Euklidska udaljenost

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id="toc-Jednodimenzionalna_udaljenost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dvodimenzionalna_udaljenost" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dvodimenzionalna_udaljenost"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Dvodimenzionalna udaljenost</span> </div> </a> <ul id="toc-Dvodimenzionalna_udaljenost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trodimenzionalna_udaljenost" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trodimenzionalna_udaljenost"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Trodimenzionalna udaljenost</span> </div> </a> <ul id="toc-Trodimenzionalna_udaljenost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-n_-_domenzionalna_udaljenost" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#n_-_domenzionalna_udaljenost"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>n - domenzionalna udaljenost</span> </div> </a> <ul id="toc-n_-_domenzionalna_udaljenost-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kvadrat_Euklidske_udaljenosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kvadrat_Euklidske_udaljenosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Kvadrat Euklidske udaljenosti</span> </div> </a> <ul id="toc-Kvadrat_Euklidske_udaljenosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Uključi/isključi sadržaj" > <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">Uključi/isključi sadržaj</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">Euklidska udaljenost</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="Idi na članak na drugom jeziku. Dostupno na 36 jezika" > <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-36" 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">36 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D9%81%D8%A9_%D8%A5%D9%82%D9%84%D9%8A%D8%AF%D9%8A%D8%A9" title="مسافة إقليدية – arapski" lang="ar" hreflang="ar" data-title="مسافة إقليدية" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Evklid_m%C9%99saf%C9%99si" title="Evklid məsafəsi – azerbejdžanski" lang="az" hreflang="az" data-title="Evklid məsafəsi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dist%C3%A0ncia_euclidiana" title="Distància euclidiana – katalonski" lang="ca" hreflang="ca" data-title="Distància euclidiana" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidovsk%C3%A1_metrika" title="Eukleidovská metrika – češki" lang="cs" hreflang="cs" data-title="Eukleidovská metrika" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BB%D0%B0_%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0" title="Евклидла метрика – čuvaški" lang="cv" hreflang="cv" data-title="Евклидла метрика" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Euklidischer_Abstand" title="Euklidischer Abstand – njemački" lang="de" hreflang="de" data-title="Euklidischer Abstand" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1_%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AE" title="Ευκλείδεια μετρική – grčki" lang="el" hreflang="el" data-title="Ευκλείδεια μετρική" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="dobar članak"><a href="https://en.wikipedia.org/wiki/Euclidean_distance" title="Euclidean distance – engleski" lang="en" hreflang="en" data-title="Euclidean distance" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/E%C5%ADklida_distanco" title="Eŭklida distanco – esperanto" lang="eo" hreflang="eo" data-title="Eŭklida distanco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distancia_euclidiana" title="Distancia euclidiana – španski" lang="es" hreflang="es" data-title="Distancia euclidiana" data-language-autonym="Español" data-language-local-name="španski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Distantzia_euklidear" title="Distantzia euklidear – baskijski" lang="eu" hreflang="eu" data-title="Distantzia euklidear" data-language-autonym="Euskara" data-language-local-name="baskijski" 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%A7%D8%B5%D9%84%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فاصله اقلیدسی – perzijski" lang="fa" hreflang="fa" data-title="فاصله اقلیدسی" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Euklidinen_metriikka" title="Euklidinen metriikka – finski" lang="fi" hreflang="fi" data-title="Euklidinen metriikka" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distance_euclidienne" title="Distance euclidienne – francuski" lang="fr" hreflang="fr" data-title="Distance euclidienne" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jarak_Euklides" title="Jarak Euklides – indonezijski" lang="id" hreflang="id" data-title="Jarak Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" 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/Distanza_euclidea" title="Distanza euclidea – italijanski" lang="it" hreflang="it" data-title="Distanza euclidea" data-language-autonym="Italiano" data-language-local-name="italijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E8%B7%9D%E9%9B%A2" title="ユークリッド距離 – japanski" lang="ja" hreflang="ja" data-title="ユークリッド距離" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</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%B1%B0%EB%A6%AC" title="유클리드 거리 – korejski" lang="ko" hreflang="ko" data-title="유클리드 거리" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Euklidinis_atstumas" title="Euklidinis atstumas – litvanski" lang="lt" hreflang="lt" data-title="Euklidinis atstumas" data-language-autonym="Lietuvių" data-language-local-name="litvanski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Gewone_metriek" title="Gewone metriek – nizozemski" lang="nl" hreflang="nl" data-title="Gewone metriek" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dist%C3%A2ncia_euclidiana" title="Distância euclidiana – portugalski" lang="pt" hreflang="pt" data-title="Distância euclidiana" data-language-autonym="Português" data-language-local-name="portugalski" 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/Distan%C8%9B%C4%83_euclidian%C4%83" title="Distanță euclidiană – rumunski" lang="ro" hreflang="ro" data-title="Distanță euclidiană" data-language-autonym="Română" data-language-local-name="rumunski" 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%B0_%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0" title="Евклидова метрика – ruski" lang="ru" hreflang="ru" data-title="Евклидова метрика" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Distanza" title="Distanza – sicilijanski" lang="scn" hreflang="scn" data-title="Distanza" data-language-autonym="Sicilianu" data-language-local-name="sicilijanski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Euklidska_udaljenost" title="Euklidska udaljenost – srpskohrvatski" lang="sh" hreflang="sh" data-title="Euklidska udaljenost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpskohrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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%AF%E0%B7%94%E0%B6%BB" title="යුක්ලිඩියානු දුර – sinhaleški" lang="si" hreflang="si" data-title="යුක්ලිඩියානු දුර" data-language-autonym="සිංහල" data-language-local-name="sinhaleški" 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_distance" title="Euclidean distance – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euclidean distance" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D1%81%D0%BA%D0%B0_%D1%80%D0%B0%D0%B7%D0%B4%D0%B0%D1%99%D0%B8%D0%BD%D0%B0" title="Еуклидска раздаљина – srpski" lang="sr" hreflang="sr" data-title="Еуклидска раздаљина" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</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%A4%E0%AF%8A%E0%AE%B2%E0%AF%88%E0%AE%B5%E0%AF%81" title="யூக்ளிடிய தொலைவு – tamilski" lang="ta" hreflang="ta" data-title="யூக்ளிடிய தொலைவு" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%A2%E0%B8%B0%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%A2%E0%B8%B8%E0%B8%84%E0%B8%A5%E0%B8%B4%E0%B8%94" title="ระยะทางแบบยุคลิด – tajlandski" lang="th" hreflang="th" data-title="ระยะทางแบบยุคลิด" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid_uzakl%C4%B1%C4%9F%C4%B1" title="Öklid uzaklığı – turski" lang="tr" hreflang="tr" data-title="Öklid uzaklığı" data-language-autonym="Türkçe" data-language-local-name="turski" 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%D0%BE%D0%B2%D0%B0_%D0%B2%D1%96%D0%B4%D1%81%D1%82%D0%B0%D0%BD%D1%8C" title="Евклідова відстань – ukrajinski" lang="uk" hreflang="uk" data-title="Евклідова відстань" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="فاصلہ (ریاضی) – urdu" lang="ur" hreflang="ur" data-title="فاصلہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="dobar članak"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_c%C3%A1ch_Euclid" title="Khoảng cách Euclid – vijetnamski" lang="vi" hreflang="vi" data-title="Khoảng cách Euclid" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link 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data-event-name="pinnable-header.vector-appearance.unpin">sakrij</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">S Wikipedije, slobodne enciklopedije</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="bs" dir="ltr"><p><b>Euklidska udaljenost</b> je najkraći razmak između dvije <a href="/wiki/Ta%C4%8Dka_(geometrija)" title="Tačka (geometrija)">tačke</a> u jednom <a href="/wiki/Prostor" title="Prostor">prostoru</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> U jednoj <a href="/wiki/Ravan" title="Ravan">ravni</a> je, primjera radi, definisana po <a href="/wiki/Pitagorina_teorema" title="Pitagorina teorema">Pitagorinoj teoremi</a><sup id="cite_ref-Matheprisma_Uni_Wuppertal_2-0" class="reference"><a href="#cite_note-Matheprisma_Uni_Wuppertal-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicija">Definicija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=1" title="Uredi odlomak "Definicija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=1" title="Edit section's source code: Definicija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Euklidova udaljenost između tačaka p i q je dužina segmenta linije koja ih povezuje (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {p} \mathbf {q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {p} \mathbf {q} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d397a90d8e00a9fbb6e7eb908cda31009fde6ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.019ex; height:2.676ex;" alt="{\displaystyle {\overline {\mathbf {p} \mathbf {q} }}}"></span>). </p><p>U Kartezijevim koordinatama, ako su <span class="mwe-math-element"><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=(p_{1},p_{2},...,p_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(p_{1},p_{2},...,p_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f8f8aa65b609b46a993e139ac8d0cb5c91f17a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.205ex; height:2.843ex;" alt="{\displaystyle p=(p_{1},p_{2},...,p_{n})}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=(q_{1},q_{2},...,q_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=(q_{1},q_{2},...,q_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1932eac73ea75b862672099d7d36018bd9151dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.619ex; height:2.843ex;" alt="{\displaystyle q=(q_{1},q_{2},...,q_{n})}"></span> dvije tačke Euklidskog n-prostora, onda je udaljenost (d) od P do Q ili od Q do P data pomoću Pitagorine formule: </p> <dl><dd> <table style="border-collapse:collapse; background:none; margin:0; border:none;"> <tbody><tr> <td style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap"><p style="margin:0;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {d} (\mathbf {p} ,\mathbf {q} )=\mathrm {d} (\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8pt]&={\sqrt {\sum _{i=1}^{n}(q_{i}-p_{i})^{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="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} (\mathbf {p} ,\mathbf {q} )=\mathrm {d} (\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8pt]&={\sqrt {\sum _{i=1}^{n}(q_{i}-p_{i})^{2}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0281a964ec758cca02ab9ef91a7f54ac00d4b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:64.975ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} (\mathbf {p} ,\mathbf {q} )=\mathrm {d} (\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8pt]&={\sqrt {\sum _{i=1}^{n}(q_{i}-p_{i})^{2}}}.\end{aligned}}}"></span></p> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0.08em;"> <p style="margin:0;"> </p><table style="border-collapse:collapse; background:none; margin:0; border:none; width:99%;"> <tbody><tr> <td style="border:none; padding:0.08em;" rowspan="2"><p style="margin:0; font-size:4pt;"> </p> </td> <td style="width:100%; border:none; padding:0.08em;"><p style="margin:0; font-size:1pt;"> </p> </td> <td style="border:none; padding:0.08em;" rowspan="2"><p style="margin:0; font-size:4pt;"> </p> </td></tr> <tr> <td style="border-left:none; border-top:0px none #e5e5e5; border-right:none; border-bottom:none; padding:0.08em;"> <p style="margin:0; font-size:1pt;"> </p> </td></tr></tbody></table> <p class="mw-empty-elt"></p> </td> <td style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap"><p style="margin:0pt;"><b>(<cite id="math_1"><span class="reference plainlinksneverexpand"><b><a href="#equation_1"> 1</a></b></span></cite>)</b></p> </td></tr></tbody></table></dd></dl> <p>Položaj tačke u Euklidskom n-prostoru je vektor, tj. p i q su Euklidski vektori. Euklidova norma ili Euklidska udaljenosti su dužine vektora: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\mathbf {p} \right\|={\sqrt {p_{1}^{2}+p_{2}^{2}+\cdots +p_{n}^{2}}}={\sqrt {\mathbf {p} \cdot \mathbf {p} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {p} \right\|={\sqrt {p_{1}^{2}+p_{2}^{2}+\cdots +p_{n}^{2}}}={\sqrt {\mathbf {p} \cdot \mathbf {p} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6127241aab8857d11f67bf9f88c23033e5852ec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.643ex; height:4.843ex;" alt="{\displaystyle \left\|\mathbf {p} \right\|={\sqrt {p_{1}^{2}+p_{2}^{2}+\cdots +p_{n}^{2}}}={\sqrt {\mathbf {p} \cdot \mathbf {p} }},}"></span></dd></dl> <p>Vektor se može opisati kao orjentisana duž u Euklidskom prostoru. Ako uzmemo u obzir da je njegova dužina od početka do kraja te duži, postaje jasno da je Euklidska norma vektora poseban slučaj Euklidove udaljenosti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} -\mathbf {p} =(q_{1}-p_{1},q_{2}-p_{2},\cdots ,q_{n}-p_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} -\mathbf {p} =(q_{1}-p_{1},q_{2}-p_{2},\cdots ,q_{n}-p_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda8f705f247ef44ccd6a8e0816eefb889f8a842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.656ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} -\mathbf {p} =(q_{1}-p_{1},q_{2}-p_{2},\cdots ,q_{n}-p_{n})}"></span></dd></dl> <p>U trodimenzionalnom prostoru (n = 3) Euklidska udaljenost između p i q je </p> <dl><dd> <table style="border-collapse:collapse; background:none; margin:0; border:none;"> <tbody><tr> <td style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap"><p style="margin:0;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {(\mathbf {q} -\mathbf {p} )\cdot (\mathbf {q} -\mathbf {p} )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {(\mathbf {q} -\mathbf {p} )\cdot (\mathbf {q} -\mathbf {p} )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367a94a87ddcfd91325bf3023af5ca77a0027e55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.916ex; height:4.843ex;" alt="{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {(\mathbf {q} -\mathbf {p} )\cdot (\mathbf {q} -\mathbf {p} )}}.}"></span></p> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0.08em;"> <p style="margin:0;"> </p><table style="border-collapse:collapse; background:none; margin:0; border:none; width:99%;"> <tbody><tr> <td style="border:none; padding:0.08em;" rowspan="2"><p style="margin:0; font-size:4pt;"> </p> </td> <td style="width:100%; border:none; padding:0.08em;"><p style="margin:0; font-size:1pt;"> </p> </td> <td style="border:none; padding:0.08em;" rowspan="2"><p style="margin:0; font-size:4pt;"> </p> </td></tr> <tr> <td style="border-left:none; border-top:0px none #e5e5e5; border-right:none; border-bottom:none; padding:0.08em;"> <p style="margin:0; font-size:1pt;"> </p> </td></tr></tbody></table> <p class="mw-empty-elt"></p> </td> <td style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap"><p style="margin:0pt;"><b>(<cite id="math_2"><span class="reference plainlinksneverexpand"><b><a href="#equation_2"> 2</a></b></span></cite>)</b></p> </td></tr></tbody></table></dd></dl> <p>ili </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {\left\|\mathbf {p} \right\|^{2}+\left\|\mathbf {q} \right\|^{2}-2\mathbf {p} \cdot \mathbf {q} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {\left\|\mathbf {p} \right\|^{2}+\left\|\mathbf {q} \right\|^{2}-2\mathbf {p} \cdot \mathbf {q} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95db6f6d65948f1d0086ac0f6f59a22a92627334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:35.218ex; height:4.843ex;" alt="{\displaystyle \left\|\mathbf {q} -\mathbf {p} \right\|={\sqrt {\left\|\mathbf {p} \right\|^{2}+\left\|\mathbf {q} \right\|^{2}-2\mathbf {p} \cdot \mathbf {q} }}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Jednodimenzionalna_udaljenost">Jednodimenzionalna udaljenost</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=2" title="Uredi odlomak "Jednodimenzionalna udaljenost"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=2" title="Edit section's source code: Jednodimenzionalna udaljenost"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>u jednodimziomalnom prostoru udaljenost između dvije tačke na realnoj pravoj je apsolutna vrijednost njihove numeričke razlike. Ako su X i Y dvije tačke prave udaljenost između nih je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {(x-y)^{2}}}=|x-y|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {(x-y)^{2}}}=|x-y|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40acb4e3dca881674b97303ffabfae6f28e3952e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.877ex; height:4.843ex;" alt="{\displaystyle {\sqrt {(x-y)^{2}}}=|x-y|.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Dvodimenzionalna_udaljenost">Dvodimenzionalna udaljenost</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=3" title="Uredi odlomak "Dvodimenzionalna udaljenost"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=3" title="Edit section's source code: Dvodimenzionalna udaljenost"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Udaljenost dvije tačke (x, y) kod jednog <a href="/wiki/Pravougli_trougao" title="Pravougli trougao">pravouglog trougla</a>: </p><p><a href="/wiki/Du%C5%BEina" title="Dužina">Dužina</a> <a href="/w/index.php?title=Horizontala&action=edit&redlink=1" class="new" title="Horizontala (stranica ne postoji)">horizontalne</a> linije je <a href="/wiki/Pitagorina_teorema" title="Pitagorina teorema">kateta</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 x=x_{1}-x_{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}-x_{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/044248f78a97d8ccd7deeb26f7f3c15b7817ba23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.07ex; height:2.343ex;" alt="{\displaystyle x=x_{1}-x_{2}\,.}"></span> <sup id="cite_ref-Matheprisma_Uni_Wuppertal_2-1" class="reference"><a href="#cite_note-Matheprisma_Uni_Wuppertal-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Dužina <a href="/w/index.php?title=Vertikala&action=edit&redlink=1" class="new" title="Vertikala (stranica ne postoji)">vertikalne</a> linije je kateta: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y_{1}-y_{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{1}-y_{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc09ae55fe4019c477a6c72c08936ffc649e8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.515ex; height:2.343ex;" alt="{\displaystyle y=y_{1}-y_{2}\,.}"></span> <sup id="cite_ref-Matheprisma_Uni_Wuppertal_2-2" class="reference"><a href="#cite_note-Matheprisma_Uni_Wuppertal-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Prema tome udaljenost je <a href="/wiki/Pitagorina_teorema" title="Pitagorina teorema">hipotenuza</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 {\sqrt {x^{2}+y^{2}}}={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+y^{2}}}={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9541e9780dbe4e11edd8e7b63fafb33d2cb57e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:39.622ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x^{2}+y^{2}}}={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,.}"></span> <sup id="cite_ref-Matheprisma_Uni_Wuppertal_2-3" class="reference"><a href="#cite_note-Matheprisma_Uni_Wuppertal-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Pojam udaljenosti, koji se upotrebljava u svakodnevnici, odnosi se upravo na Euklidsku udaljenost.<sup id="cite_ref-Matheprisma_Uni_Wuppertal_2-4" class="reference"><a href="#cite_note-Matheprisma_Uni_Wuppertal-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Ako su tačke date u polarnim koordinatama onda </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c749e7296a1a2521e0f8b91a4078b8128c54e465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.663ex; height:4.843ex;" alt="{\displaystyle {\sqrt {r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Trodimenzionalna_udaljenost">Trodimenzionalna udaljenost</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=4" title="Uredi odlomak "Trodimenzionalna udaljenost"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=4" title="Edit section's source code: Trodimenzionalna udaljenost"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U trodimenzionalnom prostoru, udaljenost je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}}.}"> <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"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d13a40a7b203b455ae6d4be8b3cce898bda625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:48.105ex; height:4.843ex;" alt="{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="n_-_domenzionalna_udaljenost">n - domenzionalna udaljenost</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=5" title="Uredi odlomak "n - domenzionalna udaljenost"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=5" title="Edit section's source code: n - domenzionalna udaljenost"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U n - dimenzionalnom prostoru, udaljenost je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}}}.}"> <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"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ef4fe055b2a51b4cca43a05e5d1cd93f758dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:71.91ex; height:4.843ex;" alt="{\displaystyle d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Kvadrat_Euklidske_udaljenosti">Kvadrat Euklidske udaljenosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=6" title="Uredi odlomak "Kvadrat Euklidske udaljenosti"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=6" title="Edit section's source code: Kvadrat Euklidske udaljenosti"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kvadrat Euklidske udaljenosti je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{2}(p,q)=(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2}(p,q)=(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16586276b11fed708c59ddb10d4c23940f4e4ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.643ex; height:3.176ex;" alt="{\displaystyle d^{2}(p,q)=(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Euklidska_udaljenost&veaction=edit&section=7" title="Uredi odlomak "Reference"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Euklidska_udaljenost&action=edit&section=7" title="Edit section's source code: Reference"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3312388">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.geoinformatik.uni-rostock.de/einzel.asp?ID=644">[https://web.archive.org/web/20160305035105/http://www.geoinformatik.uni-rostock.de/einzel.asp?ID=644 Arhivirano</a> 5. 3. 2016. na <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Euklidska udaljenost, Leksikon matematike na univerzitetu Rostock, Njemačka, <a href="/wiki/Njema%C4%8Dki_jezik" title="Njemački jezik">njem.</a>] učitano 01.01.2014</span> </li> <li id="cite_note-Matheprisma_Uni_Wuppertal-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Matheprisma_Uni_Wuppertal_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Matheprisma_Uni_Wuppertal_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Matheprisma_Uni_Wuppertal_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Matheprisma_Uni_Wuppertal_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Matheprisma_Uni_Wuppertal_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.matheprisma.uni-wuppertal.de/Module/PI/Euklid.htm">Euklidska udaljenost, Leksikon matematike na univerzitetu Wuppertal, Njemačka, </a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130621063625/http://www.matheprisma.uni-wuppertal.de/Module/PI/Euklid.htm">Arhivirano</a> 21. 6. 2013. na <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a><a href="/wiki/Njema%C4%8Dki_jezik" title="Njemački jezik">njem.</a><span> </span> učitano 01.01.2014. (<i><b>Napomena:</b> x1 i x2 - tačke na x-osi, y1 i y2 - na y-osi. Na izvoru su to drugačije označene tačke.</i>)</span> </li> </ol></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Preuzeto iz "<a dir="ltr" href="https://bs.wikipedia.org/w/index.php?title=Euklidska_udaljenost&oldid=3483174">https://bs.wikipedia.org/w/index.php?title=Euklidska_udaljenost&oldid=3483174</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorije</a>: <ul><li><a href="/w/index.php?title=Kategorija:%C5%A0abloni_za_matemati%C4%8Dko_formatiranje&action=edit&redlink=1" class="new" title="Kategorija:Šabloni za matematičko formatiranje (stranica ne postoji)">Šabloni za matematičko formatiranje</a></li><li><a href="/wiki/Kategorija:Rastojanje" title="Kategorija:Rastojanje">Rastojanje</a></li><li><a href="/wiki/Kategorija:Du%C5%BEina" title="Kategorija:Dužina">Dužina</a></li><li><a href="/wiki/Kategorija:Metri%C4%8Dka_geometrija" title="Kategorija:Metrička geometrija">Metrička geometrija</a></li><li><a href="/wiki/Kategorija:Pitagorina_teorema" title="Kategorija:Pitagorina teorema">Pitagorina teorema</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Sakrivena kategorija: <ul><li><a href="/wiki/Kategorija:Webarchive_template_wayback_links" title="Kategorija:Webarchive template wayback links">Webarchive template wayback links</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"> Ova stranica je posljednji put izmijenjena na datum 3 februar 2023 u 16:06.</li> <li id="footer-info-copyright">Tekst je dostupan pod <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/">slobodnom licencom Autorstvo-Dijeliti pod istim uvjetima</a>; mogu se primijeniti i dodatni uvjeti. 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