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/\u003E\u003C/a\u003E\u003C/span\u003E\u003C/td\u003E\n\u003Ctd class=\"mbox-text\" style=\"\"\u003E \u003Ca href=\"/wiki/Wikipedija:Kriteriji_zna%C4%8Dajnosti_po_temama\" title=\"Wikipedija:Kriteriji značajnosti po temama\"\u003ERaspravite \u003Cb\u003Erazradu po temama kriterija\u003C/b\u003E o značajnosti!\u003C/a\u003E \u003C/td\u003E\n\u003C/tr\u003E\n\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="Mrežno mjesto"> <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="Sadržaj" 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">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Povijest" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Povijest"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Povijest</span> </div> </a> <ul id="toc-Povijest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definicija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definicija</span> </div> </a> <button aria-controls="toc-Definicija-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>Sadržaj cjeline Definicija</span> </button> <ul id="toc-Definicija-sublist" class="vector-toc-list"> <li id="toc-Kartezijev_dvodimenzionalni_koordinatni_sustav" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kartezijev_dvodimenzionalni_koordinatni_sustav"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Kartezijev dvodimenzionalni koordinatni sustav</span> </div> </a> <ul id="toc-Kartezijev_dvodimenzionalni_koordinatni_sustav-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kartezijev_trodimenzionalni_koordinatni_sustav" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kartezijev_trodimenzionalni_koordinatni_sustav"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kartezijev trodimenzionalni koordinatni sustav</span> </div> </a> <ul id="toc-Kartezijev_trodimenzionalni_koordinatni_sustav-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kartezijev_višedimenzionalni_koordinatni_sustav" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kartezijev_višedimenzionalni_koordinatni_sustav"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Kartezijev višedimenzionalni koordinatni sustav</span> </div> </a> <ul id="toc-Kartezijev_višedimenzionalni_koordinatni_sustav-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Neposredne_primjene_i_svojstva" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Neposredne_primjene_i_svojstva"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Neposredne primjene i svojstva</span> </div> </a> <button aria-controls="toc-Neposredne_primjene_i_svojstva-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>Sadržaj cjeline Neposredne primjene i svojstva</span> </button> <ul id="toc-Neposredne_primjene_i_svojstva-sublist" class="vector-toc-list"> <li id="toc-Udaljenost_između_dviju_točaka" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Udaljenost_između_dviju_točaka"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Udaljenost između dviju točaka</span> </div> </a> <ul id="toc-Udaljenost_između_dviju_točaka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polovište_dužine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polovište_dužine"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Polovište dužine</span> </div> </a> <ul id="toc-Polovište_dužine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Koordinate_težišta_trokuta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Koordinate_težišta_trokuta"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Koordinate težišta trokuta</span> </div> </a> <ul id="toc-Koordinate_težišta_trokuta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translacija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Translacija"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Translacija</span> </div> </a> <ul id="toc-Translacija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uvećanje,_smanjenje" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uvećanje,_smanjenje"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Uvećanje, smanjenje</span> </div> </a> <ul id="toc-Uvećanje,_smanjenje-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prikaz_krivulja_u_koordinatnom_sustavu_u_ravnini" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prikaz_krivulja_u_koordinatnom_sustavu_u_ravnini"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Prikaz krivulja u koordinatnom sustavu u ravnini</span> </div> </a> <ul id="toc-Prikaz_krivulja_u_koordinatnom_sustavu_u_ravnini-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prikaz_vektora_u_Kartezijevim_koordinatama" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prikaz_vektora_u_Kartezijevim_koordinatama"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Prikaz vektora u Kartezijevim koordinatama</span> </div> </a> <ul id="toc-Prikaz_vektora_u_Kartezijevim_koordinatama-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Primjene" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primjene"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Primjene</span> </div> </a> <ul id="toc-Primjene-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Značaj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Značaj"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Značaj</span> </div> </a> <ul id="toc-Značaj-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-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="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Kartezijev koordinatni sustav</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 druge jezične varijante članka. Dostupan je na 74 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-74" 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">74 jezika</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/Cartesiese_ko%C3%B6rdinatestelsel" title="Cartesiese koördinatestelsel – afrikaans" lang="af" hreflang="af" data-title="Cartesiese koördinatestelsel" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D8%AF%D9%8A%D9%83%D8%A7%D8%B1%D8%AA%D9%8A%D8%A9" title="نظام إحداثيات ديكارتية – arapski" lang="ar" hreflang="ar" data-title="نظام إحداثيات ديكارتية" data-language-autonym="العربية" data-language-local-name="arapski" 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/Coordenaes_cartesianes" title="Coordenaes cartesianes – asturijski" lang="ast" hreflang="ast" data-title="Coordenaes cartesianes" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Karteziyan_koordinat_sistemi" title="Karteziyan koordinat sistemi – azerbajdžanski" lang="az" hreflang="az" data-title="Karteziyan koordinat sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D1%83%D1%80%D0%B0_%D0%BC%D3%A9%D0%B9%D3%A9%D1%88%D0%BB%D3%A9_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Тура мөйөшлө координаталар системаһы – baškirski" lang="ba" hreflang="ba" data-title="Тура мөйөшлө координаталар системаһы" data-language-autonym="Башҡортса" data-language-local-name="baškirski" 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%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Декартова координатна система – bugarski" lang="bg" hreflang="bg" data-title="Декартова координатна система" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A7%87%E0%A6%B8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%82%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="কার্তেসীয় স্থানাংক ব্যবস্থা – bangla" lang="bn" hreflang="bn" data-title="কার্তেসীয় স্থানাংক ব্যবস্থা" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Descartesov_koordinatni_sistem" title="Descartesov koordinatni sistem – bosanski" lang="bs" hreflang="bs" data-title="Descartesov koordinatni sistem" data-language-autonym="Bosanski" data-language-local-name="bosanski" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_de_coordenades_cartesianes" title="Sistema de coordenades cartesianes – katalonski" lang="ca" hreflang="ca" data-title="Sistema de coordenades cartesianes" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%D9%BE%DB%86%D8%AA%D8%A7%D9%86%DB%8C_%D8%AF%DB%8E%DA%A9%D8%A7%D8%B1%D8%AA%DB%8C" title="سیستمی پۆتانی دێکارتی – soranski kurdski" lang="ckb" hreflang="ckb" data-title="سیستمی پۆتانی دێکارتی" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kart%C3%A9zsk%C3%A1_soustava_sou%C5%99adnic" title="Kartézská soustava souřadnic – češki" lang="cs" hreflang="cs" data-title="Kartézská soustava souřadnic" 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%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D1%82%D3%B3%D1%80_%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%D0%B5_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен тӳр кĕтесле тытăмĕ – čuvaški" lang="cv" hreflang="cv" data-title="Координатсен тӳр кĕтесле тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/System_gyfesurynnol_Gartesaidd" title="System gyfesurynnol Gartesaidd – velški" lang="cy" hreflang="cy" data-title="System gyfesurynnol Gartesaidd" data-language-autonym="Cymraeg" data-language-local-name="velški" 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/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – danski" lang="da" hreflang="da" data-title="Kartesisk koordinatsystem" data-language-autonym="Dansk" data-language-local-name="danski" 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/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem – njemački" lang="de" hreflang="de" data-title="Kartesisches Koordinatensystem" 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%9A%CE%B1%CF%81%CF%84%CE%B5%CF%83%CE%B9%CE%B1%CE%BD%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Καρτεσιανό σύστημα συντεταγμένων – grčki" lang="el" hreflang="el" data-title="Καρτεσιανό σύστημα συντεταγμένων" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system – engleski" lang="en" hreflang="en" data-title="Cartesian coordinate system" 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/Kartezia_koordinato" title="Kartezia koordinato – esperanto" lang="eo" hreflang="eo" data-title="Kartezia koordinato" 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/Coordenadas_cartesianas" title="Coordenadas cartesianas – španjolski" lang="es" hreflang="es" data-title="Coordenadas cartesianas" data-language-autonym="Español" data-language-local-name="španjolski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Descartesi_koordinaadid" title="Descartesi koordinaadid – estonski" lang="et" hreflang="et" data-title="Descartesi koordinaadid" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koordenatu_kartesiar" title="Koordenatu kartesiar – baskijski" lang="eu" hreflang="eu" data-title="Koordenatu kartesiar" 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/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA_%D8%AF%DA%A9%D8%A7%D8%B1%D8%AA%DB%8C" title="دستگاه مختصات دکارتی – perzijski" lang="fa" hreflang="fa" data-title="دستگاه مختصات دکارتی" 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/Koordinaatisto#Suorakulmainen_koordinaatisto" title="Koordinaatisto – finski" lang="fi" hreflang="fi" data-title="Koordinaatisto" 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/Coordonn%C3%A9es_cart%C3%A9siennes" title="Coordonnées cartésiennes – francuski" lang="fr" hreflang="fr" data-title="Coordonnées cartésiennes" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comhordan%C3%A1id%C3%AD_Cairt%C3%A9iseacha" title="Comhordanáidí Cairtéiseacha – irski" lang="ga" hreflang="ga" data-title="Comhordanáidí Cairtéiseacha" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_coordenadas_cartesianas" title="Sistema de coordenadas cartesianas – galicijski" lang="gl" hreflang="gl" data-title="Sistema de coordenadas cartesianas" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%A6%D7%99%D7%A8%D7%99%D7%9D_%D7%A7%D7%A8%D7%98%D7%96%D7%99%D7%AA" title="מערכת צירים קרטזית – hebrejski" lang="he" hreflang="he" data-title="מערכת צירים קרטזית" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="कार्तीय निर्देशांक पद्धति – hindski" lang="hi" hreflang="hi" data-title="कार्तीय निर्देशांक पद्धति" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Descartes-f%C3%A9le_koordin%C3%A1ta-rendszer" title="Descartes-féle koordináta-rendszer – mađarski" lang="hu" hreflang="hu" data-title="Descartes-féle koordináta-rendszer" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D5%AF%D5%A1%D6%80%D5%BF%D5%B5%D5%A1%D5%B6_%D5%AF%D5%B8%D5%B8%D6%80%D5%A4%D5%AB%D5%B6%D5%A1%D5%BF%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Դեկարտյան կոորդինատների համակարգ – armenski" lang="hy" hreflang="hy" data-title="Դեկարտյան կոորդինատների համակարգ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat_Cartesius" title="Sistem koordinat Cartesius – indonezijski" lang="id" hreflang="id" data-title="Sistem koordinat Cartesius" 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-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Karteziana_koordinataro" title="Karteziana koordinataro – ido" lang="io" hreflang="io" data-title="Karteziana koordinataro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Kartes%C3%ADusarhnitakerfi%C3%B0" title="Kartesíusarhnitakerfið – islandski" lang="is" hreflang="is" data-title="Kartesíusarhnitakerfið" data-language-autonym="Íslenska" data-language-local-name="islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_riferimento_cartesiano" title="Sistema di riferimento cartesiano – talijanski" lang="it" hreflang="it" data-title="Sistema di riferimento cartesiano" data-language-autonym="Italiano" data-language-local-name="talijanski" 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/%E7%9B%B4%E4%BA%A4%E5%BA%A7%E6%A8%99%E7%B3%BB" title="直交座標系 – japanski" lang="ja" hreflang="ja" data-title="直交座標系" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%94%E1%83%99%E1%83%90%E1%83%A0%E1%83%A2%E1%83%98%E1%83%A1_%E1%83%99%E1%83%9D%E1%83%9D%E1%83%A0%E1%83%93%E1%83%98%E1%83%9C%E1%83%90%E1%83%A2%E1%83%97%E1%83%90_%E1%83%A1%E1%83%98%E1%83%A1%E1%83%A2%E1%83%94%E1%83%9B%E1%83%90" title="დეკარტის კოორდინატთა სისტემა – gruzijski" lang="ka" hreflang="ka" data-title="დეკარტის კოორდინატთა სისტემა" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Cartesian_Coordinates" title="Cartesian Coordinates – kikuyu" lang="ki" hreflang="ki" data-title="Cartesian Coordinates" data-language-autonym="Gĩkũyũ" data-language-local-name="kikuyu" class="interlanguage-link-target"><span>Gĩkũyũ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D1%82%D1%8B%D2%9B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%82%D0%B0%D1%80_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Декарттық координаттар жүйесі – kazaški" lang="kk" hreflang="kk" data-title="Декарттық координаттар жүйесі" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B0%EC%B9%B4%EB%A5%B4%ED%8A%B8_%EC%A2%8C%ED%91%9C%EA%B3%84" title="데카르트 좌표계 – korejski" lang="ko" hreflang="ko" data-title="데카르트 좌표계" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_Cartesianum_coordinatarum" title="Systema Cartesianum coordinatarum – latinski" lang="la" hreflang="la" data-title="Systema Cartesianum coordinatarum" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dekarto_koordina%C4%8Di%C5%B3_sistema" title="Dekarto koordinačių sistema – litavski" lang="lt" hreflang="lt" data-title="Dekarto koordinačių sistema" data-language-autonym="Lietuvių" data-language-local-name="litavski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dekarta_koordin%C4%81tu_sist%C4%93ma" title="Dekarta koordinātu sistēma – latvijski" lang="lv" hreflang="lv" data-title="Dekarta koordinātu sistēma" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Декартов координатен систем – makedonski" lang="mk" hreflang="mk" data-title="Декартов координатен систем" data-language-autonym="Македонски" data-language-local-name="makedonski" 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%85%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B4%82" title="അക്ഷം – malajalamski" lang="ml" hreflang="ml" data-title="അക്ഷം" data-language-autonym="മലയാളം" data-language-local-name="malajalamski" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%9F%E0%A5%87%E0%A4%B6%E0%A4%BF%E0%A4%AF%E0%A4%A8_%E0%A4%B8%E0%A4%B9%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A5%80" title="कार्टेशियन सहनिर्देशक पद्धती – marathski" lang="mr" hreflang="mr" data-title="कार्टेशियन सहनिर्देशक पद्धती" data-language-autonym="मराठी" data-language-local-name="marathski" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_Cartes" title="Sistem koordinat Cartes – malajski" lang="ms" hreflang="ms" data-title="Sistem koordinat Cartes" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AC%E1%80%90%E1%80%80%E1%80%BA%E1%80%85%E1%80%AE%E1%80%B8%E1%80%9A%E1%80%94%E1%80%BA%E1%80%B8_%E1%80%80%E1%80%AD%E1%80%AF%E1%80%A9%E1%80%92%E1%80%AD%E1%80%94%E1%80%AD%E1%80%90%E1%80%BA%E1%80%85%E1%80%94%E1%80%85%E1%80%BA" title="ကာတက်စီးယန်း ကိုဩဒိနိတ်စနစ် – burmanski" lang="my" hreflang="my" data-title="ကာတက်စီးယန်း ကိုဩဒိနိတ်စနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmanski" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Karteesch_Koordinatensystem" title="Karteesch Koordinatensystem – donjonjemački" lang="nds" hreflang="nds" data-title="Karteesch Koordinatensystem" data-language-autonym="Plattdüütsch" data-language-local-name="donjonjemački" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cartesisch_co%C3%B6rdinatenstelsel" title="Cartesisch coördinatenstelsel – nizozemski" lang="nl" hreflang="nl" data-title="Cartesisch coördinatenstelsel" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – norveški nynorsk" lang="nn" hreflang="nn" data-title="Kartesisk koordinatsystem" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem – norveški bokmål" lang="nb" hreflang="nb" data-title="Kartesisk koordinatsystem" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A8%BE%E0%A8%B0%E0%A8%9F%E0%A9%87%E0%A8%9C%E0%A8%BC%E0%A9%80_%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%95_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%AC%E0%A9%B0%E0%A8%A7" title="ਕਾਰਟੇਜ਼ੀ ਗੁਣਕ ਪ੍ਰਬੰਧ – pandžapski" lang="pa" hreflang="pa" data-title="ਕਾਰਟੇਜ਼ੀ ਗੁਣਕ ਪ੍ਰਬੰਧ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžapski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_kartezja%C5%84skich" title="Układ współrzędnych kartezjańskich – poljski" lang="pl" hreflang="pl" data-title="Układ współrzędnych kartezjańskich" data-language-autonym="Polski" data-language-local-name="poljski" 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/Sistema_de_coordenadas_cartesiano" title="Sistema de coordenadas cartesiano – portugalski" lang="pt" hreflang="pt" data-title="Sistema de coordenadas cartesiano" 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/Coordonate_carteziene" title="Coordonate carteziene – rumunjski" lang="ro" hreflang="ro" data-title="Coordonate carteziene" data-language-autonym="Română" data-language-local-name="rumunjski" 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%9F%D1%80%D1%8F%D0%BC%D0%BE%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Прямоугольная система координат – ruski" lang="ru" hreflang="ru" data-title="Прямоугольная система координат" 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/Sistema_di_rifirimentu_cartisianu" title="Sistema di rifirimentu cartisianu – sicilijski" lang="scn" hreflang="scn" data-title="Sistema di rifirimentu cartisianu" data-language-autonym="Sicilianu" data-language-local-name="sicilijski" 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/Kartezijanski_koordinatni_sistem" title="Kartezijanski koordinatni sistem – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Kartezijanski koordinatni sistem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Cartesian coordinate system" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kartezi%C3%A1nska_s%C3%BAstava_s%C3%BAradn%C3%ADc_(v_naju%C5%BE%C5%A1om_zmysle)" title="Karteziánska sústava súradníc (v najužšom zmysle) – slovački" lang="sk" hreflang="sk" data-title="Karteziánska sústava súradníc (v najužšom zmysle)" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kartezi%C4%8Dni_koordinatni_sistem" title="Kartezični koordinatni sistem – slovenski" lang="sl" hreflang="sl" data-title="Kartezični koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_koordinativ_kartezian" title="Sistemi koordinativ kartezian – albanski" lang="sq" hreflang="sq" data-title="Sistemi koordinativ kartezian" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2_%D0%BF%D1%80%D0%B0%D0%B2%D0%BE%D1%83%D0%B3%D0%BB%D0%B8_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Декартов правоугли координатни систем – srpski" lang="sr" hreflang="sr" data-title="Декартов правоугли координатни систем" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kartesiskt_koordinatsystem" title="Kartesiskt koordinatsystem – švedski" lang="sv" hreflang="sv" data-title="Kartesiskt koordinatsystem" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%80%E0%AE%9A%E0%AE%BF%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%86%E0%AE%B3%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="காட்டீசியன் ஆள்கூற்று முறைமை – tamilski" lang="ta" hreflang="ta" data-title="காட்டீசியன் ஆள்கூற்று முறைமை" 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%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94%E0%B8%84%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B8%97%E0%B8%B5%E0%B9%80%E0%B8%8B%E0%B8%B5%E0%B8%A2%E0%B8%99" title="ระบบพิกัดคาร์ทีเซียน – tajlandski" lang="th" hreflang="th" data-title="ระบบพิกัดคาร์ทีเซียน" 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/Kartezyen_koordinat_sistemi" title="Kartezyen koordinat sistemi – turski" lang="tr" hreflang="tr" data-title="Kartezyen koordinat sistemi" 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%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Декартова система координат – ukrajinski" lang="uk" hreflang="uk" data-title="Декартова система координат" 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/%DA%A9%D8%A7%D8%B1%D8%AA%DB%8C%D8%B3%DB%8C_%D9%85%D8%AA%D9%86%D8%A7%D8%B3%D9%82_%D9%86%D8%B8%D8%A7%D9%85" title="کارتیسی متناسق نظام – urdski" lang="ur" hreflang="ur" data-title="کارتیسی متناسق نظام" data-language-autonym="اردو" data-language-local-name="urdski" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dekart_koordinatalar_tizimi" title="Dekart koordinatalar tizimi – uzbečki" lang="uz" hreflang="uz" data-title="Dekart koordinatalar tizimi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_t%E1%BB%8Da_%C4%91%E1%BB%99_Descartes" title="Hệ tọa độ Descartes – vijetnamski" lang="vi" hreflang="vi" data-title="Hệ tọa độ Descartes" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" 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/%E7%AC%9B%E5%8D%A1%E5%84%BF%E5%9D%90%E6%A0%87%E7%B3%BB" title="笛卡儿坐标系 – wu kineski" lang="wuu" hreflang="wuu" data-title="笛卡儿坐标系" data-language-autonym="吴语" data-language-local-name="wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AC%9B%E5%8D%A1%E5%B0%94%E5%9D%90%E6%A0%87%E7%B3%BB" title="笛卡尔坐标系 – kineski" lang="zh" hreflang="zh" data-title="笛卡尔坐标系" data-language-autonym="中文" data-language-local-name="kineski" 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href="/wiki/Datoteka:Cartesian-coordinate-system.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/300px-Cartesian-coordinate-system.svg.png" decoding="async" width="300" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/450px-Cartesian-coordinate-system.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/600px-Cartesian-coordinate-system.svg.png 2x" data-file-width="661" data-file-height="654" /></a><figcaption>Kartezijev koordinatni sustav u <a href="/wiki/Ravnina" title="Ravnina">ravnini</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Coord_system_CA_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/300px-Coord_system_CA_0.svg.png" decoding="async" width="300" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/450px-Coord_system_CA_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/600px-Coord_system_CA_0.svg.png 2x" data-file-width="620" data-file-height="600" /></a><figcaption>Pravokutni Kartezijev koordinatni sustav u <a href="/wiki/Prostor" title="Prostor">prostoru</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Cartesian_coordinates_2D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/300px-Cartesian_coordinates_2D.svg.png" decoding="async" width="300" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/450px-Cartesian_coordinates_2D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Cartesian_coordinates_2D.svg/600px-Cartesian_coordinates_2D.svg.png 2x" data-file-width="800" data-file-height="742" /></a><figcaption>Raspored kvadranta.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Octant_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Octant_numbers.svg/300px-Octant_numbers.svg.png" decoding="async" width="300" height="318" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Octant_numbers.svg/450px-Octant_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Octant_numbers.svg/600px-Octant_numbers.svg.png 2x" data-file-width="1170" data-file-height="1242" /></a><figcaption>Raspored oktanta u prostoru (3D sustavu).</figcaption></figure> <p><b>Kartezijev koordinatni sustav</b>, <b>pravokutni koordinatni sustav</b> ili <b>pravokutni Kartezijev koordinatni sustav</b> u <a href="/wiki/Prostor" title="Prostor">prostoru</a> određen je trima međusobno okomitim pravcima <i>x, y, z</i>, koji se sijeku u ishodištu <i>O</i>, i s Kartezijevim koordinatnim sustavima na njima. Koordinate se tada zovu apscisa (na osi <i>x</i>), ordinata (na osi <i>y</i>) i aplikata (na osi <i>z</i>).<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Povijest">Povijest</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=1" title="Uredi odlomak: Povijest" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=1" title="Uredi kôd odjeljka Povijest"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zasluga za otkriće Kartezijevog koordinatnog sustava kako on danas nosi ime, pripala je francuskom matematičaru <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Reneu Descartesu</a> (1596. – 1650.) koji ga je imenovao po svojoj latinskoj inačici imena <i>Cartesius</i>. Premda je ideja bila utemeljena još 1637. godine odvojeno u dva zapisa Descartesa i <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermata</a>, potonji nije objavio svoje otkriće. Upravo je Descartes zato uveo novu zamisao određivanja položaja točke ili objekta u ravnini upotrijebivši dvije međusobno okomite osi kao mjerila. Otkriće Kartezijevog koordinatnog sustava značilo je velik napredak u matematici povezujući najprije <a href="/wiki/Euklidska_geometrija" title="Euklidska geometrija">Euklidsku geometriju</a> i algebru. <a href="/wiki/Kru%C5%BEnica" title="Kružnica">Kružnice</a>, <a href="/wiki/Elipsa" title="Elipsa">elipse</a> i druge krivulje sada su prvi puta mogle biti opisivane “kartezijskim” algebarskim jednadžbama pomoću koordinata točaka krivulje u ravnini. Razvoj kartezijevog koordinatnog sustava značajno je doprinio daljnjem razvoju matematike i omogućio <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newtonu</a> i <a href="/wiki/Gottfried_Wilhelm_Leibniz" class="mw-redirect" title="Gottfried Wilhelm Leibniz">Leibnitzu</a> skoro otkriće diferencijalnog i integralnog računa. </p> <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=Kartezijev_koordinatni_sustav&veaction=edit&section=2" title="Uredi odlomak: Definicija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=2" title="Uredi kôd odjeljka Definicija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nalik zemljopisnoj karti gdje je položaj nekog mjesta određen s dva podatka: zemljopisnom širinom i zemljopisnom dužinom, nacrtamo li dva međusobno okomita brojevna pravca, na primjer <i>x</i> i <i>y</i> - uobičajeno <i>x</i> horizontalan, a <i>y</i> vertikalan, koji se sijeku u točki O i odredimo li na pravcima <i>x</i> i <i>y</i> jedinične točke E i F, tako da je /OE/=/OF/=1, definirali smo pravokutni ili Kartezijev koordinatni sustav u ravnini. </p> <div class="mw-heading mw-heading3"><h3 id="Kartezijev_dvodimenzionalni_koordinatni_sustav">Kartezijev dvodimenzionalni koordinatni sustav</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=3" title="Uredi odlomak: Kartezijev dvodimenzionalni koordinatni sustav" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=3" title="Uredi kôd odjeljka Kartezijev dvodimenzionalni koordinatni sustav"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Točka O zove se ishodište koordinatnog sustava, brojevni pravac x zove se os x ili apscisa, a brojevni pravac y os y ili ordinata koordinatnog sustava. Katkada govorimo skrećeno o x-osi ili y-osi, odn. o osima koordinatnog sustava. Na svaku od osi smješten je brojevni pravac, gdje svaki realni broj: cijeli, racionalni ili iracionalni ima jedinstveno mjesto na osi. Svakoj točki ravnine dodijeljene su na taj način odgovarajuće koordinate koje nalazimo okomitim, odn. ortogonalnim projekcijama koje iz odgovarajuće točke povlačimo na os x, odn. os y, gdje su koordinate date u određenom broju jediničnih duljina. </p><p>Kartezijske koordinate se zapisuju u zagradama u obliku <a href="/wiki/Ure%C4%91eni_par" title="Uređeni par">uređenog para</a> brojeva gdje prvi broj označava položaj osi na x-osi, a drugi na y-osi. Na slici gore desno prikazane su tako četiri točke s njihovim odgovarajućim koordinatama u Kartezijevom koordinatnom sustavu i to: (2,3) zeleno, (−3,1) crveno, (−1.5,−2.5) plavo i ishodište (0,0) ljubičasto. </p><p>Osi koordinatnog sustava dijele ravninu na četiri beskonačno velika dijela, “kvadranta”, od kojih je svaki omeđen s dvije odgovarajuće osi i naznačen rimskim brojevima od I do IV kako je prikazano na slici desno. </p> <div class="mw-heading mw-heading3"><h3 id="Kartezijev_trodimenzionalni_koordinatni_sustav">Kartezijev trodimenzionalni koordinatni sustav</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=4" title="Uredi odlomak: Kartezijev trodimenzionalni koordinatni sustav" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=4" title="Uredi kôd odjeljka Kartezijev trodimenzionalni koordinatni sustav"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kartezijev koordinatni sustav možemo izabrati i kao o jednodimenzionalni matematički prostor, gdje će takav prostor biti određen jednom osi uz izbor orijentacije osi i jedinične dužine, a koordinata (jedna) će u tom slučaju određivati položaj točke na brojevnom pravcu koji je pridružen koordinatnoj osi. </p><p>Kartezijev dvodimenzionalni koordinatni sustav određuje položaj točke u ravnini, a kartezijev trodimenzionalni koordinatni sustav određuje položaj točke u prostoru gdje je takav koordinatni sustav definiran središtem koordinatnog sustava <i>0</i>, i tri orijentirane osi (<i>x</i>, <i>y</i> i <i>z</i>) s odgovarajućim jediničnim dužinama. Koordinate svake točke u takvom sustavu zadate su uređenim skupom od 3 broja, na primjer (3, -1, 5) koji označavaju odgovarajuće koordinate u trodimenzionalnom matematičkom prostoru, gdje su koordinate predstavljene orijentiranim okomitim udaljenostima od neke točke do odgovarajuće ravnine. U trodimenzionalnom koordinatnom sustavu nazivi osi (apscisa i ordinata) nisu uvjetovane, no ako se upotrebljavaju tada je uobičajeno treću, <i>z</i>-os, nazvati aplikata. Na isti način je uobičajeno <i>x</i>-os i <i>y</i>-os postaviti u horizontalnu ravninu, a preostalu, <i>z</i>-os postaviti okomito na njih. Konačno, trodimenzionalni koordinatni sustav dijelimo na osam područja, “oktanata”, omeđenih s odgovarajućim dijelovim ravnina. Prvi oktant je onaj gdje su sve tri poluosi pozitivne. </p> <div class="mw-heading mw-heading3"><h3 id="Kartezijev_višedimenzionalni_koordinatni_sustav"><span id="Kartezijev_vi.C5.A1edimenzionalni_koordinatni_sustav"></span>Kartezijev višedimenzionalni koordinatni sustav</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=5" title="Uredi odlomak: Kartezijev višedimenzionalni koordinatni sustav" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=5" title="Uredi kôd odjeljka Kartezijev višedimenzionalni koordinatni sustav"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Slijedeći navedeni princip općenito se mogu koordinate točke odrediti i u n-dimenzionalnom matematičkom prostoru gdje će se pomoću <i>n</i> odgovarajućih koordinata definirati orijentirana udaljenost od točke do jedne od <i>n</i> <a href="/w/index.php?title=Hiperravnina&action=edit&redlink=1" class="new" title="Hiperravnina (stranica ne postoji)">hiperravnina</a>. U četverodimenzionalnom matematičkom prostoru na primjer, postojat će četiri osi x, y, z i w, a koordinate svake točke u takvom matematičkom prostoru bit će određene uređenim skupom od četiri broja. </p> <div class="mw-heading mw-heading2"><h2 id="Neposredne_primjene_i_svojstva">Neposredne primjene i svojstva</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=6" title="Uredi odlomak: Neposredne primjene i svojstva" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=6" title="Uredi kôd odjeljka Neposredne primjene i svojstva"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Udaljenost_između_dviju_točaka"><span id="Udaljenost_izme.C4.91u_dviju_to.C4.8Daka"></span>Udaljenost između dviju točaka</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=7" title="Uredi odlomak: Udaljenost između dviju točaka" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=7" title="Uredi kôd odjeljka Udaljenost između dviju točaka"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Udaljenost dviju točaka u ravnini određenih Kartezijevim koordinatama <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_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></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 (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</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/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> 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={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</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> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba52e38f3619c2c8ff3aa5255c92920174e1d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:30.688ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"></span></dd></dl> <p>što je na neki način izraz <a href="/wiki/Pitagorin_pou%C4%8Dak" title="Pitagorin poučak">Pitagorina poučka</a> iskazanog u Kartezijevom koordinatnom sustavu. </p><p>Udaljenost dviju točaka u prostoru određenih u trodimenzionalnom Kartezijevom koordinatnom sustavu <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_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1},z_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781d7569148878d5fab8e65498162edcc5430791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1},z_{1})}"></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 (x_{2},y_{2},z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2},z_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c761bb04bd0bf983b9b8e83310e5407b7426ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2},z_{2})}"></span> 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={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</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>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba8898eb1705d746d09143f43286a12ac48bbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:42.856ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}"></span></dd></dl> <p>Ovaj izraz može se generalizirati i na bilo koji broj dimenzija. Neka su točke (<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ..., <i>a<sub>n</sub></i>) i (<i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, ..., <i>b<sub>n</sub></i>). Tada vrijedi </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={\sqrt {\sum _{k=1}^{n}{(b_{k}-a_{k})^{2}}}}={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+\ldots +(b_{n}-a_{n})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</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>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {\sum _{k=1}^{n}{(b_{k}-a_{k})^{2}}}}={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+\ldots +(b_{n}-a_{n})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999356019c948e9e9dfaa98aa13d29226d9f3ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.603ex; height:7.509ex;" alt="{\displaystyle d={\sqrt {\sum _{k=1}^{n}{(b_{k}-a_{k})^{2}}}}={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+\ldots +(b_{n}-a_{n})^{2}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Polovište_dužine"><span id="Polovi.C5.A1te_du.C5.BEine"></span>Polovište dužine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=8" title="Uredi odlomak: Polovište dužine" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=8" title="Uredi kôd odjeljka Polovište dužine"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neka je dužina zadana točkama <i>A</i> i <i>B</i> i njihovim koordinatama <i>A</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 (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span> i <i>B</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 (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</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/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> tada će polovište dužine imati koordinate </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 x_{p}={\frac {x_{1}+x_{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{p}={\frac {x_{1}+x_{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64944a73ea0f669b66e99594720abba33288253e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.932ex; height:5.009ex;" alt="{\displaystyle x_{p}={\frac {x_{1}+x_{2}}{2}}}"></span> i</dd></dl> <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 y_{p}={\frac {y_{1}+y_{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{p}={\frac {y_{1}+y_{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bc535c74bddbf6038d1eb28de8d3e02e0c5731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.36ex; height:5.176ex;" alt="{\displaystyle y_{p}={\frac {y_{1}+y_{2}}{2}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Koordinate_težišta_trokuta"><span id="Koordinate_te.C5.BEi.C5.A1ta_trokuta"></span>Koordinate težišta trokuta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=9" title="Uredi odlomak: Koordinate težišta trokuta" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=9" title="Uredi kôd odjeljka Koordinate težišta trokuta"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neka je trokut <i>ABC</i> smješten u Kartezijevom koordinatnom sustavu i određen točkama s koordinatama <i>A</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 (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span>, <i>B</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 (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</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/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> i <i>C</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 (x_{3},y_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{3},y_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f4325abeb2792b97969c8ebcf2b65a1e17d3c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{3},y_{3})}"></span>, tada će težište trokuta imati koordinate </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 x_{t}={\frac {x_{1}+x_{2}+x_{3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t}={\frac {x_{1}+x_{2}+x_{3}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154099199d5e7e902ab1e3a7ce7fcb61eb08a056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.923ex; height:5.009ex;" alt="{\displaystyle x_{t}={\frac {x_{1}+x_{2}+x_{3}}{3}}}"></span> i</dd></dl> <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 y_{t}={\frac {y_{1}+y_{2}+y_{3}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{t}={\frac {y_{1}+y_{2}+y_{3}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceda60bb9a5e9e74f4dffdd48d48943c9836b643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.161ex; height:5.176ex;" alt="{\displaystyle y_{t}={\frac {y_{1}+y_{2}+y_{3}}{3}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Translacija">Translacija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=10" title="Uredi odlomak: Translacija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=10" title="Uredi kôd odjeljka Translacija"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Skup točaka u ravnini, na primjer trokuta <i>ABC</i>, može se pomaknuti u ravnini uz očuvanje međusobnih udaljenosti i orijentacije uz dodavanje utvrđenog parova brojeva (<i>X</i>,<i>Y</i>) Kartezijevim koordinatama svake točke skupa. Ako su koordinate točaka trokuta <i>A</i>(x’, y’), <i>B</i>(x’’, y’’) i <i>C</i>(x’’’, y’’’) tada će translatirani, odn. pomaknuti trokut imati koordinate <i>A’</i>(x’+X, y’+Y), <i>B’</i>(x’’+X, y’’+Y) i <i>C’</i>(x’’’+X, y’’’+Y) </p> <div class="mw-heading mw-heading3"><h3 id="Uvećanje,_smanjenje"><span id="Uve.C4.87anje.2C_smanjenje"></span>Uvećanje, smanjenje</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=11" title="Uredi odlomak: Uvećanje, smanjenje" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=11" title="Uredi kôd odjeljka Uvećanje, smanjenje"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Želimo li u Kartezijevim koordinatama neki lik prikazati većim ili manjim tada valja sve koordinate svih točaka pomnožiti faktorom proporcionalnosti, nazovimo ga <i>m</i>. Ako su koordinate točaka koje određuju dužinu <i>AB</i>, <i>A</i>(x’, y’) i <i>B</i>(x’’, y’’) tada će nove koordinate točaka koje određuju dužinu A’B’ biti A’(mx’, my’) i B’(mx’’, my’’). Ako je <i>m</i>>1 dobiveni lik će biti veći, a ako je <i>m</i><1 dobiveni lik bit će manji od izvornog lika. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Cartesian-coordinate-system-with-circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/250px-Cartesian-coordinate-system-with-circle.svg.png" decoding="async" width="250" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/375px-Cartesian-coordinate-system-with-circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Cartesian-coordinate-system-with-circle.svg/500px-Cartesian-coordinate-system-with-circle.svg.png 2x" data-file-width="768" data-file-height="790" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Prikaz_krivulja_u_koordinatnom_sustavu_u_ravnini">Prikaz krivulja u koordinatnom sustavu u ravnini</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=12" title="Uredi odlomak: Prikaz krivulja u koordinatnom sustavu u ravnini" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=12" title="Uredi kôd odjeljka Prikaz krivulja u koordinatnom sustavu u ravnini"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U Kartezijevom koordinatnom sustavu jednostavno se prikazuju krivulje u ravnini (kružnica, elipsa, parabola i td.) te različite funkcije (linearne, polinomne, eksponencijalne, trigonometrijske i td.). </p><p>Prikazujući na primjer kružnicu u Kartezijevom koordinatnom sustavu ustanovljavamo da za svaku točku kružnice vrijedi da 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 x^{2}+y^{2}=r^{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>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2466f6ffc9fe675f7ba45a4a8e35877ea5b48cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.027ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}\,}"></span></dd></dl> <p>te će prema tome jednadžba kružnice polumjera 2 (slika desno) biti </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 x^{2}+y^{2}=2^{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> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=2^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a043c442f9160de6f431ef3ce4d3d6ce5e901193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.141ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=2^{2}\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Prikaz_vektora_u_Kartezijevim_koordinatama">Prikaz vektora u Kartezijevim koordinatama</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=13" title="Uredi odlomak: Prikaz vektora u Kartezijevim koordinatama" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=13" title="Uredi kôd odjeljka Prikaz vektora u Kartezijevim koordinatama"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Točka u prostoru opisanom Kartezijevim koordinatama može definirati vektor. Vektor pomaka, na primjer <i>r</i>, može imati hvatište u ishodištu Kartezijeva koordinatnog sustava i vrh u točki u prostoru. Strelica koja pokazuje prema vrhu vektora definira smjer vektora (smjer pomaka), a ortogonalne projekcije na osi x, y i z odgovarajući pomak u x, y ili z smjeru. Dužina samog vektora tada je apsolutna veličina pomaka u prostoru </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 |{r}|={\sqrt {x^{2}+y^{2}+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <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> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{r}|={\sqrt {x^{2}+y^{2}+z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1166814d1e0504ec443ac2d0a0a009c624b628fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.189ex; height:4.843ex;" alt="{\displaystyle |{r}|={\sqrt {x^{2}+y^{2}+z^{2}}}}"></span>,</dd></dl> <p>a također možemo zapisati da 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 \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac595da78365d804fa5967ce929fb67e99fd113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.424ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} }"></span>,</dd></dl> <p>gdje su <i><b>i</b></i>, <i><b>j</b></i> i <i><b>k</b></i> jedinični vektori u smjeru x, y i z osi. </p><p>Vektor u Kartezijevom trodinemzionalnom prostoru određen je na taj način u cijelosti uređenim skupom od četiri veličine (r, x, y, z). Ovakav prikaz vektora uveo je <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">W. R. Hamilton</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Primjene">Primjene</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=14" title="Uredi odlomak: Primjene" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=14" title="Uredi kôd odjeljka Primjene"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Svaka os može u praktičnoj primjeni prema potrebi imati različite mjerne jedinice (kilograme, sekunde, vate, itd), što znači da Kartezijevim koordinatnim sustavom možemo prikazivati ne samo krivulje, likove i geometrijska tijela u dvodimenzionalnom, odnosno trodimenzionalnom prostoru, već da možemo prikazivati i sve moguće ostale varijable (masa, vrijeme, energija, sila i mnoge druge). Premda je teško vizualizirati četvero i višedimenzionalne prostore, algebra Kartezijevih koordinata može se jednostavno proširiti na četiri ili više varijabli tako da se mogu izvršiti izračuni vrijednosti funkcija i s četiri ili više varijabli. Takva algebra definira geometriju višedimenzionalnih prostora. </p> <div class="mw-heading mw-heading2"><h2 id="Značaj"><span id="Zna.C4.8Daj"></span>Značaj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=15" title="Uredi odlomak: Značaj" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=15" title="Uredi kôd odjeljka Značaj"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kartezijeve koordinate su temelj analitičke geometrije i osiguravaju geometrijsku interpretaciju za brojna područja matematike kao što su linearna algebra, kompleksna analiza, diferencijalna geometrija i td. Jedan od najpoznatijih primjera je koncept grafičkog prikaza ili grafa funkcije. Kartezijske koordinate su osnovno oruđe u mnogim područjima koja se bave geometrijom uključujući astronomiju, fiziku, tehničke struke, ekonomiju i drugdje. </p><p>Premda je Descartes dao koordinatnom sustavu svoje ime, valja naglasiti da su se slični koordinatni sustavi koristili i prije njega uključivši <a href="/w/index.php?title=Abu_Rayhan_Biruni&action=edit&redlink=1" class="new" title="Abu Rayhan Biruni (stranica ne postoji)">Abu Rayhan Birunia </a> te Perzijsku matematiku X i XI stoljeća. </p><p>Nakon Descartesa razvijeni su i drugi koordinatni sustavi kao što su polarni, sferični, cilindrični i drugi. </p> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&veaction=edit&section=16" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kartezijev_koordinatni_sustav&action=edit&section=16" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"> <b>koordinatni sustavi</b>, <a rel="nofollow" class="external autonumber" href="http://www.enciklopedija.hr/Natuknica.aspx?ID=33043">[1]</a> "Hrvatska enciklopedija", Leksikografski zavod Miroslav Krleža, www.enciklopedija.hr, 2018.</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="">Dobavljeno iz "<a dir="ltr" href="https://hr.wikipedia.org/w/index.php?title=Kartezijev_koordinatni_sustav&oldid=6237552">https://hr.wikipedia.org/w/index.php?title=Kartezijev_koordinatni_sustav&oldid=6237552</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="/wiki/Kategorija:Analiti%C4%8Dka_geometrija" title="Kategorija:Analitička geometrija">Analitička geometrija</a></li><li><a href="/wiki/Kategorija:Ren%C3%A9_Descartes" title="Kategorija:René Descartes">René Descartes</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 posljednji je put uređivana 1. siječnja 2022. u 01:18.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencijom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hr">Creative Commons: Imenuj autora/Dijeli pod istim uvjetima</a>; mogu se primjenjivati i dodatni uvjeti. 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