 1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&3&-2\end{pmatrix}}{\begin{pmatrix}4\\2\\1\end{pmatrix}}=1\times 4+3\times 2+(-2)\times 1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558d98b7a9031879181ec85e22da17c764579ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.636ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}1&3&-2\end{pmatrix}}{\begin{pmatrix}4\\2\\1\end{pmatrix}}=1\times 4+3\times 2+(-2)\times 1=8}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="기하학적_정의"><span id=".EA.B8.B0.ED.95.98.ED.95.99.EC.A0.81_.EC.A0.95.EC.9D.98"></span>기하학적 정의</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=3" title="부분 편집: 기하학적 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라곱은 기하학적 성질인 '<a href="/wiki/%EA%B8%B8%EC%9D%B4" title="길이">길이</a>'와 '<a href="/wiki/%EA%B0%81%EB%8F%84" class="mw-redirect" title="각도">각도</a>'를 통해 다음과 같이 정의할 수 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} \cdot \mathbf {b} ={\begin{cases}\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} ={\begin{cases}\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3824b2459871d1ba1dd03994ecbe34065ea12ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.243ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} ={\begin{cases}\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}"></span></dd></dl> <p>여기서 </p> <ul><li><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 \Vert \mathbf {a} \Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Vert \mathbf {a} \Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03484f6f6ad837a567da3bacb3996c78688fcd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.624ex; height:2.843ex;" alt="{\displaystyle \Vert \mathbf {a} \Vert }"></span>는 <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/f0e552f1e95bc5e98cf3af4168af6c1592e6328a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.593ex; height:2.843ex;" alt="{\displaystyle |\mathbf {a} |}"></span>로 표기하기도 하며, 벡터 <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>의 <a href="/wiki/%EB%85%B8%EB%A6%84" class="mw-redirect" title="노름">노름</a>을 뜻한다. 이는 <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>의 길이 또는 크기를 나타낸다. <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 \Vert \mathbf {b} \Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Vert \mathbf {b} \Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e69e5f7bec0d4007dd838fc168010e792979f53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.81ex; height:2.843ex;" alt="{\displaystyle \Vert \mathbf {b} \Vert }"></span> 역시 마찬가지이다.</li> <li><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 \measuredangle (\mathbf {a} ,\mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfea8f00136d3c041f255d9732eb70596807510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.306ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )}"></span>는 두 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f1488994015f56ee267a2dfadf01a1d067e7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.819ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} }"></span> 사이의 각도이다. 이는 두 벡터가 모두 0이 아닐 때에만 정의되며, 보통 <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 [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}"></span>에서 값을 취한다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd896ba114c436e1dad87fa385048f2f28b8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.804ex; height:2.843ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )}"></span>는 <a href="/wiki/%EC%BD%94%EC%82%AC%EC%9D%B8" class="mw-redirect" title="코사인">코사인</a>이며, <a href="/wiki/%EC%A7%81%EA%B0%81_%EC%82%BC%EA%B0%81%ED%98%95" class="mw-redirect" title="직각 삼각형">직각 삼각형</a>의 이웃변과 빗변의 길이의 비로 정의하거나, <a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EA%B8%89%EC%88%98" title="테일러 급수">테일러 급수</a> 전개식을 통해 정의할 수 있다.</li></ul> <p>예를 들어, 만약 두 벡터의 길이가 모두 2이며, 둘 사이의 각도의 코사인 값이 1/2이라면, 이 두 벡터의 스칼라곱은 2 × 2 × 1/2 = 2이다. </p><p>이 정의에서 스칼라곱은 두 벡터의 길이와 위치 관계에만 의존하므로, 스칼라곱이 좌표계와 무관함이 더욱 뚜렷하다. 반대로 두 벡터를 똑같은 <a href="/wiki/%EB%93%B1%EA%B1%B0%EB%A6%AC_%EB%B3%80%ED%99%98" class="mw-redirect" title="등거리 변환">등거리 변환</a>에 의하여 변환시켰을 때, 두 벡터의 스칼라곱은 변하지 않는다는 점 역시 정의로부터 자명하다. </p><p>몇 가지 특수한 각도의 경우는 다음과 같다. </p> <ul><li>만약 <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 \measuredangle (\mathbf {a} ,\mathbf {b} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61282cab6845c06f079b83c2e3a9efee0e3c57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.567ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=0}"></span>이라면, (즉, 두 벡터의 방향이 같다면,) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f582db075a9edaa4b13f1d6b09356b9c719150c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.065ex; height:2.843ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=1}"></span>이므로, 내적은 단순히 두 벡터의 길이의 곱이다. <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} =\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3ca861c4e426d7645abdf9e184370d5d9f67a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.997ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }"></span></dd></dl></li> <li>만약 <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 \measuredangle (\mathbf {a} ,\mathbf {b} )=90^{\circ }=\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=90^{\circ }=\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c325efe1a1be1c6bce49f2aac38a6bc5619bc06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.539ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=90^{\circ }=\pi /2}"></span>라면, (즉, 두 벡터가 서로 <a href="/wiki/%EC%88%98%EC%A7%81" class="mw-redirect" title="수직">수직</a>이라면,) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4c53a3101e5a84fcc6f09aae2bceb39865ae50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.065ex; height:2.843ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=0}"></span>이므로, 내적은 0이다. <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} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c416b33910828e0941fec78eec1170c79e7ca146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.725ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}"></span></dd></dl></li> <li>만약 <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 \measuredangle (\mathbf {a} ,\mathbf {b} )=180^{\circ }=\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=180^{\circ }=\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a786972a29f78d44a620dc55b77c415b8ed199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.377ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )=180^{\circ }=\pi }"></span>라면, (즉 두 벡터의 방향이 서로 반대라면,) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65899f6df31abb45cb8fa87550dc7d3d9f5e2b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.873ex; height:2.843ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )=-1}"></span>이므로, <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>와 <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>의 내적은 다음과 같다. <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} =-\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =-\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280e12f61be6ab2dc11f437e7f807b9ba38616de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.805ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =-\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }"></span></dd></dl></li></ul> <p>또한, 이 정의로부터 두 벡터 사이의 각도를 구하는 다음과 같은 공식을 얻을 수 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 \cos \measuredangle (\mathbf {a} ,\mathbf {b} )={\frac {\mathbf {a} \cdot \mathbf {b} }{\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }}\qquad (\mathbf {a} \neq \mathbf {0} ,\;\mathbf {b} \neq \mathbf {0} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )={\frac {\mathbf {a} \cdot \mathbf {b} }{\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }}\qquad (\mathbf {a} \neq \mathbf {0} ,\;\mathbf {b} \neq \mathbf {0} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315af203b4a463f87b45d1f89d2f6a37d5749a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.962ex; height:6.176ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {a} ,\mathbf {b} )={\frac {\mathbf {a} \cdot \mathbf {b} }{\Vert \mathbf {a} \Vert \Vert \mathbf {b} \Vert }}\qquad (\mathbf {a} \neq \mathbf {0} ,\;\mathbf {b} \neq \mathbf {0} )}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=4" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>임의의 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7df13546da20cdbfc235d9c985f745af390404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.778ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{n}}"></span> 및 스칼라 <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 k\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/177754a3bb6c26dbd54cbc866337b20bafa64e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {R} }"></span>에 대하여, 다음 성질들이 성립한다.<sup id="cite_ref-Lay_1-0" class="reference"><a href="#cite_note-Lay-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/%EA%B5%90%ED%99%98_%EB%B2%95%EC%B9%99" class="mw-redirect" title="교환 법칙">교환 법칙</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} =\mathbf {b} \cdot \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d05ac1224f43e104888999aabf0e5f43a64586a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.026ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} }"></span></dd></dl></li> <li><a href="/wiki/%EC%99%BC%EC%AA%BD_%EB%B6%84%EB%B0%B0_%EB%B2%95%EC%B9%99" class="mw-redirect" title="왼쪽 분배 법칙">왼쪽 분배 법칙</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} +\mathbf {b} )\cdot \mathbf {c} =\mathbf {a} \cdot \mathbf {c} +\mathbf {b} \cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} +\mathbf {b} )\cdot \mathbf {c} =\mathbf {a} \cdot \mathbf {c} +\mathbf {b} \cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281eac971dae424426ddcb39e769309384e15b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.759ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} +\mathbf {b} )\cdot \mathbf {c} =\mathbf {a} \cdot \mathbf {c} +\mathbf {b} \cdot \mathbf {c} }"></span></dd></dl></li> <li><a href="/wiki/%EC%98%A4%EB%A5%B8%EC%AA%BD_%EB%B6%84%EB%B0%B0_%EB%B2%95%EC%B9%99" class="mw-redirect" title="오른쪽 분배 법칙">오른쪽 분배 법칙</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} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9fe2c4a91b228cdd1258a15f9a3366eae3891f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.871ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} }"></span></dd></dl></li> <li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EA%B3%B1%EC%85%88" title="스칼라 곱셈">스칼라 곱셈</a>의 보존 <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 (k\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (k\mathbf {b} )=k\mathbf {a} \cdot \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (k\mathbf {b} )=k\mathbf {a} \cdot \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1876873dbe6f09c2bfdb98f6f32c9071c964cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.841ex; height:2.843ex;" alt="{\displaystyle (k\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (k\mathbf {b} )=k\mathbf {a} \cdot \mathbf {b} }"></span></dd></dl></li> <li>위 네 가지 성질에 따라, 스칼라곱은 <a href="/wiki/%EB%8C%80%EC%B9%AD_%EC%8C%8D%EC%84%A0%ED%98%95_%ED%98%95%EC%8B%9D" class="mw-redirect" title="대칭 쌍선형 형식">대칭 쌍선형 형식</a>이다.</li> <li>자기 자신과의 스칼라곱은 음이 아닌 실수이다. <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} =\Vert \mathbf {a} \Vert ^{2}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {a} =\Vert \mathbf {a} \Vert ^{2}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed653badee892c68f68619bceca4925dd08a597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.316ex; height:3.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {a} =\Vert \mathbf {a} \Vert ^{2}\geq 0}"></span></dd></dl></li> <li>영벡터와의 스칼라곱은 0이다. <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 {0} \cdot \mathbf {a} =\mathbf {a} \cdot \mathbf {0} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>⋅</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>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} \cdot \mathbf {a} =\mathbf {a} \cdot \mathbf {0} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f399c563984d8b7c826f5871d30826e5aa5f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.99ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} \cdot \mathbf {a} =\mathbf {a} \cdot \mathbf {0} =0}"></span></dd></dl></li> <li>자기 자신과의 스칼라곱이 0인 벡터는 영벡터뿐이다. <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} =0\iff \mathbf {a} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {a} =0\iff \mathbf {a} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2d739575371076350565056369e52b7605c00f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.171ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {a} =0\iff \mathbf {a} =\mathbf {0} }"></span></dd></dl></li> <li>위 세 가지 성질에 따라, 스칼라곱은 <a href="/wiki/%EC%96%91%EC%9D%98_%EC%A0%95%EB%B6%80%ED%98%B8_%ED%98%95%EC%8B%9D" class="mw-redirect" title="양의 정부호 형식">양의 정부호 형식</a>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c416b33910828e0941fec78eec1170c79e7ca146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.725ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}"></span>일 필요충분조건은 <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} \perp \mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⊥</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \perp \mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccca89c93b1cb42769b4fc5b148095829882795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.883ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \perp \mathbf {b} }"></span>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649f39487f2fa40c3c01bae7def3e877893dd997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.725ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} >0}"></span>일 필요충분조건은 <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 \measuredangle (\mathbf {a} ,\mathbf {b} )<90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )<90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9ce762daa2fc0f870f8420fb84fd4d379c2ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.784ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )<90^{\circ }}"></span>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfbdf7cd26ae7d7b920c3d9f0d8ca65ec7aebe6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.725ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} <0}"></span>일 필요충분조건은 <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 \measuredangle (\mathbf {a} ,\mathbf {b} )>90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>></mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )>90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73a4765962be1de133c2d039a4cfb2dfdec8091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.784ex; height:2.843ex;" alt="{\displaystyle \measuredangle (\mathbf {a} ,\mathbf {b} )>90^{\circ }}"></span>이다.</li></ul> <p>반면 스칼라곱이 만족시키지 않는 성질에는 다음이 있다. </p> <ul><li><a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%B2%95%EC%B9%99" class="mw-redirect" title="결합 법칙">결합 법칙</a>은 (1차원 유클리드 공간 <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 \mathbb {R} ^{1}=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96122a79290f1c8bad3fe9a71095bed0d0938d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}=\mathbb {R} }"></span>을 제외하면) 성립하지 않는다. 이는 두 벡터의 스칼라곱이 벡터가 아닌 스칼라이므로, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8d72072d1f2a19495a141c6ce1fd560722977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.14ex; height:2.843ex;" alt="{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }"></span>나 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1baea8c615db765ff89591b667ae571aee5aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.14ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}"></span>가 무의미한 수식이기 때문이다.</li> <li><a href="/w/index.php?title=%EC%86%8C%EA%B1%B0_%EB%B2%95%EC%B9%99&action=edit&redlink=1" class="new" title="소거 법칙 (없는 문서)">소거 법칙</a>은 (1차원 유클리드 공간 <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 \mathbb {R} ^{1}=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96122a79290f1c8bad3fe9a71095bed0d0938d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}=\mathbb {R} }"></span>을 제외하면) 성립하지 않는다. 예를 들어, <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>에서, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =(1,0)}"> <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> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =(1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a946f44779888d5bf8a1d89e7e47683dcb212a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.566ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =(1,0)}"></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} =(1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =(1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abdabb02e67831460c3319aa80bd991fbb2729c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:2.843ex;" alt="{\displaystyle \mathbf {b} =(1,1)}"></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 {c} =(1,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =(1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9273faf2af3675d6ec4a9d494accf07659eb78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.263ex; height:2.843ex;" alt="{\displaystyle \mathbf {c} =(1,-1)}"></span>이라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c63b143cb06ac5bd24ca810943480ec59c776d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.99ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} =1}"></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 {a} \neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbd86f21598eb5c1c24152737ed1c22835368e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.735ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \neq \mathbf {0} }"></span>이지만, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} \neq \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \neq \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd7fe932f086de06c1b134c3a43790c215aaa2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.772ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} \neq \mathbf {c} }"></span>이다. 사실, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ef6fab228b0e35c52d1c80ed024fdfe4bc25f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.729ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }"></span>일 필요충분조건은 <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} \perp (\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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \perp (\mathbf {b} -\mathbf {c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c60e50a4b76691b230e6c0693e031e82b281a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.721ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \perp (\mathbf {b} -\mathbf {c} )}"></span>이다.</li></ul> <div class="mw-heading mw-heading2"><h2 id="응용"><span id=".EC.9D.91.EC.9A.A9"></span>응용</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=5" title="부분 편집: 응용"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="스칼라_사영"><span id=".EC.8A.A4.EC.B9.BC.EB.9D.BC_.EC.82.AC.EC.98.81"></span>스칼라 사영</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=6" title="부분 편집: 스칼라 사영"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>벡터 <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>의 벡터 <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> 위의 <b>스칼라 사영</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">scalar projection</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 {a} _{\mathbf {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathbf {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd43fd25c5c2f6f90cf2969706794ad4f55d3ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.582ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\mathbf {b} }}"></span>은 <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>를 <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>로 <a href="/w/index.php?title=%EC%88%98%EC%A7%81_%EC%82%AC%EC%98%81&action=edit&redlink=1" class="new" title="수직 사영 (없는 문서)">수직 사영</a>하여 얻는 벡터의 길이이다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\mathbf {b} }={\begin{cases}\Vert a\Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo fence="false" stretchy="false">‖</mo> <mi>a</mi> <mo fence="false" stretchy="false">‖</mo> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathbf {b} }={\begin{cases}\Vert a\Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9bc1568985057d7aabbc9d9c93e65db3be944a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.481ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} _{\mathbf {b} }={\begin{cases}\Vert a\Vert \cos \measuredangle (\mathbf {a} ,\mathbf {b} )&\mathbf {a} \neq \mathbf {0} \land \mathbf {b} \neq \mathbf {0} \\0&\mathbf {a} =\mathbf {0} \lor \mathbf {b} =\mathbf {0} \end{cases}}}"></span></dd></dl> <p>스칼라 사영은 다음과 같이 <a href="/wiki/%EB%8B%A8%EC%9C%84_%EB%B2%A1%ED%84%B0" class="mw-redirect" title="단위 벡터">단위 벡터</a>와의 스칼라곱으로 나타낼 수 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} _{\mathbf {b} }=\mathbf {a} \cdot {\frac {\mathbf {b} }{\Vert \mathbf {b} \Vert }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathbf {b} }=\mathbf {a} \cdot {\frac {\mathbf {b} }{\Vert \mathbf {b} \Vert }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2540fc7814ca22f7a6647a36107ecfeef16f552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.305ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} _{\mathbf {b} }=\mathbf {a} \cdot {\frac {\mathbf {b} }{\Vert \mathbf {b} \Vert }}}"></span></dd></dl> <p>반대로, 스칼라곱은 다음과 같이 스칼라 사영과 벡터의 길이의 곱으로 나타낼 수 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} \cdot \mathbf {b} =\Vert \mathbf {b} \Vert \mathbf {a} _{\mathbf {b} }=\Vert \mathbf {a} \Vert \mathbf {b} _{\mathbf {a} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =\Vert \mathbf {b} \Vert \mathbf {a} _{\mathbf {b} }=\Vert \mathbf {a} \Vert \mathbf {b} _{\mathbf {a} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730cb03d58b8c78d489816c7cf29175db82c3487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.314ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =\Vert \mathbf {b} \Vert \mathbf {a} _{\mathbf {b} }=\Vert \mathbf {a} \Vert \mathbf {b} _{\mathbf {a} }}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="코사인_법칙"><span id=".EC.BD.94.EC.82.AC.EC.9D.B8_.EB.B2.95.EC.B9.99"></span>코사인 법칙</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=7" title="부분 편집: 코사인 법칙"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Dot_product_cosine_rule.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/220px-Dot_product_cosine_rule.svg.png" decoding="async" width="220" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/330px-Dot_product_cosine_rule.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Dot_product_cosine_rule.svg/440px-Dot_product_cosine_rule.svg.png 2x" data-file-width="106" data-file-height="135" /></a><figcaption>삼각형의 세 변에 대응하는 세 벡터 <b>a</b>, <b>b</b>, <b>c</b>와 이들 가운데 두 벡터의 각도 <i>θ</i>.</figcaption></figure> <p>삼각형의 세 변 <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,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span>와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>가 마주보는 각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>에 대한 <a href="/wiki/%EC%BD%94%EC%82%AC%EC%9D%B8_%EB%B2%95%EC%B9%99" title="코사인 법칙">코사인 법칙</a>은 스칼라곱의 성질을 통해 유도할 수 있다. 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98735198279ea2237902abe353cfc8156f2eea0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.041ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }"></span>가 그림과 같다고 하면, 코사인 법칙은 다음과 같이 증명된다. </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}c^{2}&=\mathbf {c} \cdot \mathbf {c} \\&=(\mathbf {a} -\mathbf {b} )\cdot (\mathbf {a} -\mathbf {b} )\\&=\mathbf {a} \cdot \mathbf {a} -\mathbf {b} \cdot \mathbf {a} -\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=\mathbf {a} \cdot \mathbf {a} -2\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=a^{2}-2ab\cos \theta +b^{2}\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> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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">b</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</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>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</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>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c^{2}&=\mathbf {c} \cdot \mathbf {c} \\&=(\mathbf {a} -\mathbf {b} )\cdot (\mathbf {a} -\mathbf {b} )\\&=\mathbf {a} \cdot \mathbf {a} -\mathbf {b} \cdot \mathbf {a} -\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=\mathbf {a} \cdot \mathbf {a} -2\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=a^{2}-2ab\cos \theta +b^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7da11b6a064c16b55a8995e9b6dae8d32fa380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:32.288ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}c^{2}&=\mathbf {c} \cdot \mathbf {c} \\&=(\mathbf {a} -\mathbf {b} )\cdot (\mathbf {a} -\mathbf {b} )\\&=\mathbf {a} \cdot \mathbf {a} -\mathbf {b} \cdot \mathbf {a} -\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=\mathbf {a} \cdot \mathbf {a} -2\mathbf {a} \cdot \mathbf {b} +\mathbf {b} \cdot \mathbf {b} \\&=a^{2}-2ab\cos \theta +b^{2}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="삼중곱"><span id=".EC.82.BC.EC.A4.91.EA.B3.B1"></span>삼중곱</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=8" title="부분 편집: 삼중곱"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>3차원 유클리드 공간 <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> 속 벡터에 대한 곱셈은 그 밖에도 여럿 존재한다. 예를 들어, 두 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcadc5d7af51d48d495998a8bc4e9e155f80d62c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.392ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{3}}"></span>의 <a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</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 \mathbf {a} \times \mathbf {b} \in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times \mathbf {b} \in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d77e817fa56b74adcfaede2fd38747035b226c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.198ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \times \mathbf {b} \in \mathbb {R} ^{3}}"></span>은 스칼라곱과 달리 두 벡터로부터 또 다른 벡터를 얻는다. 그러나 이는 3차원이 아닌 유클리드 공간에서 의미를 잃는다. </p><p><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EC%82%BC%EC%A4%91%EA%B3%B1" class="mw-redirect" title="스칼라 삼중곱">스칼라 삼중곱</a>은 <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>에서 스칼라곱과 벡터곱을 사용하여 정의된다. 세 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426cce0564ac70669ebf98c58d0237eb76fa8741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.614ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}"></span>의 스칼라 삼중곱은 <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} )\in \mathbb {R} }"> <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="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31057d7dab89b3cfae93fa79b81501b93f4dd631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.82ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\in \mathbb {R} }"></span>로 정의된다. </p><p><a href="/wiki/%EB%B2%A1%ED%84%B0_%EC%82%BC%EC%A4%91%EA%B3%B1" class="mw-redirect" title="벡터 삼중곱">벡터 삼중곱</a>은 <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>에서 두 번의 벡터 곱으로 정의된다. 세 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426cce0564ac70669ebf98c58d0237eb76fa8741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.614ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} \in \mathbb {R} ^{3}}"></span>의 벡터 삼중곱은 스칼라곱을 계수로 하는 선형 결합 전개식으로 나타낼 수 있으며, 이는 다음과 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} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \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> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5ccb6c880c36b1c1e53b76d8d236f2bfeb3ad8" 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 {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="물리학"><span id=".EB.AC.BC.EB.A6.AC.ED.95.99"></span>물리학</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=9" title="부분 편집: 물리학"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EB%AC%BC%EB%A6%AC%ED%95%99" title="물리학">물리학</a>의 여러 가지 개념은 스칼라곱을 통해 정의된다. 예를 들어, <a href="/wiki/%EC%9D%BC_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일 (물리학)">일</a>은 <a href="/wiki/%ED%9E%98_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="힘 (물리학)">힘</a>과 <a href="/wiki/%EB%B3%80%EC%9C%84" title="변위">변위</a>의 스칼라곱이며, <a href="/wiki/%EC%9E%90%EA%B8%B0_%EC%84%A0%EC%86%8D" title="자기 선속">자기 선속</a>은 <a href="/wiki/%EC%9E%90%EA%B8%B0_%EC%84%A0%EC%86%8D_%EB%B0%80%EB%8F%84" class="mw-redirect" title="자기 선속 밀도">자기 선속 밀도</a>와 면적 벡터의 스칼라곱이다. 물론 변하는 힘이나 일정하지 않은 자기 선속의 경우 <a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a>을 사용한다. </p> <div class="mw-heading mw-heading2"><h2 id="일반화"><span id=".EC.9D.BC.EB.B0.98.ED.99.94"></span>일반화</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=10" title="부분 편집: 일반화"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="복소수_벡터의_경우"><span id=".EB.B3.B5.EC.86.8C.EC.88.98_.EB.B2.A1.ED.84.B0.EC.9D.98_.EA.B2.BD.EC.9A.B0"></span>복소수 벡터의 경우</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=11" title="부분 편집: 복소수 벡터의 경우"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>차원이 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>인 복소수 곱공간 <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> 속의 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f7dedb709ce5a31d3cd0aa1c8a8b06fd82ada8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.667ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}}"></span>에 대하여 스칼라곱과 비슷한 함수를 정의할 수 있으며, 이는 다음과 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 {u} \cdot \mathbf {v} =\mathbf {u} ^{*}\mathbf {v} ={\overline {u_{1}}}v_{1}+\cdots +{\overline {u_{n}}}v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} =\mathbf {u} ^{*}\mathbf {v} ={\overline {u_{1}}}v_{1}+\cdots +{\overline {u_{n}}}v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237b2f4e0b12c0402ac02230363c777343b86e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.817ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} =\mathbf {u} ^{*}\mathbf {v} ={\overline {u_{1}}}v_{1}+\cdots +{\overline {u_{n}}}v_{n}}"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12cfa608d32f313d04d0aeb1241ca330efdb423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.54ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} ^{*}}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span>의 (열벡터로서의) <a href="/wiki/%EC%BC%A4%EB%A0%88%EC%A0%84%EC%B9%98" class="mw-redirect" title="켤레전치">켤레전치</a>이며, <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 {\overline {u_{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {u_{k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d76c22260eb12491cd14527fcb74c6722f1af273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.533ex; height:2.676ex;" alt="{\displaystyle {\overline {u_{k}}}}"></span>는 <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 u_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9837644700489d04d977da272524cd5fda36f3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle u_{k}}"></span>의 <a href="/wiki/%EC%BC%A4%EB%A0%88_%EB%B3%B5%EC%86%8C%EC%88%98" title="켤레 복소수">켤레 복소수</a>이다. 이러한 함수는 <a href="/wiki/%EC%96%91%EC%9D%98_%EC%A0%95%EB%B6%80%ED%98%B8%EC%84%B1" class="mw-redirect" title="양의 정부호성">양의 정부호성</a>을 만족시킨다. 즉, 영벡터가 아닌 복소수 벡터와 자기 자신의 스칼라곱은 항상 실수이며 0보다 크다. 그러나 실수 벡터의 스칼라곱과 달리 쌍선형성을 만족시키지 않으며, 대신 다음과 같은 <a href="/wiki/%EB%B0%98%EC%8C%8D%EC%84%A0%ED%98%95%EC%84%B1" class="mw-redirect" title="반쌍선형성">반쌍선형성</a>을 만족시킨다. 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} \in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} \in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78dfc58237d19a90a6065e5840b055aafb99b5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.633ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} \in \mathbb {C} ^{n}}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8fc86339df069b29043cc911848d044ec3f670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {C} }"></span>에 대하여, </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 (c\mathbf {u} +\mathbf {v} )\cdot \mathbf {w} ={\bar {c}}\mathbf {u} \cdot \mathbf {w} +\mathbf {v} \cdot \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c\mathbf {u} +\mathbf {v} )\cdot \mathbf {w} ={\bar {c}}\mathbf {u} \cdot \mathbf {w} +\mathbf {v} \cdot \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/002dd0d14e1bbb239e4fbd40f4d45f8eb36964c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.511ex; height:2.843ex;" alt="{\displaystyle (c\mathbf {u} +\mathbf {v} )\cdot \mathbf {w} ={\bar {c}}\mathbf {u} \cdot \mathbf {w} +\mathbf {v} \cdot \mathbf {w} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot (c\mathbf {v} +\mathbf {w} )=c\mathbf {u} \cdot \mathbf {v} +\mathbf {u} \cdot \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot (c\mathbf {v} +\mathbf {w} )=c\mathbf {u} \cdot \mathbf {v} +\mathbf {u} \cdot \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533c3fe44157bc98c969f8c672fa650dd4dcc6c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.78ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} \cdot (c\mathbf {v} +\mathbf {w} )=c\mathbf {u} \cdot \mathbf {v} +\mathbf {u} \cdot \mathbf {w} }"></span></dd></dl> <p>또한 대칭성(교환 법칙) 대신 다음과 같은 <a href="/w/index.php?title=%EC%BC%A4%EB%A0%88_%EB%8C%80%EC%B9%AD%EC%84%B1&action=edit&redlink=1" class="new" title="켤레 대칭성 (없는 문서)">켤레 대칭성</a>을 만족시킨다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\overline {\mathbf {v} \cdot \mathbf {u} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\overline {\mathbf {v} \cdot \mathbf {u} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2a93cb6ee8b072015bcab5ed1d37846294e10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.364ex; height:2.343ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} ={\overline {\mathbf {v} \cdot \mathbf {u} }}}"></span></dd></dl> <p>이 경우, 영벡터가 아닌 두 복소수 벡터의 사잇각을 나타내는 공식은 다음과 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 \cos \measuredangle (\mathbf {u} ,\mathbf {v} )={\frac {\operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}{\Vert \mathbf {u} \Vert \Vert \mathbf {v} \Vert }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>∡</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \measuredangle (\mathbf {u} ,\mathbf {v} )={\frac {\operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}{\Vert \mathbf {u} \Vert \Vert \mathbf {v} \Vert }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4da8ae2f5c6b563e9f36b3661148a047a81ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.978ex; height:6.509ex;" alt="{\displaystyle \cos \measuredangle (\mathbf {u} ,\mathbf {v} )={\frac {\operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}{\Vert \mathbf {u} \Vert \Vert \mathbf {v} \Vert }}}"></span></dd></dl> <p>여기서 <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 \operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35a45c0164eb6ad5057180cf243136c220e41cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.128ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (\mathbf {u} \cdot \mathbf {v} )}"></span>는 복소수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3161cc5faca46f8bd209de2beb24159a50bc66e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.575ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} \cdot \mathbf {v} }"></span>의 <a href="/wiki/%EC%8B%A4%EC%88%98%EB%B6%80" class="mw-redirect" title="실수부">실수부</a>이다. </p><p>만약 이 함수의 정의에서 켤레 복소수를 생략한다면, 이는 쌍선형성과 대칭성을 유지하지만 양의 정부호성을 잃는다. 이는 대략 <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 i^{2}=-1<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def714194f7b882e8f8954e1cf7ba611cae041a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.187ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1<0}"></span>이기 때문이다. 사실, 양의 정부호성과 대칭성과 쌍선형성을 동시에 만족시키는 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\colon \mathbb {C} ^{n}\times \mathbb {C} ^{n}\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\colon \mathbb {C} ^{n}\times \mathbb {C} ^{n}\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251c602e8284c651adc3d2e2afd2ab11dbcc8735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.724ex; height:2.343ex;" alt="{\displaystyle B\colon \mathbb {C} ^{n}\times \mathbb {C} ^{n}\to \mathbb {C} }"></span>는 존재하지 않는다. 이는 이러한 함수의 존재가 다음과 같은 모순을 가져오기 때문이다. </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 0<B(i\mathbf {v} ,i\mathbf {v} )=i^{2}B(\mathbf {v} ,\mathbf {v} )=-B(\mathbf {v} ,\mathbf {v} )<0\qquad (\mathbf {v} \in \mathbb {C} ^{n}\setminus \{\mathbf {0} \})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> <mspace width="2em" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<B(i\mathbf {v} ,i\mathbf {v} )=i^{2}B(\mathbf {v} ,\mathbf {v} )=-B(\mathbf {v} ,\mathbf {v} )<0\qquad (\mathbf {v} \in \mathbb {C} ^{n}\setminus \{\mathbf {0} \})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32d800830ac819ad2e78ef18667b856afb96e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.735ex; height:3.176ex;" alt="{\displaystyle 0<B(i\mathbf {v} ,i\mathbf {v} )=i^{2}B(\mathbf {v} ,\mathbf {v} )=-B(\mathbf {v} ,\mathbf {v} )<0\qquad (\mathbf {v} \in \mathbb {C} ^{n}\setminus \{\mathbf {0} \})}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="내적"><span id=".EB.82.B4.EC.A0.81"></span>내적</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=12" title="부분 편집: 내적"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>유클리드 공간이나 복소수 곱공간의 스칼라곱을 일반화하여 <a href="/wiki/%EB%82%B4%EC%A0%81" class="mw-redirect" title="내적">내적</a>의 개념을 얻을 수 있다. 실수 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>에서, 두 벡터 <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 u,v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7acb96be1087c3c2f30d303aa4cac24f62f45daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.119ex; height:2.509ex;" alt="{\displaystyle u,v\in V}"></span>로부터 실수 스칼라 <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 \langle u,v\rangle \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13a8995991bd8ce250561396330786e13281e4de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.819ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle \in \mathbb {R} }"></span>를 얻는 연산이 양의 정부호성과 대칭성과 쌍선형성을 만족시킬 경우, 이를 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 위의 <a href="/wiki/%EB%82%B4%EC%A0%81" class="mw-redirect" title="내적">내적</a>이라고 한다. 복소수 벡터 공간 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>의 두 벡터 <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 u,v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7acb96be1087c3c2f30d303aa4cac24f62f45daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.119ex; height:2.509ex;" alt="{\displaystyle u,v\in V}"></span>로부터 복소수 스칼라 <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 \langle u,v\rangle \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle \in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94688076d696eba1e4646464386b6626122e025c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.819ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle \in \mathbb {C} }"></span>를 얻는 연산이 양의 정부호성과 켤레 대칭성과 반쌍선형성을 만족시킬 경우, 이를 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> 위의 내적이라고 한다. 예를 들어, <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>에 다음과 같은 함수를 정의하면, 이는 내적을 이룬다.<sup id="cite_ref-Hoffman_2-0" class="reference"><a href="#cite_note-Hoffman-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:271</sup></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=a_{1}b_{1}-a_{2}b_{1}-a_{1}b_{2}+4a_{2}b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=a_{1}b_{1}-a_{2}b_{1}-a_{1}b_{2}+4a_{2}b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9865ba45d6cb2a4285d9d3c272ced1f17e086e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.163ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})\cdot (b_{1},b_{2})=a_{1}b_{1}-a_{2}b_{1}-a_{1}b_{2}+4a_{2}b_{2}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="함수의_경우"><span id=".ED.95.A8.EC.88.98.EC.9D.98_.EA.B2.BD.EC.9A.B0"></span>함수의 경우</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=13" title="부분 편집: 함수의 경우"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>두 실숫값 함수 <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 f,g\colon [a,b]\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon [a,b]\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c9cdd885f66cd36e34a6418e28444dfc32ed27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.31ex; height:2.843ex;" alt="{\displaystyle f,g\colon [a,b]\to \mathbb {R} }"></span>의 내적 <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 \langle f,g\rangle \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908bf79c3fe524529af8b1b6765baf0b3513ade3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.757ex; height:2.843ex;" alt="{\displaystyle \langle f,g\rangle \in \mathbb {R} }"></span>은 <a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a> 대신 <a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a>을 사용하여 다음과 같이 정의할 수 있으며, 이 역시 양의 정부호성과 대칭성과 쌍선형성을 만족시킨다. </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 \langle f,g\rangle =\int _{a}^{b}f(x)g(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)g(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc17f3c4f37f3c51b0859251ac67f79eeeb6be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.343ex; height:6.343ex;" alt="{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)g(x)dx}"></span></dd></dl> <p>보다 일반적으로, 두 복소숫값 함수 <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 f,g\colon [a,b]\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon [a,b]\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d203bd5dd281940020a3cff7f9436e4aa8b34ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.31ex; height:2.843ex;" alt="{\displaystyle f,g\colon [a,b]\to \mathbb {C} }"></span>의 내적 <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 \langle f,g\rangle \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle \in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df11a0a3b989d72606320ef40346fea825e06cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.757ex; height:2.843ex;" alt="{\displaystyle \langle f,g\rangle \in \mathbb {C} }"></span>은 다음과 같으며, 이는 양의 정부호성과 켤레 대칭성과 반쌍선형성을 만족시킨다. </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 \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d030a3611ce7078660c690118fdfaa378c6ca707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.458ex; height:6.343ex;" alt="{\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)dx}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="행렬의_경우"><span id=".ED.96.89.EB.A0.AC.EC.9D.98_.EA.B2.BD.EC.9A.B0"></span>행렬의 경우</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=14" title="부분 편집: 행렬의 경우"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>사이즈가 같은 두 실수 <a href="/wiki/%ED%96%89%EB%A0%AC" title="행렬">행렬</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 A,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span>의 <b>프로베니우스 내적</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Frobenius inner product</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 A:B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f5c82f0b8222224ff55e86d571ccc59289081d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.176ex;" alt="{\displaystyle A:B}"></span>은 위치가 같은 두 성분의 곱들을 합한 결과이며, <a href="/wiki/%EB%8C%80%EA%B0%81%ED%95%A9" title="대각합">대각합</a>과 <a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a>을 통해 나타낼 수도 있다. 즉, 다음과 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 A:B=\operatorname {tr} (A^{\operatorname {T} }B)=\sum _{i,j}A_{ij}B_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>B</mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:B=\operatorname {tr} (A^{\operatorname {T} }B)=\sum _{i,j}A_{ij}B_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90807d7cb1bd671cbea527bb20bedc6234d40ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.396ex; height:5.843ex;" alt="{\displaystyle A:B=\operatorname {tr} (A^{\operatorname {T} }B)=\sum _{i,j}A_{ij}B_{ij}}"></span></dd></dl> <p>보다 일반적으로, 두 복소수 행렬 <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,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span>의 프로베니우스 내적은 다음과 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 A:B=\operatorname {tr} (A^{*}B)=\sum _{i,j}{\overline {A_{ij}}}B_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>B</mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:B=\operatorname {tr} (A^{*}B)=\sum _{i,j}{\overline {A_{ij}}}B_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b21fac938ef6573074b3c98894d672e97ff07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.147ex; height:6.009ex;" alt="{\displaystyle A:B=\operatorname {tr} (A^{*}B)=\sum _{i,j}{\overline {A_{ij}}}B_{ij}}"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>의 <a href="/wiki/%EC%BC%A4%EB%A0%88%EC%A0%84%EC%B9%98" class="mw-redirect" title="켤레전치">켤레전치</a>이다. </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=15" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%BD%94%EC%8B%9C-%EC%8A%88%EB%B0%94%EB%A5%B4%EC%B8%A0_%EB%B6%80%EB%93%B1%EC%8B%9D" title="코시-슈바르츠 부등식">코시-슈바르츠 부등식</a></li> <li><a href="/wiki/%EB%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a></li> <li><a href="/wiki/%EC%99%B8%EC%A0%81" title="외적">외적</a></li></ul> <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%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=16" 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-Lay-1"><span class="mw-cite-backlink"><a href="#cite_ref-Lay_1-0">↑</a></span> <span class="reference-text"><cite class="citation book">Lay, David C. (2012). 《Linear Algebra and Its Applications》 4판. Pearson Education. <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%B0/978-0-321-38517-8" title="특수:책찾기/978-0-321-38517-8"><bdi>978-0-321-38517-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=4&rft.pub=Pearson+Education&rft.date=2012&rft.isbn=978-0-321-38517-8&rft.aulast=Lay&rft.aufirst=David+C.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Hoffman-2"><span class="mw-cite-backlink"><a href="#cite_ref-Hoffman_2-0">↑</a></span> <span class="reference-text"><cite class="citation book">Hoffman, Kenneth (1971). <a rel="nofollow" class="external text" href="https://archive.org/details/LinearAlgebraHoffmanAndKunze">《Linear Algebra》</a> (영어) 2판. Upper Saddle River, New Jersey: Prentice-Hall. <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%B0/0-13-536797-2" title="특수:책찾기/0-13-536797-2"><bdi>0-13-536797-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.edition=2&rft.pub=Prentice-Hall&rft.date=1971&rft.isbn=0-13-536797-2&rft.aulast=Hoffman&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FLinearAlgebraHoffmanAndKunze&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&action=edit&section=17" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DotProduct.html">“Dot product”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Dot+product&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDotProduct.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://planetmath.org/DotProduct">“Dot product”</a>. 《PlanetMath》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=Dot+product&rft_id=http%3A%2F%2Fplanetmath.org%2FDotProduct&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" class="Z3988"><span style="display:none;"> </span></span></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 dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}.mw-parser-output .hlist-pipe dd:after,.mw-parser-output .hlist-pipe li:after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd:after,.mw-parser-output .hlist-hyphen li:after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd:after,.mw-parser-output .hlist-comma li:after{content:", ";font-weight:normal}.mw-parser-output .hlist-slash dd:after,.mw-parser-output .hlist-slash li:after{content:" / ";font-weight:normal}</style><style data-mw-deduplicate="TemplateStyles:r36429174">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style></div><div role="navigation" class="navbox" aria-labelledby="선형대수학" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><style data-mw-deduplicate="TemplateStyles:r34311309">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-보기"><a href="/wiki/%ED%8B%80:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="틀:선형대수학"><abbr title="이 틀을 보기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-토론"><a href="/w/index.php?title=%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99&action=edit&redlink=1" class="new" title="틀토론:선형대수학 (없는 문서)"><abbr title="이 틀에 관해 토론하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="특수:문서편집/틀:선형대수학"><abbr title="이 틀을 편집하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="선형대수학" style="font-size:114%;margin:0 4em"><a href="/wiki/%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="선형대수학">선형대수학</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">기본 개념</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_(%EC%88%98%ED%95%99)" title="스칼라 (수학)">스칼라</a></li> <li><a href="/wiki/%EB%B2%A1%ED%84%B0_(%EB%AC%BC%EB%A6%AC)" class="mw-redirect" title="벡터 (물리)">벡터</a></li> <li><a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a></li> <li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EA%B3%B1%EC%85%88" title="스칼라 곱셈">스칼라 곱셈</a></li> <li><a href="/w/index.php?title=%EB%B2%A1%ED%84%B0_%EC%82%AC%EC%98%81&action=edit&redlink=1" class="new" title="벡터 사영 (없는 문서)">벡터 사영</a></li> <li><a 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 href="/w/index.php?title=%ED%96%89_%EA%B3%B5%EA%B0%84%EA%B3%BC_%EC%97%B4_%EA%B3%B5%EA%B0%84&action=edit&redlink=1" class="new" title="행 공간과 열 공간 (없는 문서)">행 공간과 열 공간</a></li> <li><a href="/wiki/%EC%A7%81%EA%B5%90" title="직교">직교</a></li> <li><a href="/wiki/%EC%98%81%EA%B3%B5%EA%B0%84" class="mw-redirect" title="영공간">영공간</a></li> <li><a href="/wiki/%EA%B3%A0%EC%9C%B3%EA%B0%92%EA%B3%BC_%EA%B3%A0%EC%9C%A0_%EB%B2%A1%ED%84%B0" title="고윳값과 고유 벡터">고윳값과 고유 벡터</a></li> <li><a href="/wiki/%EC%99%B8%EC%A0%81" title="외적">외적</a></li> <li><a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a></li> <li><a class="mw-selflink selflink">스칼라곱</a></li> <li><a href="/wiki/%EC%A0%84%EC%B9%98%ED%96%89%EB%A0%AC" class="mw-redirect" title="전치행렬">전치행렬</a></li> <li><a href="/wiki/%EA%B7%B8%EB%9E%8C-%EC%8A%88%EB%AF%B8%ED%8A%B8_%EA%B3%BC%EC%A0%95" title="그람-슈미트 과정">그람-슈미트 과정</a></li> <li><a href="/wiki/%EC%97%B0%EB%A6%BD_%EC%9D%BC%EC%B0%A8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="연립 일차 방정식">일차 방정식</a></li> <li><a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EC%A0%95%EB%A6%AC&action=edit&redlink=1" class="new" title="선형대수학의 기본 정리 (없는 문서)">기본 정리</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">벡터 대수</th><td 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%B2%A1%ED%84%B0%EA%B3%B1" title="벡터곱">벡터곱</a></li> <li><a href="/wiki/%EC%82%BC%EC%A4%91%EA%B3%B1" title="삼중곱">삼중곱</a></li> <li><a href="/wiki/7%EC%B0%A8%EC%9B%90_%EC%99%B8%EC%A0%81" class="mw-redirect" title="7차원 외적">7차원 외적</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EB%8B%A4%EC%A4%91%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="다중선형대수학">다중선형대수학</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B8%B0%ED%95%98%EC%A0%81_%EB%8C%80%EC%88%98%ED%95%99" title="기하적 대수학">기하적 대수학</a></li> <li><a href="/wiki/%EC%99%B8%EB%8C%80%EC%88%98" title="외대수">외대수</a></li> <li><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></li> <li><a href="/w/index.php?title=%EB%8B%A4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="다중벡터 (없는 문서)">다중벡터</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C" title="텐서">텐서</a></li> <li><a href="/w/index.php?title=%EC%95%84%EC%9A%B0%ED%84%B0%EB%AA%A8%ED%94%BC%EC%A6%98&action=edit&redlink=1" class="new" title="아우터모피즘 (없는 문서)">아우터모피즘</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%ED%96%89%EB%A0%AC" title="행렬">행렬</a></th><td 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%B8%94%EB%A1%9D_%ED%96%89%EB%A0%AC" title="블록 행렬">블록 행렬</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EB%B6%84%ED%95%B4" title="행렬 분해">행렬 분해</a></li> <li><a href="/wiki/%EA%B0%80%EC%97%AD%ED%96%89%EB%A0%AC" title="가역행렬">가역행렬</a></li> <li><a href="/wiki/%EC%86%8C%ED%96%89%EB%A0%AC%EC%8B%9D" class="mw-redirect" title="소행렬식">소행렬식</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a></li> <li><a href="/wiki/%EA%B3%84%EC%88%98_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="계수 (선형대수학)">계수</a></li> <li><a href="/wiki/%EB%B3%80%ED%99%98%ED%96%89%EB%A0%AC" title="변환행렬">변환행렬</a></li> <li><a href="/wiki/%ED%81%AC%EB%9D%BC%EB%A9%94%EB%A5%B4_%EA%B3%B5%EC%8B%9D" class="mw-redirect" title="크라메르 공식">크라메르 공식</a></li> <li><a href="/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4_%EC%86%8C%EA%B1%B0%EB%B2%95" title="가우스 소거법">가우스 소거법</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%B6%94%EC%83%81%EB%8C%80%EC%88%98%ED%95%99" title="추상대수학">대수적</a> 구성</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%8C%8D%EB%8C%80_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="쌍대 공간">쌍대 공간</a></li> <li><a href="/wiki/%EC%A7%81%ED%95%A9" title="직합">직합</a></li> <li><a href="/wiki/%ED%95%A8%EC%88%98_%EA%B3%B5%EA%B0%84" title="함수 공간">함수 공간</a></li> <li><a href="/wiki/%EB%AA%AB_%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="몫 벡터 공간">몫공간</a></li> <li><a href="/wiki/%EB%B6%80%EB%B6%84_%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="부분 벡터 공간">부분공간</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C%EA%B3%B1" title="텐서곱">텐서곱</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%88%98%EC%B9%98%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="수치선형대수학">수치선형대수학</a></th><td 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 typeof="mw:File"><span title="목록 문서"><img alt="목록 문서" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99_%EC%A3%BC%EC%A0%9C_%EB%AA%A9%EB%A1%9D&action=edit&redlink=1" class="new" title="선형대수학 주제 목록 (없는 문서)">개요</a></li> <li><span typeof="mw:File"><span title="포털"><img alt="포털" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/16px-Portal.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/24px-Portal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/32px-Portal.svg.png 2x" data-file-width="36" data-file-height="32" /></span></span> <a href="/wiki/%ED%8F%AC%ED%84%B8:%EC%88%98%ED%95%99" title="포털:수학">수학 포털</a></li> <li><span typeof="mw:File"><span title="위키책"><img alt="위키책" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> <a href="https://en.wikipedia.org/wiki/wikibooks:Linear_algebra" class="extiw" title="en:wikibooks:Linear algebra">위키책</a></li> <li><span typeof="mw:File"><span title="위키배움터"><img alt="위키배움터" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/16px-Wikiversity-logo.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/24px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/32px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></span></span> <a href="https://en.wikipedia.org/wiki/wikiversity:Linear_algebra" class="extiw" title="en:wikiversity:Linear algebra">위키배움터</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36429174"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%A0%84%EA%B1%B0_%ED%86%B5%EC%A0%9C" title="위키백과:전거 통제">전거 통제</a>: 국가 <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q181365#identifiers" title="위키데이터에서 편집하기"><img alt="위키데이터에서 편집하기" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4181619-5">독일</a></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=스칼라곱&oldid=37274980">https://ko.wikipedia.org/w/index.php?title=스칼라곱&oldid=37274980</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" title="특수:분류">분류</a>: <ul><li><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">선형대수학</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EB%B2%A1%ED%84%B0" title="분류:벡터">벡터</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%B4%EC%84%9D%EA%B8%B0%ED%95%98%ED%95%99" title="분류:해석기하학">해석기하학</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%85%90%EC%84%9C" title="분류:텐서">텐서</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">숨은 분류: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%B4%EA%B2%B0%EB%90%98%EC%A7%80_%EC%95%8A%EC%9D%80_%EC%86%8D%EC%84%B1%EC%9D%B4_%EC%9E%88%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:해결되지 않은 속성이 있는 문서">해결되지 않은 속성이 있는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:CS1_-_%EC%98%81%EC%96%B4_%EC%9D%B8%EC%9A%A9_(en)" title="분류:CS1 - 영어 인용 (en)">CS1 - 영어 인용 (en)</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P18%EC%9D%84_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P18을 사용하는 문서">위키데이터 속성 P18을 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P373%EC%9D%84_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P373을 사용하는 문서">위키데이터 속성 P373을 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P227%EC%9D%84_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P227을 사용하는 문서">위키데이터 속성 P227을 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P7859%EB%A5%BC_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P7859를 사용하는 문서">위키데이터 속성 P7859를 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%98%81%EC%96%B4_%ED%91%9C%EA%B8%B0%EB%A5%BC_%ED%8F%AC%ED%95%A8%ED%95%9C_%EB%AC%B8%EC%84%9C" title="분류:영어 표기를 포함한 문서">영어 표기를 포함한 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:GND_%EC%8B%9D%EB%B3%84%EC%9E%90%EB%A5%BC_%ED%8F%AC%ED%95%A8%ED%95%9C_%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC_%EB%AC%B8%EC%84%9C" title="분류:GND 식별자를 포함한 위키백과 문서">GND 식별자를 포함한 위키백과 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EA%B5%AC%EC%8B%9D_%ED%8F%AC%EB%A7%B7%EC%9D%98_math_%ED%83%9C%EA%B7%B8%EB%A5%BC_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:구식 포맷의 math 태그를 사용하는 문서">구식 포맷의 math 태그를 사용하는 문서</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 이 문서는 2024년 6월 1일 (토) 20:32에 마지막으로 편집되었습니다.</li> <li id="footer-info-copyright">모든 문서는 <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.ko">크리에이티브 커먼즈 저작자표시-동일조건변경허락 4.0</a>에 따라 사용할 수 있으며, 추가적인 조건이 적용될 수 있습니다. 자세한 내용은 <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/ko">이용 약관</a>을 참고하십시오.<br />Wikipedia®는 미국 및 다른 국가에 등록되어 있는 <a rel="nofollow" class="external text" href="https://www.wikimediafoundation.org">Wikimedia Foundation, Inc.</a> 소유의 등록 상표입니다.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">개인정보처리방침</a></li> <li id="footer-places-about"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%86%8C%EA%B0%9C">위키백과 소개</a></li> <li id="footer-places-disclaimers"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%A9%B4%EC%B1%85_%EC%A1%B0%ED%95%AD">면책 조항</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">행동 강령</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">개발자</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/ko.wikipedia.org">통계</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">쿠키 정책</a></li> <li id="footer-places-mobileview"><a href="//ko.m.wikipedia.org/w/index.php?title=%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">모바일 보기</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-rf9q5","wgBackendResponseTime":151,"wgPageParseReport":{"limitreport":{"cputime":"0.442","walltime":"0.717","ppvisitednodes":{"value":2472,"limit":1000000},"postexpandincludesize":{"value":28046,"limit":2097152},"templateargumentsize":{"value":1010,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":21542,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 419.790 1 -total"," 32.46% 136.283 1 틀:위키데이터_속성_추적"," 23.97% 100.642 1 틀:선형대수학"," 23.46% 98.469 1 틀:둘러보기_상자"," 20.96% 88.001 1 틀:각주"," 16.29% 68.392 2 틀:서적_인용"," 7.30% 30.653 1 틀:전거_통제"," 6.58% 27.604 4 틀:Llang"," 2.69% 11.294 1 틀:두_다른_뜻"," 2.00% 8.392 2 틀:웹_인용"]},"scribunto":{"limitreport-timeusage":{"value":"0.152","limit":"10.000"},"limitreport-memusage":{"value":3919127,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-849f99967d-l8jwl","timestamp":"20241123005944","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\uc2a4\uce7c\ub77c\uacf1","url":"https:\/\/ko.wikipedia.org\/wiki\/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1","sameAs":"http:\/\/www.wikidata.org\/entity\/Q181365","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q181365","author":{"@type":"Organization","name":"\uc704\ud0a4\ubbf8\ub514\uc5b4 \ud504\ub85c\uc81d\ud2b8 \uae30\uc5ec\uc790"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-11-14T04:31:31Z","dateModified":"2024-06-01T11:32:17Z","headline":"\ubca1\ud130 \uc5f0\uc0b0\uc758 \ud558\ub098"}</script> </body> </html>

CINXE.COM

스칼라곱 - 위키백과, 우리 모두의 백과사전