Euclidean space - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Euclidean space - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);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":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"7a2480f9-8dfe-4c26-bd58-ca6d5a877018","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Euclidean_space","wgTitle":"Euclidean space","wgCurRevisionId":1275507369,"wgRevisionId":1275507369,"wgArticleId":9697,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Articles to be split from March 2023","All articles to be split","Euclidean geometry","Linear algebra","Homogeneous spaces","Norms (mathematics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Euclidean_space","wgRelevantArticleId":9697,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[], "wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q17295","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser" :false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.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","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","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"];</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=en&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=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.16"> <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 property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/1200px-Coord_system_CA_0.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1161"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/800px-Coord_system_CA_0.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="774"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/640px-Coord_system_CA_0.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="619"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Euclidean space - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Euclidean_space"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Euclidean_space&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="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Euclidean_space"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&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-Euclidean_space rootpage-Euclidean_space skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</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="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <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="Main menu" > <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">Main menu</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">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" 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="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; 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/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</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="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <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="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <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="Appearance" > <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">Appearance</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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</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=Special:CreateAccount&returnto=Euclidean+space" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</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=Special:UserLogin&returnto=Euclidean+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</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="Log in and more options" > <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="Personal tools" > <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">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Euclidean+space" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Euclidean+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</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"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</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/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</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="Site"> <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="Contents" 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">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</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">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-History_of_the_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History_of_the_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>History of the definition</span> </div> </a> <ul id="toc-History_of_the_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Motivation_of_the_modern_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motivation_of_the_modern_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Motivation of the modern definition</span> </div> </a> <ul id="toc-Motivation_of_the_modern_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Technical_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Technical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Technical definition</span> </div> </a> <ul id="toc-Technical_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Prototypical_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Prototypical_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Prototypical examples</span> </div> </a> <ul id="toc-Prototypical_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Affine_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Affine structure</span> </div> </a> <button aria-controls="toc-Affine_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Affine structure subsection</span> </button> <ul id="toc-Affine_structure-sublist" class="vector-toc-list"> <li id="toc-Subspaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subspaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Subspaces</span> </div> </a> <ul id="toc-Subspaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lines_and_segments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lines_and_segments"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Lines and segments</span> </div> </a> <ul id="toc-Lines_and_segments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parallelism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parallelism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Parallelism</span> </div> </a> <ul id="toc-Parallelism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Metric_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Metric_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Metric structure</span> </div> </a> <button aria-controls="toc-Metric_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Metric structure subsection</span> </button> <ul id="toc-Metric_structure-sublist" class="vector-toc-list"> <li id="toc-Distance_and_length" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distance_and_length"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Distance and length</span> </div> </a> <ul id="toc-Distance_and_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthogonality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Orthogonality</span> </div> </a> <ul id="toc-Orthogonality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Angle</span> </div> </a> <ul id="toc-Angle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Cartesian coordinates</span> </div> </a> <ul id="toc-Cartesian_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Other coordinates</span> </div> </a> <ul id="toc-Other_coordinates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Isometries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Isometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Isometries</span> </div> </a> <button aria-controls="toc-Isometries-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Isometries subsection</span> </button> <ul id="toc-Isometries-sublist" class="vector-toc-list"> <li id="toc-Isometry_with_prototypical_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isometry_with_prototypical_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Isometry with prototypical examples</span> </div> </a> <ul id="toc-Isometry_with_prototypical_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Euclidean group</span> </div> </a> <ul id="toc-Euclidean_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Topology</span> </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axiomatic_definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Axiomatic_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Axiomatic definitions</span> </div> </a> <ul id="toc-Axiomatic_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Usage" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Usage</span> </div> </a> <ul id="toc-Usage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_geometric_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_geometric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Other geometric spaces</span> </div> </a> <button aria-controls="toc-Other_geometric_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other geometric spaces subsection</span> </button> <ul id="toc-Other_geometric_spaces-sublist" class="vector-toc-list"> <li id="toc-Affine_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Affine space</span> </div> </a> <ul id="toc-Affine_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Projective space</span> </div> </a> <ul id="toc-Projective_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Euclidean_geometries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Euclidean_geometries"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Non-Euclidean geometries</span> </div> </a> <ul id="toc-Non-Euclidean_geometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curved_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curved_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Curved spaces</span> </div> </a> <ul id="toc-Curved_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudo-Euclidean_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudo-Euclidean_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Pseudo-Euclidean space</span> </div> </a> <ul id="toc-Pseudo-Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Euclidean space</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 64 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" 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">64 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Euklidiese_ruimte" title="Euklidiese ruimte – Afrikaans" lang="af" hreflang="af" data-title="Euklidiese ruimte" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Euklidischer_Raum" title="Euklidischer Raum – Alemannic" lang="gsw" hreflang="gsw" data-title="Euklidischer Raum" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%A5%D9%82%D9%84%D9%8A%D8%AF%D9%8A" title="فضاء إقليدي – Arabic" lang="ar" hreflang="ar" data-title="فضاء إقليدي" data-language-autonym="العربية" data-language-local-name="Arabic" 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/Espaciu_euclideu" title="Espaciu euclideu – Asturian" lang="ast" hreflang="ast" data-title="Espaciu euclideu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%87%E0%A6%89%E0%A6%95%E0%A7%8D%E0%A6%B2%E0%A6%BF%E0%A6%A1%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8" title="ইউক্লিডীয় স্থান – Bangla" lang="bn" hreflang="bn" data-title="ইউক্লিডীয় স্থান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%93%D1%8B" title="Евклид арауығы – Bashkir" lang="ba" hreflang="ba" data-title="Евклид арауығы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Евклидово пространство – Bulgarian" lang="bg" hreflang="bg" data-title="Евклидово пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_euclidi%C3%A0" title="Espai euclidià – Catalan" lang="ca" hreflang="ca" data-title="Espai euclidià" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D1%83%C3%A7%D0%BB%C4%83%D1%85%C4%95" title="Евклид уçлăхĕ – Chuvash" lang="cv" hreflang="cv" data-title="Евклид уçлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidovsk%C3%BD_prostor" title="Eukleidovský prostor – Czech" lang="cs" hreflang="cs" data-title="Eukleidovský prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_Euclidaidd" title="Gofod Euclidaidd – Welsh" lang="cy" hreflang="cy" data-title="Gofod Euclidaidd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Euklidisk_rum" title="Euklidisk rum – Danish" lang="da" hreflang="da" data-title="Euklidisk rum" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Euklidischer_Raum" title="Euklidischer Raum – German" lang="de" hreflang="de" data-title="Euklidischer Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Eukleidiline_ruum" title="Eukleidiline ruum – Estonian" lang="et" hreflang="et" data-title="Eukleidiline ruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%BF%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Ευκλείδειος χώρος – Greek" lang="el" hreflang="el" data-title="Ευκλείδειος χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo – Spanish" lang="es" hreflang="es" data-title="Espacio euclídeo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/E%C5%ADklida_spaco" title="Eŭklida spaco – Esperanto" lang="eo" hreflang="eo" data-title="Eŭklida spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Espazio_euklidear" title="Espazio euklidear – Basque" lang="eu" hreflang="eu" data-title="Espazio euklidear" data-language-autonym="Euskara" data-language-local-name="Basque" 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/%D9%81%D8%B6%D8%A7%DB%8C_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فضای اقلیدسی – Persian" lang="fa" hreflang="fa" data-title="فضای اقلیدسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_euclidien" title="Espace euclidien – French" lang="fr" hreflang="fr" data-title="Espace euclidien" data-language-autonym="Français" data-language-local-name="French" 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/Espazo_euclidiano" title="Espazo euclidiano – Galician" lang="gl" hreflang="gl" data-title="Espazo euclidiano" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간 – Korean" lang="ko" hreflang="ko" data-title="유클리드 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AF%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1%E0%A5%80%E0%A4%A8_%E0%A4%B8%E0%A4%AE%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BF" title="यूक्लिडीन समष्टि – Hindi" lang="hi" hreflang="hi" data-title="यूक्लिडीन समष्टि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" 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/Euklidski_prostor" title="Euklidski prostor – Croatian" lang="hr" hreflang="hr" data-title="Euklidski prostor" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Euklidana_spaco" title="Euklidana spaco – Ido" lang="io" hreflang="io" data-title="Euklidana spaco" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_Euklides" title="Ruang Euklides – Indonesian" lang="id" hreflang="id" data-title="Ruang Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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/Spazio_euclideo" title="Spazio euclideo – Italian" lang="it" hreflang="it" data-title="Spazio euclideo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="מרחב אוקלידי – Hebrew" lang="he" hreflang="he" data-title="מרחב אוקלידי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%B3%D1%96" title="Евклид кеңістігі – Kazakh" lang="kk" hreflang="kk" data-title="Евклид кеңістігі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Euklidin%C4%97_erdv%C4%97" title="Euklidinė erdvė – Lithuanian" lang="lt" hreflang="lt" data-title="Euklidinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euklideszi_t%C3%A9r_(line%C3%A1ris_algebra)" title="Euklideszi tér (lineáris algebra) – Hungarian" lang="hu" hreflang="hu" data-title="Euklideszi tér (lineáris algebra)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Евклидов простор – Macedonian" lang="mk" hreflang="mk" data-title="Евклидов простор" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AF%E0%B5%82%E0%B4%95%E0%B5%8D%E0%B4%B2%E0%B4%BF%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B5%BB_%E0%B4%B8%E0%B5%8D%E0%B4%AA%E0%B5%86%E0%B4%AF%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="യൂക്ലിഡിയൻ സ്പെയ്സ് – Malayalam" lang="ml" hreflang="ml" data-title="യൂക്ലിഡിയൻ സ്പെയ്സ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_Euclides" title="Ruang Euclides – Malay" lang="ms" hreflang="ms" data-title="Ruang Euclides" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD_%D0%B7%D0%B0%D0%B9" title="Евклидийн орон зай – Mongolian" lang="mn" hreflang="mn" data-title="Евклидийн орон зай" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9A%E1%80%B0%E1%80%80%E1%80%9C%E1%80%85%E1%80%BA%E1%80%92%E1%80%BA_%E1%80%85%E1%80%95%E1%80%B1%E1%80%B7%E1%80%85%E1%80%BA" title="ယူကလစ်ဒ် စပေ့စ် – Burmese" lang="my" hreflang="my" data-title="ယူကလစ်ဒ် စပေ့စ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Euclidische_ruimte" title="Euclidische ruimte – Dutch" lang="nl" hreflang="nl" data-title="Euclidische ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E7%A9%BA%E9%96%93" title="ユークリッド空間 – Japanese" lang="ja" hreflang="ja" data-title="ユークリッド空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Euklidsk_rom" title="Euklidsk rom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Euklidsk rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yevklid_fazosi" title="Yevklid fazosi – Uzbek" lang="uz" hreflang="uz" data-title="Yevklid fazosi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AF%E0%A9%81%E0%A8%95%E0%A8%B2%E0%A8%BF%E0%A8%A1%E0%A9%80%E0%A8%85%E0%A8%A8_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਯੁਕਲਿਡੀਅਨ ਸਪੇਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਯੁਕਲਿਡੀਅਨ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C_%D8%B3%D9%BE%DB%8C%D8%B3" title="اقلیدسی سپیس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="اقلیدسی سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_euklidesowa" title="Przestrzeń euklidesowa – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń euklidesowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_euclidiano" title="Espaço euclidiano – Portuguese" lang="pt" hreflang="pt" data-title="Espaço euclidiano" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Spa%C8%9Biu_euclidian" title="Spațiu euclidian – Romanian" lang="ro" hreflang="ro" data-title="Spațiu euclidian" data-language-autonym="Română" data-language-local-name="Romanian" 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%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Евклидово пространство – Russian" lang="ru" hreflang="ru" data-title="Евклидово пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_Euklidiane" title="Hapësira Euklidiane – Albanian" lang="sq" hreflang="sq" data-title="Hapësira Euklidiane" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BA%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%BD%E0%B7%92%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%85%E0%B7%80%E0%B6%9A%E0%B7%8F%E0%B7%81%E0%B6%BA" title="යුක්ලිඩියානු අවකාශය – Sinhala" lang="si" hreflang="si" data-title="යුක්ලිඩියානු අවකාශය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euclidean_space" title="Euclidean space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euclidean space" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Evklidski_prostor" title="Evklidski prostor – Slovenian" lang="sl" hreflang="sl" data-title="Evklidski prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%D8%A6%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="بۆشاییی ئیقلیدسی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بۆشاییی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Еуклидов простор – Serbian" lang="sr" hreflang="sr" data-title="Еуклидов простор" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Euklidski_prostor" title="Euklidski prostor – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Euklidski prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Euklidinen_avaruus" title="Euklidinen avaruus – Finnish" lang="fi" hreflang="fi" data-title="Euklidinen avaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Euklidiskt_rum" title="Euklidiskt rum – Swedish" lang="sv" hreflang="sv" data-title="Euklidiskt rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_Euclides" title="Espasyong Euclides – Tagalog" lang="tl" hreflang="tl" data-title="Espasyong Euclides" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AF%E0%AF%82%E0%AE%95%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%9F%E0%AE%BF%E0%AE%AF_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="யூக்ளிடிய வெளி – Tamil" lang="ta" hreflang="ta" data-title="யூக்ளிடிய வெளி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D1%84%D3%99%D0%B7%D0%B0%D1%81%D1%8B" title="Евклид фәзасы – Tatar" lang="tt" hreflang="tt" data-title="Евклид фәзасы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid_uzay%C4%B1" title="Öklid uzayı – Turkish" lang="tr" hreflang="tr" data-title="Öklid uzayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D1%96%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Евклідів простір – Ukrainian" lang="uk" hreflang="uk" data-title="Евклідів простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_Euclid" title="Không gian Euclid – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian Euclid" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间 – Wu" lang="wuu" hreflang="wuu" data-title="欧几里得空间" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%90%E5%B9%BE%E9%87%8C%E5%BE%97%E7%A9%BA%E9%96%93" title="歐幾里得空間 – Cantonese" lang="yue" hreflang="yue" data-title="歐幾里得空間" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间 – Chinese" lang="zh" hreflang="zh" data-title="欧几里得空间" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q17295#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</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="Namespaces"> <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/Euclidean_space" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Euclidean_space" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</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="Change language variant" > <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">English</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="Views"> <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/Euclidean_space"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Euclidean_space&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Euclidean_space&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Tools" > <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">Tools</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">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </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/Euclidean_space"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Euclidean_space&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Euclidean_space&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Euclidean_space" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Euclidean_space" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Euclidean_space&oldid=1275507369" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Euclidean_space&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Euclidean_space&id=1275507369&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEuclidean_space"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEuclidean_space"><span>Download QR code</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"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Euclidean_space&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Euclidean_space&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</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"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q17295" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Fundamental space of geometry</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Coord_system_CA_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/250px-Coord_system_CA_0.svg.png" decoding="async" width="250" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/375px-Coord_system_CA_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/500px-Coord_system_CA_0.svg.png 2x" data-file-width="620" data-file-height="600" /></a><figcaption>A point in three-dimensional Euclidean space can be located by three coordinates.</figcaption></figure> <p><b>Euclidean space</b> is the fundamental space of <a href="/wiki/Geometry" title="Geometry">geometry</a>, intended to represent <a href="/wiki/Physical_space" class="mw-redirect" title="Physical space">physical space</a>. Originally, in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a>, it was the <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, but in modern <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> there are <i>Euclidean spaces</i> of any positive integer <a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimension</a> <i>n</i>, which are called <b>Euclidean <i>n</i>-spaces</b> when one wants to specify their dimension.<sup id="cite_ref-FOOTNOTESolomentsev2001_1-0" class="reference"><a href="#cite_note-FOOTNOTESolomentsev2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For <i>n</i> equal to one or two, they are commonly called respectively <a href="/wiki/Euclidean_line" class="mw-redirect" title="Euclidean line">Euclidean lines</a> and <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean planes</a>. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">spaces</a> that were later considered in <a href="/wiki/Physics" title="Physics">physics</a> and modern mathematics. </p><p>Ancient <a href="/wiki/History_of_geometry#Greek_geometry" title="History of geometry">Greek geometers</a> introduced Euclidean space for modeling the physical space. Their work was collected by the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek</a> mathematician <a href="/wiki/Euclid" title="Euclid">Euclid</a> in his <i>Elements</i>,<sup id="cite_ref-FOOTNOTEBall196050–62_2-0" class="reference"><a href="#cite_note-FOOTNOTEBall196050–62-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> with the great innovation of <i><a href="/wiki/Mathematical_proof" title="Mathematical proof">proving</a></i> all properties of the space as <a href="/wiki/Theorem" title="Theorem">theorems</a>, by starting from a few fundamental properties, called <i><a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a></i>, which either were considered as evident (for example, there is exactly one <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> passing through two points), or seemed impossible to prove (<a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>). </p><p>After the introduction at the end of the 19th century of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>, the old postulates were re-formalized to define Euclidean spaces through <a href="/wiki/Axiomatic_theory" class="mw-redirect" title="Axiomatic theory">axiomatic theory</a>. Another definition of Euclidean spaces by means of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> and <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article.<sup id="cite_ref-FOOTNOTEBerger1987_3-0" class="reference"><a href="#cite_note-FOOTNOTEBerger1987-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. </p><p>There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a>. Therefore, it is usually possible to work with a specific Euclidean space, denoted <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 {E} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211e93d9fef6b12298e44e6237782f55c0be25e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.976ex; height:2.343ex;" alt="{\displaystyle \mathbf {E} ^{n}}"></span> or <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 {E} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8fa8586d428ff5706c6d0a00a7939950fad89b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {E} ^{n}}"></span>, which can be represented using <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a> as the <a href="/wiki/Real_n-space" class="mw-redirect" title="Real n-space">real <span class="texhtml mvar" style="font-style:italic;">n</span>-space</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> equipped with the standard <a href="/wiki/Dot_product" title="Dot product">dot product</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="History_of_the_definition">History of the definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=2" title="Edit section: History of the definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euclidean space was introduced by <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greeks</a> as an abstraction of our physical space. Their great innovation, appearing in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's <i>Elements</i></a> was to build and <i><a href="/wiki/Proof_(mathematics)" class="mw-redirect" title="Proof (mathematics)">prove</a></i> all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a>, or <a href="/wiki/Axiom" title="Axiom">axioms</a> in modern language. This way of defining Euclidean space is still in use under the name of <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>. </p><p>In 1637, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> introduced <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to <a href="/wiki/Algebra" title="Algebra">algebra</a> was a major change in point of view, as, until then, the <a href="/wiki/Real_number" title="Real number">real numbers</a> were defined in terms of lengths and distances. </p><p>Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> generalized Euclidean geometry to spaces of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>, using both synthetic and algebraic methods, and discovered all of the regular <a href="/wiki/Polytope" title="Polytope">polytopes</a> (higher-dimensional analogues of the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>) that exist in Euclidean spaces of any dimension.<sup id="cite_ref-FOOTNOTECoxeter1973_4-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Despite the wide use of Descartes' approach, which was called <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces. </p> <div class="mw-heading mw-heading3"><h3 id="Motivation_of_the_modern_definition">Motivation of the modern definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=3" title="Edit section: Motivation of the modern definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One way to think of the Euclidean plane is as a <a href="/wiki/Point_set" class="mw-redirect" title="Point set">set of points</a> satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as <a href="/wiki/Motion_(geometry)" title="Motion (geometry)">motions</a>) on the plane. One is <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two <a href="/wiki/Figure_(geometry)" class="mw-redirect" title="Figure (geometry)">figures</a> (usually considered as <a href="/wiki/Subset" title="Subset">subsets</a>) of the plane should be considered equivalent (<a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a>) if one can be transformed into the other by some sequence of translations, rotations and <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a> (see <a href="#Euclidean_group">below</a>). </p><p>In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in <a href="/wiki/Physics" title="Physics">physical</a> theories, Euclidean space is an <a href="/wiki/Abstraction" title="Abstraction">abstraction</a> detached from actual physical locations, specific <a href="/wiki/Frame_of_reference" title="Frame of reference">reference frames</a>, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of <a href="/wiki/Unit_of_length" title="Unit of length">units of length</a> and other <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">physical dimensions</a>: the distance in a "mathematical" space is a <a href="/wiki/Number" title="Number">number</a>, not something expressed in inches or metres. </p><p>The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a <a href="/wiki/Real_vector_space" class="mw-redirect" title="Real vector space">real vector space</a> <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> – the <i>space of translations</i> which is equipped with an <a href="/wiki/Inner_product_space" title="Inner product space">inner product</a>.<sup id="cite_ref-FOOTNOTESolomentsev2001_1-1" class="reference"><a href="#cite_note-FOOTNOTESolomentsev2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The action of translations makes the space an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, and this allows defining lines, planes, subspaces, dimension, and <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallelism</a>. The inner product allows defining distance and angles. </p><p>The set <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> of <span class="texhtml mvar" style="font-style:italic;">n</span>-tuples of real numbers equipped with the <a href="/wiki/Dot_product" title="Dot product">dot product</a> is a Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>. Conversely, the choice of a point called the <i>origin</i> and an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of the space of translations is equivalent with defining an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> between a Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span> and <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> viewed as a Euclidean space. </p><p><span class="anchor" id="Standard"></span>It follows that everything that can be said about a Euclidean space can also be said about <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> <mo>.</mo> </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/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}"></span> Therefore, many authors, especially at elementary level, call <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> the <i><b>standard Euclidean space</b></i> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>,<sup id="cite_ref-FOOTNOTEBerger1987Section_9.1_5-0" class="reference"><a href="#cite_note-FOOTNOTEBerger1987Section_9.1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> or simply <i>the</i> Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Blender3D_BW_Grid_256.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Blender3D_BW_Grid_256.png/220px-Blender3D_BW_Grid_256.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/45/Blender3D_BW_Grid_256.png 1.5x" data-file-width="256" data-file-height="256" /></a><figcaption>Origin-free illustration of the Euclidean plane</figcaption></figure> <p>A reason for introducing such an abstract definition of Euclidean spaces, and for working with <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 {E} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8fa8586d428ff5706c6d0a00a7939950fad89b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {E} ^{n}}"></span> instead of <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> is that it is often preferable to work in a <i>coordinate-free</i> and <i>origin-free</i> manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world. </p> <div class="mw-heading mw-heading3"><h3 id="Technical_definition">Technical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=4" title="Edit section: Technical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Euclidean_vector_space"></span><span class="vanchor-text">Euclidean vector space</span></span></b> is a finite-dimensional <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> over the <a href="/wiki/Real_number" title="Real number">real numbers</a>.<sup id="cite_ref-FOOTNOTEBerger1987Chapter_9_6-0" class="reference"><a href="#cite_note-FOOTNOTEBerger1987Chapter_9-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A <b>Euclidean space</b> is an <a href="/wiki/Affine_space" title="Affine space">affine space</a> over the <a href="/wiki/Real_number" title="Real number">reals</a> such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called <i>Euclidean affine spaces</i> to distinguish them from Euclidean vector spaces.<sup id="cite_ref-FOOTNOTEBerger1987Chapter_9_6-1" class="reference"><a href="#cite_note-FOOTNOTEBerger1987Chapter_9-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml mvar" style="font-style:italic;">E</span> is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted <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 {\overrightarrow {E}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0051204f558ccfb7c22f0e5fc78c63ce2188b27c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}.}"></span> The <i>dimension</i> of a Euclidean space is the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of its associated vector space. </p><p>The elements of <span class="texhtml mvar" style="font-style:italic;">E</span> are called <i>points</i>, and are commonly denoted by capital letters. The elements of <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> are called <i><a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</a></i> or <i><a href="/wiki/Free_vector" class="mw-redirect" title="Free vector">free vectors</a></i>. They are also called <i>translations</i>, although, properly speaking, a <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> is the <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a> resulting from the <a href="/wiki/Group_action" title="Group action">action</a> of a Euclidean vector on the Euclidean space. </p><p>The action of a translation <span class="texhtml mvar" style="font-style:italic;">v</span> on a point <span class="texhtml mvar" style="font-style:italic;">P</span> provides a point that is denoted <span class="texhtml"><i>P</i> + <i>v</i></span>. This action satisfies </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P+(v+w)=(P+v)+w.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>w</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P+(v+w)=(P+v)+w.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a8add975310f3732261bf70355c6bbd6d3dc39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.8ex; height:2.843ex;" alt="{\displaystyle P+(v+w)=(P+v)+w.}"></span> </p><p><b>Note:</b> The second <span class="texhtml">+</span> in the left-hand side is a vector addition; each other <span class="texhtml">+</span> denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of <span class="texhtml">+</span>, it suffices to look at the nature of its left argument. </p><p>The fact that the action is free and transitive means that, for every pair of points <span class="texhtml">(<i>P</i>, <i>Q</i>)</span>, there is exactly one <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">displacement vector</a> <span class="texhtml mvar" style="font-style:italic;">v</span> such that <span class="texhtml"><i>P</i> + <i>v</i> = <i>Q</i></span>. This vector <span class="texhtml mvar" style="font-style:italic;">v</span> is denoted <span class="texhtml"><i>Q</i> − <i>P</i></span> or <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 {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">)</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f583db1f1e17ac70565116ef8fdadb22e8e1facf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.815ex; width:5.176ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.}"></span> </p><p>As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in <a href="#Affine_structure">§ Affine structure</a> and its subsections. The properties resulting from the inner product are explained in <a href="#Metric_structure">§ Metric structure</a> and its subsections. </p> <div class="mw-heading mw-heading2"><h2 id="Prototypical_examples">Prototypical examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=5" title="Edit section: Prototypical examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space. </p><p>A typical case of Euclidean vector space is <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> viewed as a vector space equipped with the <a href="/wiki/Dot_product" title="Dot product">dot product</a> as an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> to it. More precisely, given a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>, the choice of a point, called an <i>origin</i> and an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> defines an isomorphism of Euclidean spaces from <span class="texhtml mvar" style="font-style:italic;">E</span> to <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> <mo>.</mo> </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/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}"></span> </p><p>As every Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span> is isomorphic to it, the Euclidean space <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> is sometimes called the <i>standard Euclidean space</i> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>.<sup id="cite_ref-FOOTNOTEBerger1987Section_9.1_5-1" class="reference"><a href="#cite_note-FOOTNOTEBerger1987Section_9.1-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Affine_structure">Affine structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=6" title="Edit section: Affine structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Affine_space" title="Affine space">Affine space</a></div> <p>Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an <a href="/wiki/Affine_space" title="Affine space">affine space</a>. They are called <a href="/wiki/Affine_property" class="mw-redirect" title="Affine property">affine properties</a> and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections. </p> <div class="mw-heading mw-heading3"><h3 id="Subspaces">Subspaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=7" title="Edit section: Subspaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Flat_(geometry)" title="Flat (geometry)">Flat (geometry)</a></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">E</span> be a Euclidean space and <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> its associated vector space. </p><p>A <i>flat</i>, <i>Euclidean subspace</i> or <i>affine subspace</i> of <span class="texhtml mvar" style="font-style:italic;">E</span> is a subset <span class="texhtml mvar" style="font-style:italic;">F</span> of <span class="texhtml mvar" style="font-style:italic;">E</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>P</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> <mi>Q</mi> <mo>∈</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4850b22390a7723fda61357997ccf5caa1c07d5c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:29.77ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}}"></span> </p><p>as the associated vector space of <span class="texhtml mvar" style="font-style:italic;">F</span> is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> (vector subspace) of <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 {\overrightarrow {E}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0051204f558ccfb7c22f0e5fc78c63ce2188b27c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}.}"></span> A Euclidean subspace <span class="texhtml mvar" style="font-style:italic;">F</span> is a Euclidean space with <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 {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2b5875f3997034668b7b97ed82ae892697ba9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {F}}}"></span> as the associated vector space. This linear subspace <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 {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2b5875f3997034668b7b97ed82ae892697ba9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {F}}}"></span> is also called the <i>direction</i> of <span class="texhtml mvar" style="font-style:italic;">F</span>. </p><p>If <span class="texhtml mvar" style="font-style:italic;">P</span> is a point of <span class="texhtml mvar" style="font-style:italic;">F</span> then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>P</mi> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbecdd4c82b1bc17ffa09d935af2624757f9be2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.883ex; height:5.176ex;" alt="{\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.}"></span> </p><p>Conversely, if <span class="texhtml mvar" style="font-style:italic;">P</span> is a point of <span class="texhtml mvar" style="font-style:italic;">E</span> and <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 {\overrightarrow {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139bd3df44a432d607ef688933115bc807b917ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {V}}}"></span> is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <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 {\overrightarrow {E}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc2d2ae2efe2b4092a3b8feac33a91c240943fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {E}},}"></span> then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>P</mi> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>v</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bbb45f835fac9af9d049450cdee9f9acad758b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.613ex; height:5.176ex;" alt="{\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}}"></span> </p><p>is a Euclidean subspace of direction <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 {\overrightarrow {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139bd3df44a432d607ef688933115bc807b917ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {V}}}"></span>. (The associated vector space of this subspace is <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 {\overrightarrow {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139bd3df44a432d607ef688933115bc807b917ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {V}}}"></span>.) </p><p>A Euclidean vector space <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> (that is, a Euclidean space that is equal to <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span>) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector. </p> <div class="mw-heading mw-heading3"><h3 id="Lines_and_segments">Lines and segments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=8" title="Edit section: Lines and segments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Line_(geometry)" title="Line (geometry)">Line (geometry)</a></div> <p>In a Euclidean space, a <i>line</i> is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>P</mi> <mo>+</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e23593a21e32705b7cfdd74f9f96760d02b215" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:23.063ex; height:5.676ex;" alt="{\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> are two distinct points of the Euclidean space as a part of the line. </p><p>It follows that <i>there is exactly one line that passes through (contains) two distinct points.</i> This implies that two distinct lines intersect in at most one point. </p><p>A more symmetric representation of the line passing through <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>O</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b693b21ffeb47db3ed54d1b9ae27adcb13b9cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:36.94ex; height:5.676ex;" alt="{\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">O</span> is an arbitrary point (not necessary on the line). </p><p>In a Euclidean vector space, the zero vector is usually chosen for <span class="texhtml mvar" style="font-style:italic;">O</span>; this allows simplifying the preceding formula into </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo>+</mo> <mi>λ</mi> <mi>Q</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7462898e76f4d9744ed30d585af91ce373594e0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.115ex; height:3.176ex;" alt="{\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.}"></span> </p><p>A standard convention allows using this formula in every Euclidean space, see <a href="/wiki/Affine_space#Affine_combinations_and_barycenter" title="Affine space">Affine space § Affine combinations and barycenter</a>. </p><p>The <i><a href="/wiki/Line_segment" title="Line segment">line segment</a></i>, or simply <i>segment</i>, joining the points <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span> is the subset of points such that <span class="texhtml">0 ≤ <i>𝜆</i> ≤ 1</span> in the preceding formulas. It is denoted <span class="texhtml mvar" style="font-style:italic;">PQ</span> or <span class="texhtml mvar" style="font-style:italic;">QP</span>; that is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>Q</mi> <mo>=</mo> <mi>Q</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>P</mi> <mo>+</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mn>0</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48128eae6b07d3e00f37e09aaac4d773bab3be32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:40.43ex; height:5.676ex;" alt="{\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Parallelism">Parallelism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=9" title="Edit section: Parallelism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel (geometry)</a></div> <p>Two subspaces <span class="texhtml mvar" style="font-style:italic;">S</span> and <span class="texhtml mvar" style="font-style:italic;">T</span> of the same dimension in a Euclidean space are <i>parallel</i> if they have the same direction (i.e., the same associated vector space).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Equivalently, they are parallel, if there is a translation vector <span class="texhtml mvar" style="font-style:italic;">v</span> that maps one to the other: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=S+v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>S</mi> <mo>+</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=S+v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8805725d38fba90073a4e871067bd83aee8b2dcc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.849ex; height:2.343ex;" alt="{\displaystyle T=S+v.}"></span> </p><p>Given a point <span class="texhtml mvar" style="font-style:italic;">P</span> and a subspace <span class="texhtml mvar" style="font-style:italic;">S</span>, there exists exactly one subspace that contains <span class="texhtml mvar" style="font-style:italic;">P</span> and is parallel to <span class="texhtml mvar" style="font-style:italic;">S</span>, which is <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 P+{\overrightarrow {S}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo>→</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P+{\overrightarrow {S}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ba6cd887aae30b32e27b4d35e4d1d82fc96930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.798ex; height:3.843ex;" alt="{\displaystyle P+{\overrightarrow {S}}.}"></span> In the case where <span class="texhtml mvar" style="font-style:italic;">S</span> is a line (subspace of dimension one), this property is <a href="/wiki/Playfair%27s_axiom" title="Playfair's axiom">Playfair's axiom</a>. </p><p>It follows that in a Euclidean plane, two lines either meet in one point or are parallel. </p><p>The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other. </p><p><span class="anchor" id="Euclidean_norm"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Metric_structure">Metric structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=10" title="Edit section: Metric structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vector space <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> associated to a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span> is an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>. This implies a <a href="/wiki/Symmetric_bilinear_form" title="Symmetric bilinear form">symmetric bilinear form</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e843bdf3ac9868ad2e0faed30591af7511e69286" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.647ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}}"></span> </p><p>that is <a href="/wiki/Positive_definite" class="mw-redirect" title="Positive definite">positive definite</a> (that is <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 x,x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f081a4822ec7bdff2661ee2070b70a3406cf70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle }"></span> is always positive for <span class="texhtml"><i>x</i> ≠ 0</span>). </p><p>The inner product of a Euclidean space is often called <i>dot product</i> and denoted <span class="texhtml"><i>x</i> ⋅ <i>y</i></span>. This is specially the case when a <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> has been chosen, as, in this case, the inner product of two vectors is the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of their <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vectors</a>. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is <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 x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> will be denoted <span class="texhtml"><i>x</i> ⋅ <i>y</i></span> in the remainder of this article. </p><p>The <b>Euclidean norm</b> of a vector <span class="texhtml mvar" style="font-style:italic;">x</span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|={\sqrt {x\cdot x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {x\cdot x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40156c4d98b51295509cdb202b8cdaba09132bc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.674ex; height:3.009ex;" alt="{\displaystyle \|x\|={\sqrt {x\cdot x}}.}"></span> </p><p>The inner product and the norm allows expressing and proving <a href="/wiki/Metric_space" title="Metric space">metric</a> and <a href="/wiki/Topology" title="Topology">topological</a> properties of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. The next subsection describe the most fundamental ones. <i>In these subsections,</i> <span class="texhtml mvar" style="font-style:italic;">E</span> <i>denotes an arbitrary Euclidean space, and <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> denotes its vector space of translations.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Distance_and_length">Distance and length</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=11" title="Edit section: Distance and length"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></div> <p>The <i>distance</i> (more precisely the <i>Euclidean distance</i>) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\Bigr \|}.{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo symmetric="true" maxsize="1.623em" minsize="1.623em">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo symmetric="true" maxsize="1.623em" minsize="1.623em">‖</mo> </mrow> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\Bigr \|}.{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13771a6a329524c6034fe78fb2de0d8e6c85563" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:19.276ex; height:5.676ex;" alt="{\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\Bigr \|}.{\vphantom {\frac {(}{}}}}"></span> </p><p>The <i>length</i> of a <a href="#Lines_and_segments">segment</a> <span class="texhtml"><i>PQ</i></span> is the distance <span class="texhtml"><i>d</i>(<i>P</i>, <i>Q</i>)</span> between its endpoints <i>P</i> and <i>Q</i>. It is often denoted <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 |PQ|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |PQ|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bce9b5b9a3507edea5dd51e5fe92e446776e4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.877ex; height:2.843ex;" alt="{\displaystyle |PQ|}"></span>. </p><p>The distance is a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, as it is positive definite, symmetric, and satisfies the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8671f5098e666ba3a4aadf5ed61f89372e7a345" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.459ex; height:2.843ex;" alt="{\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}"></span> </p><p>Moreover, the equality is true if and only if a point <span class="texhtml mvar" style="font-style:italic;">R</span> belongs to the segment <span class="texhtml"><i>PQ</i></span>. This inequality means that the length of any edge of a <a href="/wiki/Triangle" title="Triangle">triangle</a> is smaller than the sum of the lengths of the other edges. This is the origin of the term <i>triangle inequality</i>. </p><p>With the Euclidean distance, every Euclidean space is a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Orthogonality">Orthogonality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=12" title="Edit section: Orthogonality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a> and <a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></div> <p>Two nonzero vectors <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> of <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> (the associated vector space of a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span>) are <i>perpendicular</i> or <i>orthogonal</i> if their inner product is zero: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\cdot v=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>⋅</mo> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\cdot v=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/746abed16e54d81e5990af958ae5f3ac24a03a45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.397ex; height:2.176ex;" alt="{\displaystyle u\cdot v=0}"></span> </p><p>Two linear subspaces of <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector. </p><p>Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said <i>perpendicular</i>. </p><p>Two segments <span class="texhtml"><i>AB</i></span> and <span class="texhtml"><i>AC</i></span> that share a common endpoint <span class="texhtml"><i>A</i></span> are <i>perpendicular</i> or <i>form a <a href="/wiki/Right_angle" title="Right angle">right angle</a></i> if the vectors <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 {\overrightarrow {AB}}{\vphantom {\frac {)}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">)</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {)}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea379802af18b7933c6c32beb9697a8d71c8fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:5.099ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {)}{}}}}"></span> and <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 {\overrightarrow {AC}}{\vphantom {\frac {)}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">)</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AC}}{\vphantom {\frac {)}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897cb4b6f0854e47bece00856ed3560c749dddb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:5.17ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {AC}}{\vphantom {\frac {)}{}}}}"></span> are orthogonal. </p><p>If <span class="texhtml"><i>AB</i></span> and <span class="texhtml"><i>AC</i></span> form a right angle, one has </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165ec53cf622f8113176f013835a474d2d1a0319" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.176ex; height:3.343ex;" alt="{\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.}"></span> </p><p>This is the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}{\vphantom {\frac {(}{}}}\\[2mu]&={\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\cdot {\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\\[4mu]&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\[6mu]&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\[6mu]&=|AB|^{2}+|AC|^{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="0.411em 0.522em 0.633em 0.633em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>A</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>A</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>A</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>B</mi> <mi>A</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>C</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}{\vphantom {\frac {(}{}}}\\[2mu]&={\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\cdot {\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\\[4mu]&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\[6mu]&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\[6mu]&=|AB|^{2}+|AC|^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790603023e87e170e286a7d92e7233d452fb1e4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.848ex; margin-bottom: -0.323ex; width:43.643ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}{\vphantom {\frac {(}{}}}\\[2mu]&={\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\cdot {\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\\[4mu]&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\[6mu]&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\[6mu]&=|AB|^{2}+|AC|^{2}.\end{aligned}}}"></span> </p><p>Here, <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 {\overrightarrow {AB}}\cdot {\overrightarrow {AC}}=0{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AB}}\cdot {\overrightarrow {AC}}=0{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4639833383df33b1ffc22b1256a454698c1fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:14.747ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {AB}}\cdot {\overrightarrow {AC}}=0{\vphantom {\frac {(}{}}}}"></span> is used since these two vectors are orthogonal. </p> <div class="mw-heading mw-heading3"><h3 id="Angle">Angle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=13" title="Edit section: Angle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Angle" title="Angle">Angle</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:45,_-315,_and_405_co-terminal_angles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/45%2C_-315%2C_and_405_co-terminal_angles.svg/220px-45%2C_-315%2C_and_405_co-terminal_angles.svg.png" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/45%2C_-315%2C_and_405_co-terminal_angles.svg/330px-45%2C_-315%2C_and_405_co-terminal_angles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/45%2C_-315%2C_and_405_co-terminal_angles.svg/440px-45%2C_-315%2C_and_405_co-terminal_angles.svg.png 2x" data-file-width="896" data-file-height="976" /></a><figcaption>Positive and negative angles on the oriented plane</figcaption></figure> <p>The (non-oriented) <i>angle</i> <span class="texhtml mvar" style="font-style:italic;">θ</span> between two nonzero vectors <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arccos \left({\frac {x\cdot y}{|x|\,|y|}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arccos \left({\frac {x\cdot y}{|x|\,|y|}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3092c780e79339c85a5c8f387bcc14ebb96db75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.123ex; height:6.343ex;" alt="{\displaystyle \theta =\arccos \left({\frac {x\cdot y}{|x|\,|y|}}\right)}"></span> </p><p>where <span class="texhtml">arccos</span> is the <a href="/wiki/Principal_value" title="Principal value">principal value</a> of the <a href="/wiki/Arccosine" class="mw-redirect" title="Arccosine">arccosine</a> function. By <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>, the argument of the arccosine is in the interval <span class="texhtml">[−1, 1]</span>. Therefore <span class="texhtml mvar" style="font-style:italic;">θ</span> is real, and <span class="texhtml">0 ≤ <i>θ</i> ≤ <i>π</i></span> (or <span class="texhtml">0 ≤ <i>θ</i> ≤ 180</span> if angles are measured in degrees). </p><p>Angles are not useful in a Euclidean line, as they can be only 0 or <span class="texhtml mvar" style="font-style:italic;">π</span>. </p><p>In an <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">oriented</a> Euclidean plane, one can define the <i><a href="/wiki/Oriented_angle" class="mw-redirect" title="Oriented angle">oriented angle</a></i> of two vectors. The oriented angle of two vectors <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> is then the opposite of the oriented angle of <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">x</span>. In this case, the angle of two vectors can have any value <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> an integer multiple of <span class="texhtml">2<i>π</i></span>. In particular, a <a href="/wiki/Reflex_angle" class="mw-redirect" title="Reflex angle">reflex angle</a> <span class="texhtml"><i>π</i> < <i>θ</i> < 2<i>π</i></span> equals the negative angle <span class="texhtml">−<i>π</i> < <i>θ</i> − 2<i>π</i> < 0</span>. </p><p>The angle of two vectors does not change if they are <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">multiplied</a> by positive numbers. More precisely, if <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are two vectors, and <span class="texhtml mvar" style="font-style:italic;">λ</span> and <span class="texhtml mvar" style="font-style:italic;">μ</span> are real numbers, then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>angle</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>angle</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> have the same sign</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> <mo>−</mo> <mi>angle</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb0b7db5952cf104314c78a317bb73b73cfbe64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.828ex; height:6.176ex;" alt="{\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}"></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, and <span class="texhtml mvar" style="font-style:italic;">C</span> are three points in a Euclidean space, the angle of the segments <span class="texhtml"><i>AB</i></span> and <span class="texhtml"><i>AC</i></span> is the angle of the vectors <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 {\overrightarrow {AB}}{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2ecb54aefa43c40310bcb48b8bd645d5d926f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:5.099ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {(}{}}}}"></span> and <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 {\overrightarrow {AC}}.{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {AC}}.{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034cc40af64b8223081a3935e16f943b3b1bc0ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:6.203ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {AC}}.{\vphantom {\frac {(}{}}}}"></span> As the multiplication of vectors by positive numbers do not change the angle, the angle of two <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">half-lines</a> with initial point <span class="texhtml mvar" style="font-style:italic;">A</span> can be defined: it is the angle of the segments <span class="texhtml"><i>AB</i></span> and <span class="texhtml"><i>AC</i></span>, where <span class="texhtml mvar" style="font-style:italic;">B</span> and <span class="texhtml mvar" style="font-style:italic;">C</span> are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point. </p><p>The angle of two lines is defined as follows. If <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either <span class="texhtml mvar" style="font-style:italic;">θ</span> or <span class="texhtml"><i>π</i> − <i>θ</i></span>. One of these angles is in the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">[0, <i>π</i>/2]</span>, and the other being in <span class="texhtml">[<i>π</i>/2, <i>π</i>]</span>. The <i>non-oriented angle</i> of the two lines is the one in the interval <span class="texhtml">[0, <i>π</i>/2]</span>. In an oriented Euclidean plane, the <i>oriented angle</i> of two lines belongs to the interval <span class="texhtml">[−<i>π</i>/2, <i>π</i>/2]</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Cartesian_coordinates">Cartesian coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=14" title="Edit section: Cartesian coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></div> <p>Every Euclidean vector space has an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a <a href="/wiki/Basis_(vector_space)" class="mw-redirect" title="Basis (vector space)">basis</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 (e_{1},\dots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{1},\dots ,e_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ccf3398cfc17c5754c3ad49d1cb98191660797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{1},\dots ,e_{n})}"></span> of <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</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 \|e_{i}\|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|e_{i}\|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3d6d071d5a49ab2075aac7e2315e7d03c2d9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.469ex; height:2.843ex;" alt="{\displaystyle \|e_{i}\|=1}"></span>) that are pairwise orthogonal (<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 e_{i}\cdot e_{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}\cdot e_{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c518289cb3483d625b7c3de26177733fd2036c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.816ex; height:2.843ex;" alt="{\displaystyle e_{i}\cdot e_{j}=0}"></span> for <span class="texhtml"><i>i</i> ≠ <i>j</i></span>). More precisely, given any <a href="/wiki/Basis_(vector_space)" class="mw-redirect" title="Basis (vector space)">basis</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 (b_{1},\dots ,b_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{1},\dots ,b_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb950795aab63cdd91c98f8cdaf13889563b1a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.902ex; height:2.843ex;" alt="{\displaystyle (b_{1},\dots ,b_{n}),}"></span> the <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a> computes an orthonormal basis such that, for every <span class="texhtml mvar" style="font-style:italic;">i</span>, the <a href="/wiki/Linear_span" title="Linear span">linear spans</a> of <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 (e_{1},\dots ,e_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{1},\dots ,e_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1a9b068ed5d25e798ba9a4c107460dde33a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.008ex; height:2.843ex;" alt="{\displaystyle (e_{1},\dots ,e_{i})}"></span> and <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_{1},\dots ,b_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{1},\dots ,b_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f423888d30c2de535fd9185abc06cc0e049d963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.837ex; height:2.843ex;" alt="{\displaystyle (b_{1},\dots ,b_{i})}"></span> are equal.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Given a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span>, a <i><a href="/wiki/Cartesian_frame" class="mw-redirect" title="Cartesian frame">Cartesian frame</a></i> is a set of data consisting of an orthonormal basis of <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 {\overrightarrow {E}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc2d2ae2efe2b4092a3b8feac33a91c240943fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {E}},}"></span> and a point of <span class="texhtml mvar" style="font-style:italic;">E</span>, called the <i>origin</i> and often denoted <span class="texhtml mvar" style="font-style:italic;">O</span>. A Cartesian frame <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 (O,e_{1},\dots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (O,e_{1},\dots ,e_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca285afbda8f7b2accf6c570da843479986bc6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.234ex; height:2.843ex;" alt="{\displaystyle (O,e_{1},\dots ,e_{n})}"></span> allows defining Cartesian coordinates for both <span class="texhtml mvar" style="font-style:italic;">E</span> and <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> in the following way. </p><p>The Cartesian coordinates of a vector <span class="texhtml mvar" style="font-style:italic;">v</span> of <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> are the coefficients of <span class="texhtml mvar" style="font-style:italic;">v</span> on the orthonormal basis <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 e_{1},\dots ,e_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\dots ,e_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbcbf740f83c80519addd87572250282a9b9666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.265ex; height:2.009ex;" alt="{\displaystyle e_{1},\dots ,e_{n}.}"></span> For example, the Cartesian coordinates of a vector <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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> on an orthonormal basis <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 (e_{1},e_{2},e_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{1},e_{2},e_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9174b8943b9b8379276d0cf9b61fbbf21c0d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.29ex; height:2.843ex;" alt="{\displaystyle (e_{1},e_{2},e_{3})}"></span> (that may be named as <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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> as a convention) in a 3-dimensional Euclidean space is <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 (\alpha _{1},\alpha _{2},\alpha _{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a83837e8fc0d80ad0d33831a3696cde317f8c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.503ex; height:2.843ex;" alt="{\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})}"></span> if <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=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd69d0612f12e25512988496541eec1a51cef8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.946ex; height:2.343ex;" alt="{\displaystyle v=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}"></span>. As the basis is orthonormal, the <span class="texhtml mvar" style="font-style:italic;">i</span>-th coefficient <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 \alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> is equal to the dot product <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\cdot e_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>⋅</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\cdot e_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81f3936ed6eb556bfcc1e862ea1eb3d9b2bdae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.337ex; height:2.009ex;" alt="{\displaystyle v\cdot e_{i}.}"></span> </p><p>The Cartesian coordinates of a point <span class="texhtml mvar" style="font-style:italic;">P</span> of <span class="texhtml mvar" style="font-style:italic;">E</span> are the Cartesian coordinates of the vector <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 {\overrightarrow {OP}}.{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {OP}}.{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65b4a50cd233868fdd1ca1c669a0498cac1579f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:6.309ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {OP}}.{\vphantom {\frac {(}{}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Other_coordinates">Other coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=15" title="Edit section: Other coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Repere_espace.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Repere_espace.png/192px-Repere_espace.png" decoding="async" width="192" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Repere_espace.png/288px-Repere_espace.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Repere_espace.png/384px-Repere_espace.png 2x" data-file-width="442" data-file-height="319" /></a><figcaption>3-dimensional skew coordinates</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></div> <p>As a Euclidean space is an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, one can consider an <a href="/wiki/Affine_frame" class="mw-redirect" title="Affine frame">affine frame</a> on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define <a href="/wiki/Affine_coordinates" class="mw-redirect" title="Affine coordinates">affine coordinates</a>, sometimes called <i>skew coordinates</i> for emphasizing that the basis vectors are not pairwise orthogonal. </p><p>An <a href="/wiki/Affine_basis" class="mw-redirect" title="Affine basis">affine basis</a> of a Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span> is a set of <span class="texhtml"><i>n</i> + 1</span> points that are not contained in a hyperplane. An affine basis define <a href="/wiki/Barycentric_coordinates" class="mw-redirect" title="Barycentric coordinates">barycentric coordinates</a> for every point. </p><p>Many other coordinates systems can be defined on a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>, in the following way. Let <span class="texhtml mvar" style="font-style:italic;">f</span> be a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> (or, more often, a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a>) from a <a href="/wiki/Dense_subset" class="mw-redirect" title="Dense subset">dense</a> <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of <span class="texhtml mvar" style="font-style:italic;">E</span> to an open subset of <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> <mo>.</mo> </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/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}"></span> The <i>coordinates</i> of a point <span class="texhtml mvar" style="font-style:italic;">x</span> of <span class="texhtml mvar" style="font-style:italic;">E</span> are the components of <span class="texhtml"><i>f</i>(<i>x</i>)</span>. The <a href="/wiki/Polar_coordinate_system" title="Polar coordinate system">polar coordinate system</a> (dimension 2) and the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical</a> and <a href="/wiki/Cylindrical_coordinate_system" title="Cylindrical coordinate system">cylindrical</a> coordinate systems (dimension 3) are defined this way. </p><p>For points that are outside the domain of <span class="texhtml mvar" style="font-style:italic;">f</span>, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the <a href="/wiki/Antimeridian" class="mw-redirect" title="Antimeridian">antimeridian</a>, the longitude passes discontinuously from –180° to +180°. </p><p>This way of defining coordinates extends easily to other mathematical structures, and in particular to <a href="/wiki/Manifold" title="Manifold">manifolds</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Isometries">Isometries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=16" title="Edit section: Isometries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Isometry" title="Isometry">isometry</a> between two <a href="/wiki/Metric_space" title="Metric space">metric spaces</a> is a <a href="/wiki/Bijection" title="Bijection">bijection</a> preserving the distance,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> that is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f(x),f(y))=d(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f(x),f(y))=d(x,y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a5227bf8483b2930cb17d4508111bed0c8064d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.01ex; height:2.843ex;" alt="{\displaystyle d(f(x),f(y))=d(x,y).}"></span> </p><p>In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f(x)\|=\|x\|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f(x)\|=\|x\|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee480a67ac04238241cc612ab3aac82ff90f397a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.142ex; height:2.843ex;" alt="{\displaystyle \|f(x)\|=\|x\|,}"></span> </p><p>since the norm of a vector is its distance from the zero vector. It preserves also the inner product </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\cdot f(y)=x\cdot y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\cdot f(y)=x\cdot y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f029e6827dbdb7045d6fad96d7200205bf3c3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.25ex; height:2.843ex;" alt="{\displaystyle f(x)\cdot f(y)=x\cdot y,}"></span> </p><p>since </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y={\tfrac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y={\tfrac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b7f37e6fb9d952d83fcb418cf5ba5f9227637e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.101ex; height:3.509ex;" alt="{\displaystyle x\cdot y={\tfrac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}"></span> </p><p>An isometry of Euclidean vector spaces is a <a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBerger1987Proposition_9.1.3_11-0" class="reference"><a href="#cite_note-FOOTNOTEBerger1987Proposition_9.1.3-11"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>An isometry <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\colon E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon E\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e588635af63181c20287f8ea6f18389b5eaec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.443ex; height:2.509ex;" alt="{\displaystyle f\colon E\to F}"></span> of Euclidean spaces defines an isometry <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 {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627766bc0f5b60d7580e2debdad939ba6445953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.684ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"></span> of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span> are Euclidean spaces, <span class="texhtml"><i>O</i> ∈ <i>E</i></span>, <span class="texhtml"><i>O</i><span class="nowrap" style="padding-left:0.15em;">′</span> ∈ <i>F</i></span>, and <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 {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6627766bc0f5b60d7580e2debdad939ba6445953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.684ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}"></span> is an isometry, then the map <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\colon E\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon E\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e588635af63181c20287f8ea6f18389b5eaec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.443ex; height:2.509ex;" alt="{\displaystyle f\colon E\to F}"></span> defined by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(P)=O'+{\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>O</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(P)=O'+{\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4308ab829a2fc1248ea435eb33046aa69de24ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:24.083ex; height:5.676ex;" alt="{\displaystyle f(P)=O'+{\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}{\vphantom {\frac {(}{}}}}"></span> </p><p>is an isometry of Euclidean spaces. </p><p>It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces. </p> <div class="mw-heading mw-heading3"><h3 id="Isometry_with_prototypical_examples">Isometry with prototypical examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=17" title="Edit section: Isometry with prototypical examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml mvar" style="font-style:italic;">E</span> is a Euclidean space, its associated vector space <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 {\overrightarrow {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d664d9d5a70d17f42f3b274eb2e79037cf3bcae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {E}}}"></span> can be considered as a Euclidean space. Every point <span class="texhtml"><i>O</i> ∈ <i>E</i></span> defines an isometry of Euclidean spaces </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto {\overrightarrow {OP}},{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\mapsto {\overrightarrow {OP}},{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282c509bb28a8b12a4ba78b544fc1852fcb67c41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:11.669ex; height:5.676ex;" alt="{\displaystyle P\mapsto {\overrightarrow {OP}},{\vphantom {\frac {(}{}}}}"></span> </p><p>which maps <span class="texhtml mvar" style="font-style:italic;">O</span> to the zero vector and has the identity as associated linear map. The inverse isometry is the map </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mapsto O+v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">↦</mo> <mi>O</mi> <mo>+</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\mapsto O+v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79689a05dcda91cba24c86d3a1d36d56600fe29a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.13ex; height:2.343ex;" alt="{\displaystyle v\mapsto O+v.}"></span> </p><p>A Euclidean frame <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (O,e_{1},\dots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (O,e_{1},\dots ,e_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca285afbda8f7b2accf6c570da843479986bc6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.234ex; height:2.843ex;" alt="{\displaystyle (O,e_{1},\dots ,e_{n})}"></span>⁠</span> allows defining the map </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto {\Bigl (}e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}{\Bigr )},{\vphantom {\frac {(}{}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto {\Bigl (}e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}{\Bigr )},{\vphantom {\frac {(}{}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246a1ee498194709e38420703eb95c6b9bbf8baf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-right: -1.074ex; width:31.628ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto {\Bigl (}e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}{\Bigr )},{\vphantom {\frac {(}{}}}\end{aligned}}}"></span> </p><p>which is an isometry of Euclidean spaces. The inverse isometry is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow> <mo>(</mo> <mrow> <mi>O</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77121fbb42812032b1af0678f6ddf36bf344a7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.871ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}}"></span> </p><p><i>This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.</i> </p><p>This justifies that many authors talk of <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> as <i>the</i> Euclidean space of dimension <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Euclidean_group">Euclidean group <span class="anchor" id="Rotations_and_reflections"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=18" title="Edit section: Euclidean group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a> and <a href="/wiki/Rigid_transformation" title="Rigid transformation">Rigid transformation</a></div> <p>An isometry from a Euclidean space onto itself is called <i>Euclidean isometry</i>, <i>Euclidean transformation</i> or <i>rigid transformation</i>. The rigid transformations of a Euclidean space form a group (under <a href="/wiki/Function_composition" title="Function composition">composition</a>), called the <i>Euclidean group</i> and often denoted <span class="texhtml">E(<i>n</i>)</span> of <span class="texhtml">ISO(<i>n</i>)</span>. </p><p>The simplest Euclidean transformations are <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to P+v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>P</mi> <mo>+</mo> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to P+v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90daa33a19ee0a08f820506486e34b66889e4b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.72ex; height:2.343ex;" alt="{\displaystyle P\to P+v.}"></span> </p><p>They are in bijective correspondence with vectors. This is a reason for calling <i>space of translations</i> the vector space associated to a Euclidean space. The translations form a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of the Euclidean group. </p><p>A Euclidean isometry <span class="texhtml mvar" style="font-style:italic;">f</span> of a Euclidean space <span class="texhtml mvar" style="font-style:italic;">E</span> defines a linear isometry <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 {\overrightarrow {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa88ab873934603aafa809d533b4c49fc16adda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.802ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {f}}}"></span> of the associated vector space (by <i>linear isometry</i>, it is meant an isometry that is also a <a href="/wiki/Linear_map" title="Linear map">linear map</a>) in the following way: denoting by <span class="texhtml"><i>Q</i> – <i>P</i></span> the vector <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 {\overrightarrow {PQ}},{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>P</mi> <mi>Q</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {PQ}},{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8e956340c4e6cf1e3c78760b83449726bacdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:6.21ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {PQ}},{\vphantom {\frac {(}{}}}}"></span> if <span class="texhtml mvar" style="font-style:italic;">O</span> is an arbitrary point of <span class="texhtml mvar" style="font-style:italic;">E</span>, one has </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}=f(P)-f(O).{\vphantom {\frac {(}{}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mfrac> <mo stretchy="false">(</mo> <mrow /> </mfrac> </mphantom> </mpadded> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}=f(P)-f(O).{\vphantom {\frac {(}{}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b99350c1874e636de72c63948c6f03be2fa658db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -1.462ex; width:27.521ex; height:5.676ex;" alt="{\displaystyle {\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}=f(P)-f(O).{\vphantom {\frac {(}{}}}}"></span> </p><p>It is straightforward to prove that this is a linear map that does not depend from the choice of <span class="texhtml mvar" style="font-style:italic;">O.</span> </p><p>The map <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\to {\overrightarrow {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\to {\overrightarrow {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9060d749d709ae651471ae7e821f79bc6b8cf0e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.695ex; height:4.009ex;" alt="{\displaystyle f\to {\overrightarrow {f}}}"></span> is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from the Euclidean group onto the group of linear isometries, called the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group. </p><p>The isometries that fix a given point <span class="texhtml mvar" style="font-style:italic;">P</span> form the <a href="/wiki/Stabilizer_subgroup" class="mw-redirect" title="Stabilizer subgroup">stabilizer subgroup</a> of the Euclidean group with respect to <span class="texhtml mvar" style="font-style:italic;">P</span>. The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">P</span> be a point, <span class="texhtml mvar" style="font-style:italic;">f</span> an isometry, and <span class="texhtml mvar" style="font-style:italic;">t</span> the translation that maps <span class="texhtml mvar" style="font-style:italic;">P</span> to <span class="texhtml"><i>f</i>(<i>P</i>)</span>. The isometry <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 g=t^{-1}\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=t^{-1}\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73fa2538db70a7c77fdcd2b40cf3112886ba494f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.86ex; height:3.009ex;" alt="{\displaystyle g=t^{-1}\circ f}"></span> fixes <span class="texhtml mvar" style="font-style:italic;">P</span>. So <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=t\circ g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>t</mi> <mo>∘</mo> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=t\circ g,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b863d8bf2aa412652919fe83c80e151031b559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.174ex; height:2.509ex;" alt="{\displaystyle f=t\circ g,}"></span> and <i>the Euclidean group is the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of the translation group and the orthogonal group.</i> </p><p>The <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> is the normal subgroup of the orthogonal group that preserves <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">handedness</a>. It is a subgroup of <a href="/wiki/Index_(group_theory)" class="mw-redirect" title="Index (group theory)">index</a> two of the orthogonal group. Its inverse image by the group homomorphism <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\to {\overrightarrow {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\to {\overrightarrow {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9060d749d709ae651471ae7e821f79bc6b8cf0e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.695ex; height:4.009ex;" alt="{\displaystyle f\to {\overrightarrow {f}}}"></span> is a normal subgroup of index two of the Euclidean group, which is called the <i>special Euclidean group</i> or the <i>displacement group</i>. Its elements are called <i>rigid motions</i> or <i>displacements</i>. </p><p>Rigid motions include the <a href="/wiki/Identity_function" title="Identity function">identity</a>, translations, <a href="/wiki/Rotation" title="Rotation">rotations</a> (the rigid motions that fix at least a point), and also <a href="/wiki/Screw_axis" title="Screw axis">screw motions</a>. </p><p>Typical examples of rigid transformations that are not rigid motions are <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame. </p><p>As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection <span class="texhtml mvar" style="font-style:italic;">r</span>, every rigid transformation that is not a rigid motion is the product of <span class="texhtml mvar" style="font-style:italic;">r</span> and a rigid motion. A <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflection</a> is an example of a rigid transformation that is not a rigid motion or a reflection. </p><p>All groups that have been considered in this section are <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Topology">Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=19" title="Edit section: Topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Real_n-space#Topological_properties" class="mw-redirect" title="Real n-space">Real n-space § Topological properties</a></div> <p>The Euclidean distance makes a Euclidean space a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, and thus a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. This topology is called the <a href="/wiki/Euclidean_topology" title="Euclidean topology">Euclidean topology</a>. In the case of <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> <mo>,</mo> </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/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></span> this topology is also the <a href="/wiki/Product_topology" title="Product topology">product topology</a>. </p><p>The <a href="/wiki/Open_set" title="Open set">open sets</a> are the subsets that contains an <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a> around each of their points. In other words, open balls form a <a href="/wiki/Base_(topology)" title="Base (topology)">base of the topology</a>. </p><p>The <a href="/wiki/Topological_dimension" class="mw-redirect" title="Topological dimension">topological dimension</a> of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a>. Moreover, the theorem of <a href="/wiki/Invariance_of_domain" title="Invariance of domain">invariance of domain</a> asserts that a subset of a Euclidean space is open (for the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension. </p><p>Euclidean spaces are <a href="/wiki/Complete_metric" class="mw-redirect" title="Complete metric">complete</a> and <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>. That is, a closed subset of a Euclidean space is compact if it is <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> (that is, contained in a ball). In particular, closed balls are compact. </p> <div class="mw-heading mw-heading2"><h2 id="Axiomatic_definitions">Axiomatic definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=20" title="Edit section: Axiomatic definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definition of Euclidean spaces that has been described in this article differs fundamentally of <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>. </p><p>Two different approaches have been used. <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> suggested to define geometries through their <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a>. The presentation of Euclidean spaces given in this article, is essentially issued from his <a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a>, with the emphasis given on the groups of translations and isometries. </p><p>On the other hand, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> proposed a set of <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">axioms</a>, inspired by <a href="/wiki/Euclid%27s_postulates" class="mw-redirect" title="Euclid's postulates">Euclid's postulates</a>. They belong to <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>, as they do not involve any definition of <a href="/wiki/Real_number" title="Real number">real numbers</a>. Later <a href="/wiki/G._D._Birkhoff" class="mw-redirect" title="G. D. Birkhoff">G. D. Birkhoff</a> and <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> proposed simpler sets of axioms, which use <a href="/wiki/Real_number" title="Real number">real numbers</a> (see <a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's axioms</a> and <a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a>). </p><p>In <i><a href="/wiki/Geometric_Algebra_(book)" title="Geometric Algebra (book)">Geometric Algebra</a></i>, <a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> has proved that all these definitions of a Euclidean space are equivalent.<sup id="cite_ref-FOOTNOTEArtin1988_12-0" class="reference"><a href="#cite_note-FOOTNOTEArtin1988-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruence</a> is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on segments. One can thus define the <i>length</i> of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert. </p> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=21" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greeks</a>, Euclidean space has been used for modeling <a href="/wiki/Shape" title="Shape">shapes</a> in the physical world. It is thus used in many <a href="/wiki/Science" title="Science">sciences</a>, such as <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, and <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as <a href="/wiki/Architecture" title="Architecture">architecture</a>, <a href="/wiki/Geodesy" title="Geodesy">geodesy</a>, <a href="/wiki/Topography" title="Topography">topography</a>, <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Industrial_design" title="Industrial design">industrial design</a>, or <a href="/wiki/Technical_drawing" title="Technical drawing">technical drawing</a>. </p><p>Space of dimensions higher than three occurs in several modern theories of physics; see <a href="/wiki/Higher_dimension" class="mw-redirect" title="Higher dimension">Higher dimension</a>. They occur also in <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration spaces</a> of <a href="/wiki/Physical_system" title="Physical system">physical systems</a>. </p><p>Beside <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, Euclidean spaces are also widely used in other areas of mathematics. <a href="/wiki/Tangent_space" title="Tangent space">Tangent spaces</a> of <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a> are Euclidean vector spaces. More generally, a <a href="/wiki/Manifold" title="Manifold">manifold</a> is a space that is locally approximated by Euclidean spaces. Most <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a> can be modeled by a manifold, and <a href="/wiki/Embedding" title="Embedding">embedded</a> in a Euclidean space of higher dimension. For example, an <a href="/wiki/Elliptic_space" class="mw-redirect" title="Elliptic space">elliptic space</a> can be modeled by an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a>. It is common to represent in a Euclidean space mathematical objects that are <i>a priori</i> not of a geometrical nature. An example among many is the usual representation of <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_geometric_spaces">Other geometric spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=22" title="Edit section: Other geometric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/50px-Split-arrows.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/75px-Split-arrows.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/100px-Split-arrows.svg.png 2x" data-file-width="60" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">It has been suggested that this section be <a href="/wiki/Wikipedia:Splitting" title="Wikipedia:Splitting">split</a> out into another article titled <i><a href="/wiki/Geometric_space" class="mw-redirect" title="Geometric space">Geometric space</a></i>. (<a href="/wiki/Talk:Geometric_space#Splitting_from_Euclidean_n-space" title="Talk:Geometric space">Discuss</a>) <small><i>(March 2023)</i></small></div></td></tr></tbody></table> <p>Since the introduction, at the end of 19th century, of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical <a href="/wiki/Axiom" title="Axiom">axioms</a>, <a href="/wiki/Embedding" title="Embedding">embedding</a> the space in a Euclidean space is a standard way for proving <a href="/wiki/Consistency" title="Consistency">consistency</a> of its definition, or, more precisely for proving that its theory is consistent, if <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is consistent (which cannot be proved). </p> <div class="mw-heading mw-heading3"><h3 id="Affine_space">Affine space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=23" title="Edit section: Affine space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Affine_space" title="Affine space">Affine space</a></div> <p>A Euclidean space is an affine space equipped with a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>. Affine spaces have many other uses in mathematics. In particular, as they are defined over any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, they allow doing geometry in other contexts. </p><p>As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> as an extension of Euclidean spaces. For example, a <a href="/wiki/Circle" title="Circle">circle</a> and a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> is built in complex affine spaces and affine spaces over <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed fields</a>. The shapes that are studied in algebraic geometry in these affine spaces are therefore called <a href="/wiki/Affine_algebraic_variety" class="mw-redirect" title="Affine algebraic variety">affine algebraic varieties</a>. </p><p>Affine spaces over the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> and more generally over <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> provide a link between (algebraic) geometry and <a href="/wiki/Number_theory" title="Number theory">number theory</a>. For example, the <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> can be stated "a <a href="/wiki/Fermat_curve" title="Fermat curve">Fermat curve</a> of degree higher than two has no point in the affine plane over the rationals." </p><p>Geometry in affine spaces over a <a href="/wiki/Finite_fields" class="mw-redirect" title="Finite fields">finite fields</a> has also been widely studied. For example, <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a> over finite fields are widely used in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Projective_space">Projective space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=24" title="Edit section: Projective space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Projective_space" title="Projective space">Projective space</a></div> <p>Originally, projective spaces have been introduced by adding "<a href="/wiki/Points_at_infinity" class="mw-redirect" title="Points at infinity">points at infinity</a>" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two <a href="/wiki/Coplanar" class="mw-redirect" title="Coplanar">coplanar</a> lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a>, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the <a href="/wiki/Vector_line" class="mw-redirect" title="Vector line">vector lines</a> in a <a href="/wiki/Vector_space" title="Vector space">vector space</a> of dimension one more. </p><p>As for affine spaces, projective spaces are defined over any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, and are fundamental spaces of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Non-Euclidean_geometries">Non-Euclidean geometries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=25" title="Edit section: Non-Euclidean geometries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a></div> <p><i>Non-Euclidean geometry</i> refers usually to geometrical spaces where the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> is false. They include <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>, where the sum of the angles of a triangle is more than 180°, and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is <a href="/wiki/Consistency" title="Consistency">consistent</a> (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the <a href="/wiki/Foundational_crisis_in_mathematics" class="mw-redirect" title="Foundational crisis in mathematics">foundational crisis in mathematics</a> of the beginning of 20th century, and motivated the systematization of <a href="/wiki/Axiomatic_theory" class="mw-redirect" title="Axiomatic theory">axiomatic theories</a> in mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Curved_spaces">Curved spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=26" title="Edit section: Curved spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Curved_space" title="Curved space">Curved space</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Manifold" title="Manifold">Manifold</a> and <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></div> <p>A <a href="/wiki/Manifold" title="Manifold">manifold</a> is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, such that each point has a <a href="/wiki/Neighborhood" class="mw-redirect" title="Neighborhood">neighborhood</a> that is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifolds</a>, <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>, <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a>, and <a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic manifolds</a>. However, none of these types of "resemblance" respect distances and angles, even approximately. </p><p>Distances and angles can be defined on a smooth manifold by providing a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smoothly varying</a> Euclidean metric on the <a href="/wiki/Tangent_space" title="Tangent space">tangent spaces</a> at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. Generally, <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight lines</a> do not exist in a Riemannian manifold, but their role is played by <a href="/wiki/Geodesic" title="Geodesic">geodesics</a>, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent. </p><p>Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a <a href="/wiki/Sphere" title="Sphere">sphere</a>. In this case, geodesics are <a href="/wiki/Great_circle" title="Great circle">arcs of great circle</a>, which are called <a href="/wiki/Orthodrome" class="mw-redirect" title="Orthodrome">orthodromes</a> in the context of <a href="/wiki/Navigation" title="Navigation">navigation</a>. More generally, the spaces of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a> can be realized as Riemannian manifolds. </p> <div class="mw-heading mw-heading3"><h3 id="Pseudo-Euclidean_space">Pseudo-Euclidean space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=27" title="Edit section: Pseudo-Euclidean space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> of a real vector space is a <a href="/wiki/Positive_definite_bilinear_form" class="mw-redirect" title="Positive definite bilinear form">positive definite bilinear form</a>, and so characterized by a <a href="/wiki/Bilinear_form#Derived_quadratic_form" title="Bilinear form">positive definite quadratic form</a>. A <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a> is an affine space with an associated real vector space equipped with a <a href="/wiki/Non-degenerate" class="mw-redirect" title="Non-degenerate">non-degenerate</a> <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> (that may be <a href="/wiki/Indefinite_quadratic_form" class="mw-redirect" title="Indefinite quadratic form">indefinite</a>). </p><p>A fundamental example of such a space is the <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, which is the <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a> of <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. It is a four-dimensional space, where the metric is defined by the <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7aae8805cc9e3217e23eb11a985354987f7bc25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.805ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}"></span> </p><p>where the last coordinate (<i>t</i>) is temporal, and the other three (<i>x</i>, <i>y</i>, <i>z</i>) are spatial. </p><p>To take <a href="/wiki/Gravity" title="Gravity">gravity</a> into account, <a href="/wiki/General_relativity" title="General relativity">general relativity</a> uses a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> that has Minkowski spaces as <a href="/wiki/Tangent_space" title="Tangent space">tangent spaces</a>. The <a href="/wiki/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">curvature</a> of this manifold at a point is a function of the value of the <a href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</a> at this point. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, a generalization to <a href="/wiki/Infinity" title="Infinity">infinite</a> <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a>, used in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a></li> <li><a href="/wiki/Position_space" class="mw-redirect" title="Position space">Position space</a>, an application in <a href="/wiki/Physics" title="Physics">physics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=29" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">It may depend on the context or the author whether a subspace is parallel to itself</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">If the condition of being a bijection is removed, a function preserving the distance is necessarily injective, and is an isometry from its domain to its image.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Proof: one must prove that <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(\lambda x+\mu y)-\lambda f(x)-\mu f(y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>x</mi> <mo>+</mo> <mi>μ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\lambda x+\mu y)-\lambda f(x)-\mu f(y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f497c938958d2108a27505eb8e3030d5ecd03311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.53ex; height:2.843ex;" alt="{\displaystyle f(\lambda x+\mu y)-\lambda f(x)-\mu f(y)=0}"></span>. For that, it suffices to prove that the square of the norm of the left-hand side is zero. Using the bilinearity of the inner product, this squared norm can be expanded into a linear combination of <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(x)\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f(x)\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7343c3097cb2c6a3825105c266bc9903a43effc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.444ex; height:3.176ex;" alt="{\displaystyle \|f(x)\|^{2},}"></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 \|f(y)\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f(y)\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af814df703194e7124723051fc7bf0c876634d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:3.176ex;" alt="{\displaystyle \|f(y)\|^{2},}"></span> and <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(x)\cdot f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\cdot f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd6d031172f3e8e377badc634416698f81f684a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.987ex; height:2.843ex;" alt="{\displaystyle f(x)\cdot f(y).}"></span> As <span class="texhtml mvar" style="font-style:italic;">f</span> is an isometry, this gives a linear combination of <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 \|x\|^{2},\|y\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|^{2},\|y\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6a607d218ede38ad69c32158fa5ba89366ac50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.924ex; height:3.176ex;" alt="{\displaystyle \|x\|^{2},\|y\|^{2},}"></span> and <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 x\cdot y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699802977820dd08f3741b402f6a5d2ae097dc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.811ex; height:2.009ex;" alt="{\displaystyle x\cdot y,}"></span> which simplifies to zero.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euclidean_space&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTESolomentsev2001-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESolomentsev2001_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESolomentsev2001_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSolomentsev2001">Solomentsev 2001</a>.</span> </li> <li id="cite_note-FOOTNOTEBall196050–62-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBall196050–62_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBall1960">Ball 1960</a>, pp. 50–62.</span> </li> <li id="cite_note-FOOTNOTEBerger1987-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerger1987_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerger1987">Berger 1987</a>.</span> </li> <li id="cite_note-FOOTNOTECoxeter1973-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1973_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter 1973</a>.</span> </li> <li id="cite_note-FOOTNOTEBerger1987Section_9.1-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBerger1987Section_9.1_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerger1987Section_9.1_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBerger1987">Berger 1987</a>, Section 9.1.</span> </li> <li id="cite_note-FOOTNOTEBerger1987Chapter_9-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBerger1987Chapter_9_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerger1987Chapter_9_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBerger1987">Berger 1987</a>, Chapter 9.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFAnton1987">Anton (1987</a>, pp. 209–215)</span> </li> <li id="cite_note-FOOTNOTEBerger1987Proposition_9.1.3-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerger1987Proposition_9.1.3_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerger1987">Berger 1987</a>, Proposition 9.1.3.</span> </li> <li id="cite_note-FOOTNOTEArtin1988-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEArtin1988_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFArtin1988">Artin 1988</a>.</span> </li> </ol></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAnton1987" class="citation cs2">Anton, Howard (1987), <i>Elementary Linear Algebra</i> (5th ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-84819-0" title="Special:BookSources/0-471-84819-0"><bdi>0-471-84819-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.place=New+York&rft.edition=5th&rft.pub=Wiley&rft.date=1987&rft.isbn=0-471-84819-0&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1988" class="citation cs2">Artin, Emil (1988) [1957], <i><a href="/wiki/Geometric_Algebra_(book)" title="Geometric Algebra (book)">Geometric Algebra</a></i>, Wiley Classics Library, New York: John Wiley & Sons Inc., pp. x+214, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781118164518">10.1002/9781118164518</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-60839-4" title="Special:BookSources/0-471-60839-4"><bdi>0-471-60839-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1009557">1009557</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Algebra&rft.place=New+York&rft.series=Wiley+Classics+Library&rft.pages=x%2B214&rft.pub=John+Wiley+%26+Sons+Inc.&rft.date=1988&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1009557%23id-name%3DMR&rft_id=info%3Adoi%2F10.1002%2F9781118164518&rft.isbn=0-471-60839-4&rft.aulast=Artin&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall1960" class="citation book cs1"><a href="/wiki/W._W._Rouse_Ball" title="W. W. Rouse Ball">Ball, W.W. Rouse</a> (1960) [1908]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi0000ball"><i>A Short Account of the History of Mathematics</i></a></span> (4th ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-20630-0" title="Special:BookSources/0-486-20630-0"><bdi>0-486-20630-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Short+Account+of+the+History+of+Mathematics&rft.edition=4th&rft.pub=Dover+Publications&rft.date=1960&rft.isbn=0-486-20630-0&rft.aulast=Ball&rft.aufirst=W.W.+Rouse&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshortaccountofhi0000ball&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger1987" class="citation cs2"><a href="/wiki/Marcel_Berger" title="Marcel Berger">Berger, Marcel</a> (1987), <i>Geometry I</i>, Berlin: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-11658-3" title="Special:BookSources/3-540-11658-3"><bdi>3-540-11658-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+I&rft.place=Berlin&rft.pub=Springer&rft.date=1987&rft.isbn=3-540-11658-3&rft.aulast=Berger&rft.aufirst=Marcel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1973" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1973) [1948]. <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)"><i>Regular Polytopes</i></a> (3rd ed.). New York: Dover. <q>Schläfli ... discovered them before 1853 -- a time when Cayley, Grassman and Möbius were the only other people who had ever conceived of the possibility of geometry in more than three dimensions.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Polytopes&rft.place=New+York&rft.edition=3rd&rft.pub=Dover&rft.date=1973&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomentsev2001" class="citation cs2">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Euclidean_space">"Euclidean space"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Euclidean+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Solomentsev&rft.aufirst=E.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DEuclidean_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEuclidean+space" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.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}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Dimension155" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.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}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Dimension_topics" title="Template:Dimension topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Dimension_topics" title="Template talk:Dimension topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Dimension_topics" title="Special:EditPage/Template:Dimension topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Dimension155" style="font-size:114%;margin:0 4em"><a href="/wiki/Dimension" title="Dimension">Dimension</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensional spaces</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/Dimension_(vector_space)" title="Dimension (vector space)">Vector space</a></li> <li><a class="mw-selflink selflink">Euclidean space</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine space</a></li> <li><a href="/wiki/Projective_space" title="Projective space">Projective space</a></li> <li><a href="/wiki/Free_module" title="Free module">Free module</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</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/File:Tesseract.gif" class="mw-file-description" title="Animated tesseract"><img alt="Animated tesseract" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/75px-Tesseract.gif" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/113px-Tesseract.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/150px-Tesseract.gif 2x" data-file-width="256" data-file-height="256" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other dimensions</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/Krull_dimension" title="Krull dimension">Krull</a></li> <li><a href="/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering</a></li> <li><a href="/wiki/Inductive_dimension" title="Inductive dimension">Inductive</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li> <li><a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Minkowski</a></li> <li><a href="/wiki/Fractal_dimension" title="Fractal dimension">Fractal</a></li> <li><a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">Degrees of freedom</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polytope" title="Polytope">Polytopes</a> and <a href="/wiki/Shape" title="Shape">shapes</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/Hyperplane" title="Hyperplane">Hyperplane</a></li> <li><a href="/wiki/Hypersurface" title="Hypersurface">Hypersurface</a></li> <li><a href="/wiki/Hypercube" title="Hypercube">Hypercube</a></li> <li><a href="/wiki/Hyperrectangle" title="Hyperrectangle">Hyperrectangle</a></li> <li><a href="/wiki/Demihypercube" title="Demihypercube">Demihypercube</a></li> <li><a href="/wiki/N-sphere" title="N-sphere">Hypersphere</a></li> <li><a href="/wiki/Cross-polytope" title="Cross-polytope">Cross-polytope</a></li> <li><a href="/wiki/Simplex" title="Simplex">Simplex</a></li> <li><a href="/wiki/Hyperpyramid" title="Hyperpyramid">Hyperpyramid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Number systems</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/Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a></li> <li><a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensions by number</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/Zero-dimensional_space" title="Zero-dimensional space">Zero</a></li> <li><a href="/wiki/One-dimensional_space" title="One-dimensional space">One</a></li> <li><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two</a></li> <li><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three</a></li> <li><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a></li> <li><a href="/wiki/Five-dimensional_space" title="Five-dimensional space">Five</a></li> <li><a href="/wiki/Six-dimensional_space" title="Six-dimensional space">Six</a></li> <li><a href="/wiki/Seven-dimensional_space" title="Seven-dimensional space">Seven</a></li> <li><a href="/wiki/Eight-dimensional_space" title="Eight-dimensional space">Eight</a></li> <li><a href="/wiki/Dimension" title="Dimension"><i>n</i>-dimensions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</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/Hyperspace" title="Hyperspace">Hyperspace</a></li> <li><a href="/wiki/Codimension" title="Codimension">Codimension</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><a href="/wiki/Category:Dimension" title="Category:Dimension">Category</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1379" 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/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q17295#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/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/en/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/en/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"><span class="rt-commentedText tooltip tooltip-dotted" title="euklidischer Raum"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4309127-1">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Espaces euclidiens"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb122864798">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Espaces euclidiens"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb122864798">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00562065">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="euklidovské prostory"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph120032&CON_LNG=ENG">Czech Republic</a></span></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Euclidean_space&oldid=1275507369">https://en.wikipedia.org/w/index.php?title=Euclidean_space&oldid=1275507369</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Euclidean_geometry" title="Category:Euclidean geometry">Euclidean geometry</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Homogeneous_spaces" title="Category:Homogeneous spaces">Homogeneous spaces</a></li><li><a href="/wiki/Category:Norms_(mathematics)" title="Category:Norms (mathematics)">Norms (mathematics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_to_be_split_from_March_2023" title="Category:Articles to be split from March 2023">Articles to be split from March 2023</a></li><li><a href="/wiki/Category:All_articles_to_be_split" title="Category:All articles to be split">All articles to be split</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"> This page was last edited on 13 February 2025, at 13:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Euclidean_space&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</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" lang="en" 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"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div 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="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Euclidean space</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>64 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </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-b766959bd-jld8k","wgBackendResponseTime":122,"wgPageParseReport":{"limitreport":{"cputime":"0.651","walltime":"4.556","ppvisitednodes":{"value":6564,"limit":1000000},"postexpandincludesize":{"value":58309,"limit":2097152},"templateargumentsize":{"value":7314,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":21,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":48710,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 572.463 1 -total"," 13.70% 78.441 3 Template:Citation"," 12.31% 70.442 1 Template:Short_description"," 11.94% 68.380 1 Template:Dimension_topics"," 11.65% 66.673 1 Template:Navbox"," 8.94% 51.204 45 Template:Math"," 8.81% 50.443 11 Template:Sfn"," 8.43% 48.275 2 Template:Pagetype"," 7.03% 40.252 1 Template:Split_section"," 7.01% 40.157 1 Template:Authority_control"]},"scribunto":{"limitreport-timeusage":{"value":"0.318","limit":"10.000"},"limitreport-memusage":{"value":7973595,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAnton1987\"] = 1,\n [\"CITEREFArtin1988\"] = 1,\n [\"CITEREFBall1960\"] = 1,\n [\"CITEREFBerger1987\"] = 1,\n [\"CITEREFCoxeter1973\"] = 1,\n [\"CITEREFSolomentsev\"] = 1,\n [\"Euclidean_norm\"] = 1,\n [\"Rotations_and_reflections\"] = 1,\n [\"Standard\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Anchor\"] = 3,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 3,\n [\"Cite book\"] = 2,\n [\"Closed-closed\"] = 5,\n [\"Dimension topics\"] = 1,\n [\"Efn\"] = 3,\n [\"Further\"] = 1,\n [\"Harvtxt\"] = 1,\n [\"Lambda\"] = 1,\n [\"Main\"] = 9,\n [\"Main article\"] = 5,\n [\"Math\"] = 40,\n [\"Mvar\"] = 112,\n [\"Notelist\"] = 1,\n [\"Pi\"] = 1,\n [\"Portal\"] = 1,\n [\"Prime\"] = 1,\n [\"Reflist\"] = 1,\n [\"See also\"] = 1,\n [\"Sfn\"] = 11,\n [\"Short description\"] = 1,\n [\"Slink\"] = 3,\n [\"Split section\"] = 1,\n [\"Springer\"] = 1,\n [\"Tmath\"] = 1,\n [\"Vanchor\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-b766959bd-49j22","timestamp":"20250214040527","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Euclidean space","url":"https:\/\/en.wikipedia.org\/wiki\/Euclidean_space","sameAs":"http:\/\/www.wikidata.org\/entity\/Q17295","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q17295","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-08-27T19:16:37Z","dateModified":"2025-02-13T13:07:50Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/6\/69\/Coord_system_CA_0.svg","headline":"generalization of Euclidean geometry to higher-dimensional vector spaces"}</script> </body> </html>

CINXE.COM

Euclidean space - Wikipedia