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[n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <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="Contingut" 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">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Definicions_i_postulats_d'Euclides" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicions_i_postulats_d'Euclides"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicions i postulats d'Euclides</span> </div> </a> <ul id="toc-Definicions_i_postulats_d'Euclides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Característiques_de_la_recta" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Característiques_de_la_recta"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Característiques de la recta</span> </div> </a> <button aria-controls="toc-Característiques_de_la_recta-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>Commuta la subsecció Característiques de la recta</span> </button> <ul id="toc-Característiques_de_la_recta-sublist" class="vector-toc-list"> <li id="toc-Semirecta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semirecta"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Semirecta</span> </div> </a> <ul id="toc-Semirecta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semirecta_oposada" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semirecta_oposada"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Semirecta oposada</span> </div> </a> <ul id="toc-Semirecta_oposada-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Posicions_relatives_de_les_rectes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Posicions_relatives_de_les_rectes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Posicions relatives de les rectes</span> </div> </a> <ul id="toc-Posicions_relatives_de_les_rectes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Les_rectes_en_geometria" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Les_rectes_en_geometria"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Les rectes en geometria</span> </div> </a> <button aria-controls="toc-Les_rectes_en_geometria-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>Commuta la subsecció Les rectes en geometria</span> </button> <ul id="toc-Les_rectes_en_geometria-sublist" class="vector-toc-list"> <li id="toc-Altres_propietats_de_les_rectes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Altres_propietats_de_les_rectes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Altres propietats de les rectes</span> </div> </a> <ul id="toc-Altres_propietats_de_les_rectes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Les_rectes_en_matemàtiques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Les_rectes_en_matemàtiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Les rectes en matemàtiques</span> </div> </a> <button aria-controls="toc-Les_rectes_en_matemàtiques-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>Commuta la subsecció Les rectes en matemàtiques</span> </button> <ul id="toc-Les_rectes_en_matemàtiques-sublist" class="vector-toc-list"> <li id="toc-La_recta_en_R²" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_recta_en_R²"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>La recta en R²</span> </div> </a> <ul id="toc-La_recta_en_R²-sublist" class="vector-toc-list"> <li id="toc-Equacions_de_la_recta_en_R²" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Equacions_de_la_recta_en_R²"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Equacions de la recta en R²</span> </div> </a> <ul id="toc-Equacions_de_la_recta_en_R²-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rectes_notables_R²" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rectes_notables_R²"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Rectes notables R²</span> </div> </a> <ul id="toc-Rectes_notables_R²-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paral·lelisme_i_perpendicularitat" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Paral·lelisme_i_perpendicularitat"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.3</span> <span>Paral·lelisme i perpendicularitat</span> </div> </a> <ul id="toc-Paral·lelisme_i_perpendicularitat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angles_i_distàncies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Angles_i_distàncies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.4</span> <span>Angles i distàncies</span> </div> </a> <ul id="toc-Angles_i_distàncies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Equació_de_la_recta_a_l'espai" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equació_de_la_recta_a_l'espai"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Equació de la recta a l'espai</span> </div> </a> <button aria-controls="toc-Equació_de_la_recta_a_l'espai-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>Commuta la subsecció Equació de la recta a l'espai</span> </button> <ul id="toc-Equació_de_la_recta_a_l'espai-sublist" class="vector-toc-list"> <li id="toc-Recta_determinada_mitjançant_un_sistema_d'equacions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recta_determinada_mitjançant_un_sistema_d'equacions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Recta determinada mitjançant un sistema d'equacions</span> </div> </a> <ul id="toc-Recta_determinada_mitjançant_un_sistema_d'equacions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recta_determinada_mitjançant_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recta_determinada_mitjançant_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Recta determinada mitjançant vectors</span> </div> </a> <ul id="toc-Recta_determinada_mitjançant_vectors-sublist" class="vector-toc-list"> <li id="toc-Posicions_relatives_entre_rectes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Posicions_relatives_entre_rectes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Posicions relatives entre rectes</span> </div> </a> <ul id="toc-Posicions_relatives_entre_rectes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-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="Contingut" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Taula de continguts" > <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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Recta</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="Vés a un article en una altra llengua. Disponible en 112 llengües" > <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-112" 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">112 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Gerade" title="Gerade - alemany suís" lang="gsw" hreflang="gsw" data-title="Gerade" data-language-autonym="Alemannisch" data-language-local-name="alemany suís" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%98%E1%88%B5%E1%88%98%E1%88%AD" title="መስመር - amhàric" lang="am" hreflang="am" data-title="መስመር" data-language-autonym="አማርኛ" data-language-local-name="amhàric" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%82%D9%8A%D9%85_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مستقيم (رياضيات) - àrab" lang="ar" hreflang="ar" data-title="مستقيم (رياضيات)" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A7%B0%E0%A7%87%E0%A6%96%E0%A6%BE" title="ৰেখা - assamès" lang="as" hreflang="as" data-title="ৰেখা" data-language-autonym="অসমীয়া" data-language-local-name="assamès" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Reuta" title="Reuta - asturià" lang="ast" hreflang="ast" data-title="Reuta" data-language-autonym="Asturianu" data-language-local-name="asturià" 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/D%C3%BCz_x%C9%99tt" title="Düz xətt - azerbaidjanès" lang="az" hreflang="az" data-title="Düz xətt" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AF%D9%88%D8%B2_%D8%AE%D8%B7" title="دوز خط - South Azerbaijani" lang="azb" hreflang="azb" data-title="دوز خط" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</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_%D2%BB%D1%8B%D2%99%D1%8B%D2%A1" title="Тура һыҙыҡ - baixkir" lang="ba" hreflang="ba" data-title="Тура һыҙыҡ" data-language-autonym="Башҡортса" data-language-local-name="baixkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Linya" title="Linya - Central Bikol" lang="bcl" hreflang="bcl" data-title="Linya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%BC%D0%B0%D1%8F" title="Прамая - belarús" lang="be" hreflang="be" data-title="Прамая" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%8F" title="Простая - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Простая" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Права (геометрия) - búlgar" lang="bg" hreflang="bg" data-title="Права (геометрия)" data-language-autonym="Български" data-language-local-name="búlgar" 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%B0%E0%A7%87%E0%A6%96%E0%A6%BE" title="রেখা - bengalí" lang="bn" hreflang="bn" data-title="রেখা" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Eeunenn_(geometriezh)" title="Eeunenn (geometriezh) - bretó" lang="br" hreflang="br" data-title="Eeunenn (geometriezh)" data-language-autonym="Brezhoneg" data-language-local-name="bretó" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prava_(geometrija)" title="Prava (geometrija) - bosnià" lang="bs" hreflang="bs" data-title="Prava (geometrija)" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%DB%8E%DA%B5_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="ھێڵ (ئەندازە) - kurd central" lang="ckb" hreflang="ckb" data-title="ھێڵ (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="kurd central" 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/P%C5%99%C3%ADmka" title="Přímka - txec" lang="cs" hreflang="cs" data-title="Přímka" data-language-autonym="Čeština" data-language-local-name="txec" 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%A2%D3%B3%D1%80%C4%95_%D0%B9%C4%95%D1%80" title="Тӳрĕ йĕр - txuvaix" lang="cv" hreflang="cv" data-title="Тӳрĕ йĕр" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" 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/Llinell" title="Llinell - gal·lès" lang="cy" hreflang="cy" data-title="Llinell" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" 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/Linje" title="Linje - danès" lang="da" hreflang="da" data-title="Linje" data-language-autonym="Dansk" data-language-local-name="danès" 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/Gerade" title="Gerade - alemany" lang="de" hreflang="de" data-title="Gerade" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%B8%CE%B5%CE%AF%CE%B1" title="Ευθεία - grec" lang="el" hreflang="el" data-title="Ευθεία" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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/Line_(geometry)" title="Line (geometry) - anglès" lang="en" hreflang="en" data-title="Line (geometry)" data-language-autonym="English" data-language-local-name="anglès" 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/Rekto" title="Rekto - esperanto" lang="eo" hreflang="eo" data-title="Rekto" 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/Recta" title="Recta - espanyol" lang="es" hreflang="es" data-title="Recta" data-language-autonym="Español" data-language-local-name="espanyol" 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/Sirge" title="Sirge - estonià" lang="et" hreflang="et" data-title="Sirge" data-language-autonym="Eesti" data-language-local-name="estonià" 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/Zuzen_(geometria)" title="Zuzen (geometria) - basc" lang="eu" hreflang="eu" data-title="Zuzen (geometria)" data-language-autonym="Euskara" data-language-local-name="basc" 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%AE%D8%B7_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="خط (هندسه) - persa" lang="fa" hreflang="fa" data-title="خط (هندسه)" data-language-autonym="فارسی" data-language-local-name="persa" 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/Suora" title="Suora - finès" lang="fi" hreflang="fi" data-title="Suora" data-language-autonym="Suomi" data-language-local-name="finès" 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/Droite_(math%C3%A9matiques)" title="Droite (mathématiques) - francès" lang="fr" hreflang="fr" data-title="Droite (mathématiques)" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Geraad" title="Geraad - frisó septentrional" lang="frr" hreflang="frr" data-title="Geraad" data-language-autonym="Nordfriisk" data-language-local-name="frisó septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/L%C3%ADne_(geoim%C3%A9adracht)" title="Líne (geoiméadracht) - irlandès" lang="ga" hreflang="ga" data-title="Líne (geoiméadracht)" data-language-autonym="Gaeilge" data-language-local-name="irlandès" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%B7%9A" title="線 - xinès gan" lang="gan" hreflang="gan" data-title="線" data-language-autonym="贛語" data-language-local-name="xinès gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Dr%C3%A8t_(mat%C3%A9matik)" title="Drèt (matématik) - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Drèt (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Loidhne" title="Loidhne - gaèlic escocès" lang="gd" hreflang="gd" data-title="Loidhne" data-language-autonym="Gàidhlig" data-language-local-name="gaèlic escocès" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Recta" title="Recta - gallec" lang="gl" hreflang="gl" data-title="Recta" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B0%E0%AB%87%E0%AA%96%E0%AA%BE" title="રેખા - gujarati" lang="gu" hreflang="gu" data-title="રેખા" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Chh%E1%B9%B3%CC%8Dt-sien" title="Chhṳ̍t-sien - xinès hakka" lang="hak" hreflang="hak" data-title="Chhṳ̍t-sien" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="xinès hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A9%D7%A8" title="ישר - hebreu" lang="he" hreflang="he" data-title="ישר" data-language-autonym="עברית" data-language-local-name="hebreu" 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%B8%E0%A4%B0%E0%A4%B2_%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="सरल रेखा - hindi" lang="hi" hreflang="hi" data-title="सरल रेखा" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pravac" title="Pravac - croat" lang="hr" hreflang="hr" data-title="Pravac" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Dwat_(adjektif)" title="Dwat (adjektif) - crioll d’Haití" lang="ht" hreflang="ht" data-title="Dwat (adjektif)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="crioll d’Haití" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egyenes" title="Egyenes - hongarès" lang="hu" hreflang="hu" data-title="Egyenes" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%82%D5%B2%D5%AB%D5%B2" title="Ուղիղ - armeni" lang="hy" hreflang="hy" data-title="Ուղիղ" data-language-autonym="Հայերեն" data-language-local-name="armeni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Recta" title="Recta - interlingua" lang="ia" hreflang="ia" data-title="Recta" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Garis_(geometri)" title="Garis (geometri) - indonesi" lang="id" hreflang="id" data-title="Garis (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" 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/Lineo_(matematiko)" title="Lineo (matematiko) - ido" lang="io" hreflang="io" data-title="Lineo (matematiko)" 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/L%C3%ADna_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Lína (rúmfræði) - islandès" lang="is" hreflang="is" data-title="Lína (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="islandès" 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/Retta" title="Retta - italià" lang="it" hreflang="it" data-title="Retta" data-language-autonym="Italiano" data-language-local-name="italià" 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%E7%B7%9A" title="直線 - japonès" lang="ja" hreflang="ja" data-title="直線" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lain_(matimatix)" title="Lain (matimatix) - crioll anglès de Jamaica" lang="jam" hreflang="jam" data-title="Lain (matimatix)" data-language-autonym="Patois" data-language-local-name="crioll anglès de Jamaica" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Garis_(g%C3%A9om%C3%A8tri)" title="Garis (géomètri) - javanès" lang="jv" hreflang="jv" data-title="Garis (géomètri)" data-language-autonym="Jawa" data-language-local-name="javanès" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%A0%E1%83%A4%E1%83%94" title="წრფე - georgià" lang="ka" hreflang="ka" data-title="წრფე" data-language-autonym="ქართული" data-language-local-name="georgià" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Ama%C9%A3ad_(tusnakt)" title="Amaɣad (tusnakt) - cabilenc" lang="kab" hreflang="kab" data-title="Amaɣad (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="cabilenc" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kcg mw-list-item"><a href="https://kcg.wikipedia.org/wiki/Lang" title="Lang - tyap" lang="kcg" hreflang="kcg" data-title="Lang" data-language-autonym="Tyap" data-language-local-name="tyap" class="interlanguage-link-target"><span>Tyap</span></a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/M%C5%A9hari_(line)" title="Mũhari (line) - kikuiu" lang="ki" hreflang="ki" data-title="Mũhari (line)" data-language-autonym="Gĩkũyũ" data-language-local-name="kikuiu" 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%A2%D2%AF%D0%B7%D1%83" title="Түзу - kazakh" lang="kk" hreflang="kk" data-title="Түзу" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9E%93%E1%9F%92%E1%9E%91%E1%9E%B6%E1%9E%8F%E1%9F%8B_(%E1%9E%92%E1%9E%9A%E1%9E%8E%E1%9E%B8%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A)" title="បន្ទាត់ (ធរណីមាត្រ) - khmer" lang="km" hreflang="km" data-title="បន្ទាត់ (ធរណីមាត្រ)" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B2%B0%E0%B2%B3%E0%B2%B0%E0%B3%87%E0%B2%96%E0%B3%86" title="ಸರಳರೇಖೆ - kannada" lang="kn" hreflang="kn" data-title="ಸರಳರೇಖೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%81%EC%84%A0" title="직선 - coreà" lang="ko" hreflang="ko" data-title="직선" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Dirust" title="Dirust - kurd" lang="ku" hreflang="ku" data-title="Dirust" data-language-autonym="Kurdî" data-language-local-name="kurd" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Linea_(mathematica)" title="Linea (mathematica) - llatí" lang="la" hreflang="la" data-title="Linea (mathematica)" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Reta" title="Reta - llombard" lang="lmo" hreflang="lmo" data-title="Reta" data-language-autonym="Lombard" data-language-local-name="llombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Monk%C9%94%CC%81l%C9%94%CC%81t%C9%94%CC%81" title="Monkɔ́lɔ́tɔ́ - lingala" lang="ln" hreflang="ln" data-title="Monkɔ́lɔ́tɔ́" data-language-autonym="Lingála" data-language-local-name="lingala" class="interlanguage-link-target"><span>Lingála</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ties%C4%97" title="Tiesė - lituà" lang="lt" hreflang="lt" data-title="Tiesė" data-language-autonym="Lietuvių" data-language-local-name="lituà" 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/Taisne" title="Taisne - letó" lang="lv" hreflang="lv" data-title="Taisne" data-language-autonym="Latviešu" data-language-local-name="letó" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Hitsy_(je%C3%B4metria)" title="Hitsy (jeômetria) - malgaix" lang="mg" hreflang="mg" data-title="Hitsy (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgaix" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Права (геометрија) - macedoni" lang="mk" hreflang="mk" data-title="Права (геометрија)" data-language-autonym="Македонски" data-language-local-name="macedoni" 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%A8%E0%B5%87%E0%B5%BC%E2%80%8C%E0%B4%B0%E0%B5%87%E0%B4%96" title="നേർ‌രേഖ - malaiàlam" lang="ml" hreflang="ml" data-title="നേർ‌രേഖ" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A8%D1%83%D0%BB%D1%83%D1%83%D0%BD_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA)" title="Шулуун (математик) - mongol" lang="mn" hreflang="mn" data-title="Шулуун (математик)" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Garis_(geometri)" title="Garis (geometri) - malai" lang="ms" hreflang="ms" data-title="Garis (geometri)" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" 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%99%E1%80%BB%E1%80%89%E1%80%BA%E1%80%B8" title="မျဉ်း - birmà" lang="my" hreflang="my" data-title="မျဉ်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmà" 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/Lien_(Mathematik)" title="Lien (Mathematik) - baix alemany" lang="nds" hreflang="nds" data-title="Lien (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="baix alemany" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%A7%E0%A5%8D%E0%A4%B5%E0%A4%83" title="ध्वः - newari" lang="new" hreflang="new" data-title="ध्वः" data-language-autonym="नेपाल भाषा" data-language-local-name="newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lijn_(meetkunde)" title="Lijn (meetkunde) - neerlandès" lang="nl" hreflang="nl" data-title="Lijn (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandès" 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/Linje" title="Linje - noruec nynorsk" lang="nn" hreflang="nn" data-title="Linje" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec 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/Linje" title="Linje - noruec bokmål" lang="nb" hreflang="nb" data-title="Linje" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Drecha_(matematicas)" title="Drecha (matematicas) - occità" lang="oc" hreflang="oc" data-title="Drecha (matematicas)" data-language-autonym="Occitan" data-language-local-name="occità" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Sarara" title="Sarara - oromo" lang="om" hreflang="om" data-title="Sarara" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%B0%E0%A8%B2_%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BE" title="ਸਰਲ ਰੇਖਾ - panjabi" lang="pa" hreflang="pa" data-title="ਸਰਲ ਰੇਖਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pfl mw-list-item"><a href="https://pfl.wikipedia.org/wiki/Grad_(Linie)" title="Grad (Linie) - alemany palatí" lang="pfl" hreflang="pfl" data-title="Grad (Linie)" data-language-autonym="Pälzisch" data-language-local-name="alemany palatí" class="interlanguage-link-target"><span>Pälzisch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prosta" title="Prosta - polonès" lang="pl" hreflang="pl" data-title="Prosta" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%A9%D8%B1%DA%9A%D9%87_(%D9%85%DB%90%DA%86%D9%BE%D9%88%D9%87%D9%86%D9%87)" title="کرښه (مېچپوهنه) - paixtu" lang="ps" hreflang="ps" data-title="کرښه (مېچپوهنه)" data-language-autonym="پښتو" data-language-local-name="paixtu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Reta" title="Reta - portuguès" lang="pt" hreflang="pt" data-title="Reta" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Siwk_siq%27i" title="Siwk siq'i - quítxua" lang="qu" hreflang="qu" data-title="Siwk siq'i" data-language-autonym="Runa Simi" data-language-local-name="quítxua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Dreapt%C4%83" title="Dreaptă - romanès" lang="ro" hreflang="ro" data-title="Dreaptă" data-language-autonym="Română" data-language-local-name="romanès" 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%B0%D1%8F" title="Прямая - rus" lang="ru" hreflang="ru" data-title="Прямая" data-language-autonym="Русский" data-language-local-name="rus" 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/L%C3%ACnia_ritta" title="Lìnia ritta - sicilià" lang="scn" hreflang="scn" data-title="Lìnia ritta" data-language-autonym="Sicilianu" data-language-local-name="sicilià" 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/Pravac" title="Pravac - serbocroat" lang="sh" hreflang="sh" data-title="Pravac" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" 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/Line" title="Line - Simple English" lang="en-simple" hreflang="en-simple" data-title="Line" 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/Priamka" title="Priamka - eslovac" lang="sk" hreflang="sk" data-title="Priamka" data-language-autonym="Slovenčina" data-language-local-name="eslovac" 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/Premica" title="Premica - eslovè" lang="sl" hreflang="sl" data-title="Premica" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mutsetse" title="Mutsetse - shona" lang="sn" hreflang="sn" data-title="Mutsetse" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Drejt%C3%ABza" title="Drejtëza - albanès" lang="sq" hreflang="sq" data-title="Drejtëza" data-language-autonym="Shqip" data-language-local-name="albanès" 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%9F%D1%80%D0%B0%D0%B2%D0%B0_(%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D0%B0)" title="Права (линија) - serbi" lang="sr" hreflang="sr" data-title="Права (линија)" data-language-autonym="Српски / srpski" data-language-local-name="serbi" 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/R%C3%A4t_linje" title="Rät linje - suec" lang="sv" hreflang="sv" data-title="Rät linje" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mstari_mnyoofu" title="Mstari mnyoofu - suahili" lang="sw" hreflang="sw" data-title="Mstari mnyoofu" data-language-autonym="Kiswahili" data-language-local-name="suahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="கோடு (வடிவவியல்) - tàmil" lang="ta" hreflang="ta" data-title="கோடு (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="tàmil" 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%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%87" title="เส้นตรง - tai" lang="th" hreflang="th" data-title="เส้นตรง" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Guhit_(heometriya)" title="Guhit (heometriya) - tagal" lang="tl" hreflang="tl" data-title="Guhit (heometriya)" data-language-autonym="Tagalog" data-language-local-name="tagal" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Fru_(geometri)" title="Doğru (geometri) - turc" lang="tr" hreflang="tr" data-title="Doğru (geometri)" data-language-autonym="Türkçe" data-language-local-name="turc" 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%9F%D1%80%D1%8F%D0%BC%D0%B0" title="Пряма - ucraïnès" lang="uk" hreflang="uk" data-title="Пряма" data-language-autonym="Українська" data-language-local-name="ucraïnès" 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/%D8%AE%D8%B7_%D9%85%D8%B3%D8%AA%D9%82%DB%8C%D9%85" title="خط مستقیم - urdú" lang="ur" hreflang="ur" data-title="خط مستقیم" data-language-autonym="اردو" data-language-local-name="urdú" 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/To%CA%BBg%CA%BBri_chiziq" title="Toʻgʻri chiziq - uzbek" lang="uz" hreflang="uz" data-title="Toʻgʻri chiziq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_th%E1%BA%B3ng" title="Đường thẳng - vietnamita" lang="vi" hreflang="vi" data-title="Đường thẳng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Bagis" title="Bagis - waray" lang="war" hreflang="war" data-title="Bagis" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%9B%B4%E7%BA%BF" title="直线 - xinès wu" lang="wuu" hreflang="wuu" data-title="直线" data-language-autonym="吴语" data-language-local-name="xinès wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A9%D7%98%D7%A8%D7%99%D7%9A" title="שטריך - ídix" lang="yi" hreflang="yi" data-title="שטריך" data-language-autonym="ייִדיש" data-language-local-name="ídix" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%9B%B4%E7%BA%BF" title="直线 - xinès" lang="zh" hreflang="zh" data-title="直线" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%B7%9A" title="線 - xinès clàssic" lang="lzh" hreflang="lzh" data-title="線" data-language-autonym="文言" data-language-local-name="xinès clàssic" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ti%CC%8Dt-ch%C5%8Da" title="Ti̍t-chōa - xinès min del sud" lang="nan" hreflang="nan" data-title="Ti̍t-chōa" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9B%B4%E7%B6%AB" title="直綫 - cantonès" lang="yue" hreflang="yue" data-title="直綫" data-language-autonym="粵語" data-language-local-name="cantonès" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q37105#sitelinks-wikipedia" title="Modificau enllaços interlingües" class="wbc-editpage">Modifica els enllaços</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espais de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Recta" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussi%C3%B3:Recta&action=edit&redlink=1" rel="discussion" class="new" title="Discussió sobre el contingut d'aquesta pàgina (encara no existeix) [t]" accesskey="t"><span>Discussió</span></a></li> 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mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Recta"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Recta&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Recta&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" 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vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Recta"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Recta&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Recta&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Recta" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Recta" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Recta&oldid=34505252" title="Enllaç permanent a aquesta revisió de la pàgina"><span>Enllaç permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Recta&action=info" title="Més informació sobre aquesta pàgina"><span>Informació de la pàgina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citau&page=Recta&id=34505252&wpFormIdentifier=titleform" title="Informació sobre com citar aquesta pàgina"><span>Citau aquest article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FRecta"><span>Obtén una URL abreujada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FRecta"><span>Descarrega el codi QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Recta"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Recta&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Recta&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Lines" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q37105" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:FuncionLineal01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/FuncionLineal01.svg/220px-FuncionLineal01.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/FuncionLineal01.svg/330px-FuncionLineal01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/FuncionLineal01.svg/440px-FuncionLineal01.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Les línies vermella i blava d'aquest gràfic tenen el mateix <a href="/wiki/Pendent_(matem%C3%A0tiques)" title="Pendent (matemàtiques)">pendent</a>; les línies vermella i verda tenen la mateixa <a href="/wiki/Intersecci%C3%B3_de_rectes" title="Intersecció de rectes">intersecció</a> amb l'eix <i>y</i> (creuen l'eix <i>y</i> en el mateix lloc).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:1D_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/1D_line.svg/220px-1D_line.svg.png" decoding="async" width="220" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/1D_line.svg/330px-1D_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/1D_line.svg/440px-1D_line.svg.png 2x" data-file-width="270" data-file-height="25" /></a><figcaption>Representació d'un <a href="/wiki/Segment_lineal" title="Segment lineal">segment</a> de recta.</figcaption></figure> <p>Una <b>recta</b>, o <b>línia recta</b>, és un objecte geomètric format per un conjunt d'<a href="/wiki/Infinit" title="Infinit">infinits</a> <a href="/wiki/Punt_(geometria)" title="Punt (geometria)">punts</a>, infinitament <a href="/wiki/Llarg" class="mw-redirect" title="Llarg">llarg</a> i infinitament prim, que no té <a href="/wiki/Curvatura" title="Curvatura">curvatura</a>.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> També es diu que els punts d'una recta estan <b>alineats</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicions_i_postulats_d'Euclides"><span id="Definicions_i_postulats_d.27Euclides"></span>Definicions i postulats d'Euclides</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=1" title="Modifica la secció: Definicions i postulats d'Euclides"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Euclides" title="Euclides">Euclides</a>, en el seu tractat anomenat <i><a href="/wiki/Elements_d%27Euclides" title="Elements d'Euclides">Els Elements</a>,</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> estableix diverses definicions relacionades amb la línia i la línia recta: </p> <ul><li>Una línia és una <a href="/wiki/Longitud" title="Longitud">longitud</a> sense <a href="/wiki/Amplada" title="Amplada">amplada</a> (Llibre I, definició 2).</li> <li>Els extrems d'una línia són punts (Llibre I, definició 3).</li> <li>Una línia recta és aquella que jeu per igual respecte dels punts que hi són (Llibre I, definició 4).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Característiques_de_la_recta"><span id="Caracter.C3.ADstiques_de_la_recta"></span>Característiques de la recta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=2" title="Modifica la secció: Característiques de la recta"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>La recta es prolonga indefinidament en ambdós sentits.</li> <li>En <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">geometria euclidiana</a>, la distància més curta entre dos punts és la línia recta.</li> <li>La recta es pot definir com el conjunt de punts situats al llarg de la intersecció de dos plans.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Semirecta">Semirecta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=3" title="Modifica la secció: Semirecta"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Faisceau_divergent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Faisceau_divergent.svg/220px-Faisceau_divergent.svg.png" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Faisceau_divergent.svg/330px-Faisceau_divergent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Faisceau_divergent.svg/440px-Faisceau_divergent.svg.png 2x" data-file-width="1024" data-file-height="525" /></a><figcaption><a href="/wiki/Feix_(matem%C3%A0tiques)" title="Feix (matemàtiques)">Feix</a> de raigs.</figcaption></figure> <p>S'anomena <b>semirecta</b><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>nota 1<span class="cite-bracket">]</span></a></sup> cadascuna de les dues parts en què queda dividida una recta en ser tallada a qualsevol dels seus punts. És la part d'una recta conformada per tots els punts que s'ubiquen cap a un costat d'un punt fix de la recta, anomenat origen, a partir del qual s'estén indefinidament en un sol sentit. </p> <div class="mw-heading mw-heading3"><h3 id="Semirecta_oposada">Semirecta oposada</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=4" title="Modifica la secció: Semirecta oposada"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <b>semirecta oposada</b> d'una semirecta és l'altra sortida semirecta de la recta que defineix la primera.<sup id="cite_ref-c_8-1" class="reference"><a href="#cite_note-c-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-d_9-1" class="reference"><a href="#cite_note-d-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Cada semirecta només té una semirecta oposada.</li> <li>Una semirecta i la seva semirecta oposada tenen el mateix origen.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Posicions_relatives_de_les_rectes">Posicions relatives de les rectes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=5" title="Modifica la secció: Posicions relatives de les rectes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Dues rectes són <b>coplanàries</b> si poden estar contingudes en un mateix pla. En cas contrari es diu que les rectes <b><a href="/wiki/Rectes_que_es_creuen" title="Rectes que es creuen">es creuen</a></b>.</li> <li>Dues rectes són <b>paral·leles</b> si són coplanàries i no tenen cap punt en comú.</li> <li>Dues rectes són <b>secants</b> si tenen un sol punt en comú. En aquest cas, també són coplanàries.</li> <li>Dues rectes són <b>perpendiculars</b> si es tallen formant <a href="/wiki/Angle" title="Angle">angles</a> de 90°.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Les_rectes_en_geometria">Les rectes en geometria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=6" title="Modifica la secció: Les rectes en geometria"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Geometria" title="Geometria">geometria</a>, la recta és un <a href="/wiki/Conjunt" title="Conjunt">conjunt</a> d'infinits <a href="/wiki/Punt_(geometria)" title="Punt (geometria)">punts</a>, <a href="/wiki/Subconjunt" title="Subconjunt">subconjunt</a> parcial dels infinits punts que formen un <a href="/wiki/Pla" title="Pla">pla</a> i que compleix unes determinades <a href="/wiki/Propietat_(ontologia)" title="Propietat (ontologia)">propietats</a>. És un ens fonamental (juntament amb el punt i el pla) que no admet una <a href="/wiki/Definici%C3%B3" title="Definició">definició</a> més concreta. Simplement, s'enuncien les propietats i se n'accepta l'existència de forma <a href="/wiki/Axiom%C3%A0tica" class="mw-redirect" title="Axiomàtica">axiomàtica</a>. Aquestes propietats (no <a href="/wiki/Demostraci%C3%B3_(matem%C3%A0tiques)" title="Demostració (matemàtiques)">demostrables</a>) són les següents: </p><p>1. Per dos punts diferents hi passa una recta i només una. </p><p>2. Si dos punts d'una recta estan en un pla, llavors tots els altres punts de la recta també estan continguts en aquest pla. </p><p>3. La recta és un conjunt de punts linealment ordenat, obert i dens, on: </p> <dl><dd><dl><dd><ul><li><i>Linealment ordenat</i> significa que, donada una terna de punts, A, B i C, si A precedeix B i B precedeix a C, llavors A precedeix a C.</li> <li><i>Obert</i> significa que no existeix ni un primer ni un últim punt.</li> <li><i>Dens</i> significa que entre dos punts d'una recta sempre n'hi ha infinits més, de manera que no existeixen punts consecutius.</li></ul></dd></dl></dd></dl> <p>4. Tota recta continguda en un pla estableix una divisió dels punts del pla no continguts en la recta en dues úniques regions tals que tot punt del pla exterior a la recta pertany a una o altra regió, i de manera que, escollits dos punts que pertanyin a diferents regions, la recta que els conté té un punt situat entre ells que pertany a la recta original i viceversa. </p><p>5. Per un punt exterior a una recta, hi passa una (i només una) recta tal que les dues estan contingudes en un mateix pla i no tenen entre elles cap punt en comú (<a href="/wiki/Paral%C2%B7lela" class="mw-redirect" title="Paral·lela">paral·lela</a>). </p><p>6. Donada una classificació dels punts d'una recta en dues regions que compleix: </p> <dl><dd><dl><dd><ul><li>Existeixen punts de la recta d'una i altra regió.</li> <li>Tot punt de la recta pertany a una o altra regió.</li> <li>Tot punt d'una regió precedeix a tot punt de l'altra regió.</li></ul></dd> <dd>llavors existeix un sol punt de la recta tal que tots els punts que el precedeixen pertanyen a la primera regió i tots els punts que el segueixen pertanyen a la segona regió.</dd></dl></dd></dl> <p><i>Temes relacionats amb els punts 4 i 6</i>: <a href="/wiki/Semipl%C3%A0" title="Semiplà">semiplà</a>, <a href="/wiki/Semirecta" title="Semirecta">semirecta</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Altres_propietats_de_les_rectes">Altres propietats de les rectes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=7" title="Modifica la secció: Altres propietats de les rectes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Una recta i un punt no contingut en ella determinen un pla que passa per ells.</li></ul> <dl><dd><b>Demostració</b>: per definició de recta, aquesta està formada per infinits punts, dels quals se'n consideren dos. Així, es té un conjunt de 3 punts no alineats. Una de les propietats axiomàtiques del <a href="/wiki/Pla" title="Pla">pla</a> és que per tres punts no alineats hi passa un pla i només un. Aquest pla que passa pels tres punts passa pel punt no contingut en la recta i també per la recta, ja que una de les propietats axiomàtiques de la recta (la segona) diu que si dos punts d'una recta estan en un pla, llavors tots els altres punts de la recta també estan continguts en aquest pla, <a href="/wiki/Quod_erat_demonstrandum" title="Quod erat demonstrandum">QED.</a></dd></dl> <ul><li>Dues rectes que es tallen determinen un pla que passa per elles (i per tant, són coplanàries).</li></ul> <dl><dd><b>Demostració</b>: per definició de recta, aquesta està formada per infinits punts, dels quals se'n considera un d'una de les rectes que interseca. Aplicant la propietat anterior, se segueix que existeix un pla que passa per aquest punt i que conté tota l'altra recta, en particular el punt comú entre les dues rectes. Aquest pla, també conté la primera recta, ja que una de les propietats axiomàtiques de la recta (la segona) diu que si dos punts d'una recta estan en un pla, llavors tots els altres punts de la recta també estan continguts en aquest pla, QED.</dd></dl> <ul><li>Si una recta (<i>a</i>) talla a una altra recta <i>b</i>, llavors <i>a</i> també talla totes les paral·leles a <i>b</i> contingudes en el pla que determinen <i>a</i> i <i>b</i>.</li></ul> <dl><dd><b>Demostració</b>, per <a href="/wiki/Reducci%C3%B3_a_l%27absurd" title="Reducció a l'absurd">reducció a l'absurd</a>: se suposa <i>a</i> que no talla a una paral·lela de <i>b</i> (<i>c</i>), continguda en el mateix pla que defineixen <i>a</i> i <i>b</i>, que intersequen al punt I. Llavors, pel punt I, exterior a <i>c</i> existirien dues rectes paral·leles a <i>c</i>, cosa que entra en contradicció amb una propietat axiomàtica de la recta (la cinquena).</dd></dl> <ul><li>Si dues rectes (<i>a</i> i <i>b</i>) són paral·leles a una tercera (<i>c</i>), llavors són paral·leles entre si (<a href="/wiki/Propietat_transitiva" class="mw-redirect" title="Propietat transitiva">propietat transitiva</a>).</li></ul> <dl><dd><b>Demostració</b> (encara no disponible a Viquipèdia).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Les_rectes_en_matemàtiques"><span id="Les_rectes_en_matem.C3.A0tiques"></span>Les rectes en matemàtiques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=8" title="Modifica la secció: Les rectes en matemàtiques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En un <a href="/wiki/Espai_vectorial" title="Espai vectorial">espai vectorial</a> (per exemple <b>R</b><sup><i>2</i></sup> o <b>R</b><sup><i>3</i></sup>) es defineix la recta <i>r</i> com: </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=\{{\vec {a}}+t{\vec {b}}\mid t\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∣</mo> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\{{\vec {a}}+t{\vec {b}}\mid t\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e00552512e4d769e765419895cd284155db537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.771ex; height:3.343ex;" alt="{\displaystyle r=\{{\vec {a}}+t{\vec {b}}\mid t\in \mathbb {R} \}}" /></span></dd></dl> <p>on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" /></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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" /></span> són <a href="/wiki/Vector_(Matem%C3%A0tiques)" class="mw-redirect" title="Vector (Matemàtiques)">vectors</a> (per exemple de <b>R</b><sup><i>2</i></sup> o <b>R</b><sup><i>3</i></sup>) fixos 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 {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" /></span> és no nul. <i>t</i> és un <a href="/wiki/Par%C3%A0metre" title="Paràmetre">paràmetre</a> real lliure que és el paràmetre arc quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" /></span> és unitari. El vector <b>b</b> descriu la direcció de la recta 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 {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" /></span> és un punt de la recta. Aquesta <a href="/wiki/Equaci%C3%B3" title="Equació">equació</a> és l'anomenada equació vectorial d'una recta. </p><p>De forma més abstracte, hom sovint assumeix que els punts d'una recta es corresponen d'un a un amb els <a href="/wiki/Nombre_real" title="Nombre real">nombres reals</a>. </p> <div class="mw-heading mw-heading3"><h3 id="La_recta_en_R²"><span id="La_recta_en_R.C2.B2"></span>La recta en R²</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=9" title="Modifica la secció: La recta en R²"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Equacions_de_la_recta_en_R²"><span id="Equacions_de_la_recta_en_R.C2.B2"></span>Equacions de la recta en R²</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=10" title="Modifica la secció: Equacions de la recta en R²"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fitxer:Recta.png" class="mw-file-description" title="Recta"><img alt="Recta" src="//upload.wikimedia.org/wikipedia/commons/6/6f/Recta.png" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a></span> </p> <ul><li><b>Equació vectorial</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:(x,y)=(x_{0},y_{0})+{\lambda }\cdot (v_{1},v_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:(x,y)=(x_{0},y_{0})+{\lambda }\cdot (v_{1},v_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0dead1e9c6e53d9bb0aa5c20c4ba5cfb3e2fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.915ex; height:2.843ex;" alt="{\displaystyle r:(x,y)=(x_{0},y_{0})+{\lambda }\cdot (v_{1},v_{2})}" /></span> on: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c7b4383bfe8c664f9feae5c2c132db4bafce3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.166ex; height:2.843ex;" alt="{\displaystyle P(x_{0},y_{0})}" /></span> és un punt per on passa la recta.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80912e657954ba7a08dc822fac02fd5fd61067c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle {\lambda }}" /></span> és un paràmetre tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\lambda }\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\lambda }\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f0649c0961ffabfebf9f8c162e4bebf4937ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle {\lambda }\in \mathbb {R} }" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=(v_{1},v_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=(v_{1},v_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3558d60f162272f0f743bdeb6ddd11a94f0734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.481ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}=(v_{1},v_{2})}" /></span> és un vector que dona la direcció de la recta i s'anomena <b>vector director</b> de la recta.</li></ul></li> <li><b>Equacions paramètriques</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:\left\{{\begin{matrix}x=x_{0}+{\lambda }\cdot v_{1}\\y=y_{0}+{\lambda }\cdot v_{2}\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:\left\{{\begin{matrix}x=x_{0}+{\lambda }\cdot v_{1}\\y=y_{0}+{\lambda }\cdot v_{2}\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6105f23c0f9a66bc3f060aec0be49f234189e0fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.349ex; height:6.176ex;" alt="{\displaystyle r:\left\{{\begin{matrix}x=x_{0}+{\lambda }\cdot v_{1}\\y=y_{0}+{\lambda }\cdot v_{2}\end{matrix}}\right.}" /></span>.</li></ul> <dl><dd>S'obtenen directament de desglossar l'equació vectorial.<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl> <ul><li><b>Equació contínua</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:{\frac {x-x_{0}}{v_{1}}}={\frac {y-y_{0}}{v_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:{\frac {x-x_{0}}{v_{1}}}={\frac {y-y_{0}}{v_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01bcbbe50954f8258f26c6fba3bb9aa698b62d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.5ex; height:5.509ex;" alt="{\displaystyle r:{\frac {x-x_{0}}{v_{1}}}={\frac {y-y_{0}}{v_{2}}}}" /></span></li></ul> <dl><dd>S'obté de plantejar la igualació del paràmetre λ de les dues equacions paramètriques. Les rectes paral·leles a algun dels <a href="/wiki/Eixos_coordenats" class="mw-redirect" title="Eixos coordenats">eixos coordenats</a>, no es poden expressar amb aquesta forma, ja que tindrien algun denominador nul.</dd></dl> <ul><li><b>Equació general o implícita</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:A\cdot x+B\cdot y+C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>A</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:A\cdot x+B\cdot y+C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12b83b7e4d37c843a8daf5516cf852fe3510198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.044ex; height:2.509ex;" alt="{\displaystyle r:A\cdot x+B\cdot y+C=0}" /></span></li></ul> <dl><dd>amb <i>A</i>, <i>B</i> i <i>C</i>, coeficients reals fixos tals que <i>A</i> i <i>B</i> són no nuls.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd> <dd>Aquesta equació s'obté de l'equació contínua considerant:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=v_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a4c8f17faaad7dee5a0180a7f911e310967d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.023ex; height:2.509ex;" alt="{\displaystyle A=v_{2}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=-v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=-v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fe19e14c2a9e09621060922d51f16aecf9a6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.852ex; height:2.509ex;" alt="{\displaystyle B=-v_{1}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=v_{1}\cdot y_{0}-v_{2}\cdot x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=v_{1}\cdot y_{0}-v_{2}\cdot x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17cfda53b99b7236cab6037db1b679e0c74e534d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.004ex; height:2.509ex;" alt="{\displaystyle C=v_{1}\cdot y_{0}-v_{2}\cdot x_{0}}" /></span>.</dd></dl> <ul><li><b>Equació explícita</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:y=m\cdot x+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:y=m\cdot x+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62dd1d19ab846838c1bb76e65674281d3e1de6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.524ex; height:2.343ex;" alt="{\displaystyle r:y=m\cdot x+n}" /></span></li></ul> <dl><dd>S'obté aïllant <i>y</i> en l'equació anterior i considerant:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-{\frac {A}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>B</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-{\frac {A}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f894c3ebacf7e43cce8bc853167cc04f914428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.547ex; height:5.343ex;" alt="{\displaystyle m=-{\frac {A}{B}}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=-{\frac {C}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>B</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=-{\frac {C}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e1bf98730657405eae4f27f94f0885b40d304e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.904ex; height:5.343ex;" alt="{\displaystyle n=-{\frac {C}{B}}}" /></span>, on <dl><dd><ul><li><i>m</i> s'anomena <b><a href="/wiki/Pendent_(matem%C3%A0tiques)" title="Pendent (matemàtiques)">pendent</a></b> de la recta i el seu valor és el de la <a href="/wiki/Tangent" title="Tangent">tangent</a> de l'<a href="/wiki/Angle_(geometria)" class="mw-redirect" title="Angle (geometria)">angle</a> (α) que forma la recta amb l'eix <i>x</i>.</li> <li><i>n</i> és l'ordenada del punt d'intersecció entre la recta i l'eix <i>y</i>.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul></dd></dl></dd></dl> <ul><li><b>Equació canònica</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:{\frac {x}{p}}+{\frac {y}{n}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>p</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:{\frac {x}{p}}+{\frac {y}{n}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d240a51af20e58c000e11d18b1c2516f166e1a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.484ex; height:5.343ex;" alt="{\displaystyle r:{\frac {x}{p}}+{\frac {y}{n}}=1}" /></span></li></ul> <dl><dd>Obtinguda dividint l'equació implícita per <i>C</i> i anomenant</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=-{\frac {C}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=-{\frac {C}{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a756468434b7903298458a888ba98808b53d5074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:8.768ex; height:5.509ex;" alt="{\displaystyle p=-{\frac {C}{A}}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=-{\frac {C}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>B</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=-{\frac {C}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e1bf98730657405eae4f27f94f0885b40d304e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.904ex; height:5.343ex;" alt="{\displaystyle n=-{\frac {C}{B}}}" /></span>.</dd> <dd>Els paràmetres resulten ser tals que la recta interseca amb els eixos de coordenades en els punts <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b63cd35d0b2ba89c90f43af1ffe40b0132c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.175ex; height:2.843ex;" alt="{\displaystyle (p,0)}" /></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 (0,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9516a58dc3bd99e060e9ec8565620a57a3a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (0,n)}" /></span>.</dd></dl> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fitxer:RCanonica.png" class="mw-file-description" title="Equació canònica de la recta"><img alt="Equació canònica de la recta" src="//upload.wikimedia.org/wikipedia/commons/b/b7/RCanonica.png" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a></span> </p> <ul><li><b>Equació punt-pendent</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:y-y_{0}=m\cdot (x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:y-y_{0}=m\cdot (x-x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9d9d9b61db21c8f909abba86d1c56dfc4a3401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.356ex; height:2.843ex;" alt="{\displaystyle r:y-y_{0}=m\cdot (x-x_{0})}" /></span></li></ul> <dl><dd>S'obté aïllant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y-y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62512df70faca0230914a2d6ad1cf791e6b4a3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.189ex; height:2.343ex;" alt="{\displaystyle y-y_{0}}" /></span> de l'equació contínua i considerant:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {v_{2}}{v_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {v_{2}}{v_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82660af6d0ceaf3be57a4144a2e1d0dac49c3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.157ex; height:5.009ex;" alt="{\displaystyle m={\frac {v_{2}}{v_{1}}}}" /></span>, que torna a ser el pendent de la recta ja que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=v_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a4c8f17faaad7dee5a0180a7f911e310967d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.023ex; height:2.509ex;" alt="{\displaystyle A=v_{2}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=-v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=-v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fe19e14c2a9e09621060922d51f16aecf9a6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.852ex; height:2.509ex;" alt="{\displaystyle B=-v_{1}}" /></span> i s'havia definit <i>m</i> com <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-{\frac {A}{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>B</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=-{\frac {A}{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f894c3ebacf7e43cce8bc853167cc04f914428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.547ex; height:5.343ex;" alt="{\displaystyle m=-{\frac {A}{B}}}" /></span></dd> <dd>Aquesta equació és especialment útil, ja que permet escriure directament l'equació d'una recta, coneguts el pendent i un punt pertanyent a la recta.</dd></dl> <p><br /> </p> <div style="float:right; padding-left:30px;"> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fitxer:La_recta_en_coordenadas_cartesianas.png" class="mw-file-description" title="La recta en coordenades cartesianes"><img alt="La recta en coordenades cartesianes" src="//upload.wikimedia.org/wikipedia/commons/e/e7/La_recta_en_coordenadas_cartesianas.png" decoding="async" width="345" height="308" class="mw-file-element" data-file-width="345" data-file-height="308" /></a></span> </p> </div> <div class="mw-heading mw-heading4"><h4 id="Rectes_notables_R²"><span id="Rectes_notables_R.C2.B2"></span>Rectes notables R²</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=11" title="Modifica la secció: Rectes notables R²"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>L'equació d'una recta vertical, com la <b>v</b>, respon a l'equació</li></ul> <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=x_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094523d1b9042f136f06d6d660856f9bafd83c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.787ex; height:2.009ex;" alt="{\displaystyle x=x_{v}}" /></span> (constant).</dd></dl> <ul><li>L'equació d'una recta horitzontal, com la <b>h</b>, respon a l'equació</li></ul> <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=y_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b37ae75dac871bff1e1c3e1315dbfb77fbcfc80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.572ex; height:2.009ex;" alt="{\displaystyle y=y_{h}}" /></span> (constant).</dd></dl> <ul><li>Una recta qualsevol que passi per l'origen O (0,0), com la <b>s</b>, complirà la condició <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}" /></span>, i la seva equació explícita serà de la forma <span class="mwe-math-element"><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:y=m\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:y=m\cdot x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8971e6d0a85725e3b811820beec09772ca5891dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.289ex; height:2.009ex;" alt="{\displaystyle r:y=m\cdot x}" /></span>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Paral·lelisme_i_perpendicularitat"><span id="Paral.C2.B7lelisme_i_perpendicularitat"></span>Paral·lelisme i perpendicularitat</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=12" title="Modifica la secció: Paral·lelisme i perpendicularitat"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dues rectes qualssevol (una representada amb primes (') sobre els seus paràmetres i l'altre no) </p> <ul><li>Seran paral·leles si i només si (els següents tres punts són equivalents): <ul><li>els seus vectors directors són paral·lels, cosa que passa quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v_{1}}{v'_{1}}}={\frac {v_{2}}{v'_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v_{1}}{v'_{1}}}={\frac {v_{2}}{v'_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ca96bf0c7512ed53efc0c46df7365d50784d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.134ex; height:5.509ex;" alt="{\displaystyle {\frac {v_{1}}{v'_{1}}}={\frac {v_{2}}{v'_{2}}}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A}{A'}}={\frac {B}{B'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A}{A'}}={\frac {B}{B'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e2d25b8b7dab79ad443df5f6b1418c4b8e4cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.647ex; height:5.509ex;" alt="{\displaystyle {\frac {A}{A'}}={\frac {B}{B'}}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=m'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=m'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b8fdd100704f01b50c0af08880f80f5190ade5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.864ex; height:2.509ex;" alt="{\displaystyle m=m'}" /></span></li></ul></li> <li>Seran <a href="/wiki/Perpendiculars" class="mw-redirect" title="Perpendiculars">perpendiculars</a> (o ortogonals) si i només si (els següents punts també són equivalents) <ul><li>els seus vectors directors són ortogonals, cosa que passa quan el seu <a href="/wiki/Producte_escalar" title="Producte escalar">producte escalar</a> és nul: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}\cdot v'_{1}+v_{2}\cdot v'_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}\cdot v'_{1}+v_{2}\cdot v'_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52c18917de3089e4c93e9617bd3f64f9697c7ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.187ex; height:2.843ex;" alt="{\displaystyle v_{1}\cdot v'_{1}+v_{2}\cdot v'_{2}=0}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot A'+B\cdot B'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>B</mi> <mo>⋅</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot A'+B\cdot B'=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842b9f6627e01dd97181ec31cf377c3a8e13964a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.843ex; height:2.676ex;" alt="{\displaystyle A\cdot A'+B\cdot B'=0}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot m'=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot m'=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8a988d8d5b7715909b6e87272f9d229676b3b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.514ex; height:2.676ex;" alt="{\displaystyle m\cdot m'=-1}" /></span></li></ul></li></ul> <div class="mw-heading mw-heading4"><h4 id="Angles_i_distàncies"><span id="Angles_i_dist.C3.A0ncies"></span>Angles i distàncies</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=13" title="Modifica la secció: Angles i distàncies"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Angle entre dues rectes</b>: Dues rectes que es tallen defineixen quatre angles iguals dos a dos. L'angle (α) que formen les rectes es defineix tal que està entre 0 i 90°. Coneguts els vectors directors amb l'<a href="/wiki/Expressi%C3%B3_algebraica" title="Expressió algebraica">expressió</a> que defineix el que formen els seus vectors directors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" /></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 {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}" /></span>, es pot calcular l'angle que formen amb l'expressió: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \alpha ={\frac {|{\vec {v}}\cdot {\vec {w}}|}{|{\vec {v}}||{\vec {w}}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \alpha ={\frac {|{\vec {v}}\cdot {\vec {w}}|}{|{\vec {v}}||{\vec {w}}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f463b4de84e842e5e0a460f5b278d33ebd002e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.733ex; height:6.509ex;" alt="{\displaystyle \cos \alpha ={\frac {|{\vec {v}}\cdot {\vec {w}}|}{|{\vec {v}}||{\vec {w}}|}}}" /></span></dd></dl> <p><b>Distància entre un punt i una recta</b>: La distància entre una recta (<span class="mwe-math-element"><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:A\cdot x+B\cdot y+C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>A</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:A\cdot x+B\cdot y+C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12b83b7e4d37c843a8daf5516cf852fe3510198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.044ex; height:2.509ex;" alt="{\displaystyle r:A\cdot x+B\cdot y+C=0}" /></span> ) i un punt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(p_{1},p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(p_{1},p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516982ffa733a34f9631a9081e978eed983e44ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.036ex; height:2.843ex;" alt="{\displaystyle P(p_{1},p_{2})}" /></span> exterior a <i>r</i> és la menor de les <a href="/wiki/Dist%C3%A0ncia" title="Distància">distàncies</a> entre el punt <i>P</i> i qualsevol dels punts de la recta <i>r</i>. Aquesta distància es minimitza amb la <a href="/wiki/Projecci%C3%B3_ortogonal" title="Projecció ortogonal">projecció ortogonal</a> de <i>P</i> sobre <i>r</i>, i l'expressió que dona la distància entre <i>P</i> i <i>r'</i> és: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,r)={\frac {|A\cdot p_{1}+B\cdot p_{2}+C|}{\sqrt {A^{2}+B^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>B</mi> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,r)={\frac {|A\cdot p_{1}+B\cdot p_{2}+C|}{\sqrt {A^{2}+B^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3720fb15fe28c91becf18b1bc949504cba588957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.841ex; height:7.009ex;" alt="{\displaystyle d(P,r)={\frac {|A\cdot p_{1}+B\cdot p_{2}+C|}{\sqrt {A^{2}+B^{2}}}}}" /></span></dd></dl> <p><b>Distància entre dues rectes</b>: la distància entre dues rectes és la menor distància entre punts d'una i altra recta. Si les rectes tenen algun punt en comú (si són secants), la distància és 0; si són paral·leles, la distància entre elles ve donada per la distància entre un punt d'una recta i l'altra recta, ja que aquest valor és independent del punt triat. </p> <dl><dd>Si <span class="mwe-math-element"><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:A\cdot x+B\cdot y+C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>A</mi> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:A\cdot x+B\cdot y+C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12b83b7e4d37c843a8daf5516cf852fe3510198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.044ex; height:2.509ex;" alt="{\displaystyle r:A\cdot x+B\cdot y+C=0}" /></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 r':A'\cdot x+B'\cdot y+C'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>:</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <msup> <mi>C</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r':A'\cdot x+B'\cdot y+C'=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6222d7d7a558ef0eaa759652d4addeb4187517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.815ex; height:2.843ex;" alt="{\displaystyle r':A'\cdot x+B'\cdot y+C'=0}" /></span> són dues rectes paral·leles del pla 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 P(p_{1},p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(p_{1},p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516982ffa733a34f9631a9081e978eed983e44ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.036ex; height:2.843ex;" alt="{\displaystyle P(p_{1},p_{2})}" /></span> un punt pertanyent a r, llavors la distància entre r i r' és:</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 d(r,r')=d(P,r')={\frac {\left|A'\cdot p_{1}+B'\cdot p_{2}+C'\right|}{\sqrt {(A')^{2}+({B}')^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>⋅</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> <mo>|</mo> </mrow> <msqrt> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(r,r')=d(P,r')={\frac {\left|A'\cdot p_{1}+B'\cdot p_{2}+C'\right|}{\sqrt {(A')^{2}+({B}')^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa3cec4b812bf2937da557e6883e01bdc54591e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:43.551ex; height:8.509ex;" alt="{\displaystyle d(r,r')=d(P,r')={\frac {\left|A'\cdot p_{1}+B'\cdot p_{2}+C'\right|}{\sqrt {(A')^{2}+({B}')^{2}}}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Equació_de_la_recta_a_l'espai"><span id="Equaci.C3.B3_de_la_recta_a_l.27espai"></span>Equació de la recta a l'espai</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=14" title="Modifica la secció: Equació de la recta a l'espai"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Recta_determinada_mitjançant_un_sistema_d'equacions"><span id="Recta_determinada_mitjan.C3.A7ant_un_sistema_d.27equacions"></span>Recta determinada mitjançant un sistema d'equacions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=15" title="Modifica la secció: Recta determinada mitjançant un sistema d'equacions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Intersecci%C3%B3nEspacioVectorial.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/170px-Intersecci%C3%B3nEspacioVectorial.gif" decoding="async" width="170" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/255px-Intersecci%C3%B3nEspacioVectorial.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Intersecci%C3%B3nEspacioVectorial.gif/340px-Intersecci%C3%B3nEspacioVectorial.gif 2x" data-file-width="417" data-file-height="500" /></a><figcaption>Sistema de 2 equacions i 3 variables.</figcaption></figure> <p>Recta a l'espai usant un <a href="/wiki/Sistema_d%27equacions_lineals" title="Sistema d'equacions lineals">sistema de 2 equacions</a> i 3 incògnites: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{matrix}x&+y&+z&=4\\x&-y&+3z&=7\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mo>=</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mn>3</mn> <mi>z</mi> </mtd> <mtd> <mo>=</mo> <mn>7</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{matrix}x&+y&+z&=4\\x&-y&+3z&=7\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2066eff3c9c122b9ad2d9f1a7fe6e3fce7579b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.43ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{matrix}x&+y&+z&=4\\x&-y&+3z&=7\end{matrix}}\right.}" /></span></dd></dl> <ul><li>Aquesta equació equival a la intersecció de dos <a href="/wiki/Pla_(geometria)" class="mw-redirect" title="Pla (geometria)">plans</a> a l'espai.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Recta_determinada_mitjançant_vectors"><span id="Recta_determinada_mitjan.C3.A7ant_vectors"></span>Recta determinada mitjançant vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=16" title="Modifica la secció: Recta determinada mitjançant vectors"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Recta a l'espai usant un punt, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=(p_{x},p_{y},p_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(p_{x},p_{y},p_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6835967e31168a5689f398fadf765e627c7998f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:14.966ex; height:3.009ex;" alt="{\displaystyle p=(p_{x},p_{y},p_{z})}" /></span>, i un vector, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=(u_{x},u_{y},u_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=(u_{x},u_{y},u_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a2351d447a86943cf053810dd9643a58e4e461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.518ex; height:3.009ex;" alt="{\displaystyle u=(u_{x},u_{y},u_{z})}" /></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)=(p_{x},p_{y},p_{z})+\lambda (u_{x},u_{y},u_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(p_{x},p_{y},p_{z})+\lambda (u_{x},u_{y},u_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70629465caa1a3c485713962d8511a29c98872da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.443ex; height:3.009ex;" alt="{\displaystyle (x,y,z)=(p_{x},p_{y},p_{z})+\lambda (u_{x},u_{y},u_{z})}" /></span></dd></dl> <ul><li>Al vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880f91e25cd451d89d1f6d0d06852b56a7b74a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle u\,}" /></span> se'n diu vector director.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Posicions_relatives_entre_rectes">Posicions relatives entre rectes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=17" title="Modifica la secció: Posicions relatives entre rectes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Dues rectes seran <a href="/wiki/Paral%C2%B7lelisme_(matem%C3%A0tica)" class="mw-redirect" title="Paral·lelisme (matemàtica)">paral·leles</a> si tenen vectors directors paral·lels.</li> <li>Dues rectes seran coincidents si comparteixen almenys dos punts diferents.</li> <li>Dues rectes s'<a href="/wiki/Recta_secant" title="Recta secant">intersequen</a> si no són paral·leles i tenen un punt en comú.</li> <li>Dues rectes seran <a href="/wiki/Coplanaritat" title="Coplanaritat">coplanàries</a><sup id="cite_ref-c_8-2" class="reference"><a href="#cite_note-c-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> si estan contingudes en algun pla. <ul><li>Dues rectes són coplanàries si i només si o bé són coincidents o bé s'intersequen o bé són paral·leles.</li></ul></li> <li>Dues rectes es <a href="/wiki/Rectes_que_es_creuen" title="Rectes que es creuen">creuen</a> si no són paral·leles ni tenen punts comuns.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=18" title="Modifica la secció: Notes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text">També s'usa <b>raig</b> el qual és un possible anglicisme de <i>ray</i><sup id="cite_ref-a_5-0" class="reference"><a href="#cite_note-a-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> a Hispanoamèrica. En alguns textos és esmentat com a raig o semirecta<sup id="cite_ref-b_6-0" class="reference"><a href="#cite_note-b-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> però predomina l'ús de semirecta en bibliografia abundant<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-c_8-0" class="reference"><a href="#cite_note-c-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-d_9-0" class="reference"><a href="#cite_note-d-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> que no recullen cap altra alternativa.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=19" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:1_1-0">1,0</a></sup> <sup><a href="#cite_ref-:1_1-1">1,1</a></sup></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Line.html">Line</a> a MathWorld <style data-mw-deduplicate="TemplateStyles:r33711417">.mw-parser-output .languageicon{font-size:0.95em;color:#555;background-color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .languageicon{background-color:inherit;color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .languageicon{background-color:inherit;color:white}}</style><span class="languageicon" title="En anglès">(anglès)</span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_2-0">2,0</a></sup> <sup><a href="#cite_ref-:0_2-1">2,1</a></sup></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.britannica.com/science/line-mathematics">line | mathematics | Britannica</a>» (en anglès). [Consulta: 21 gener 2022].</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://estudyando.com/que-es-una-linea-recta-definicion-y-ejemplos/">▷ ¿Qué es una línea recta? - Definición y ejemplos</a>» (en castellà), 23-09-2020. [Consulta: 21 gener 2022].</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">www.euclides.org: <i>Los Elementos</i> <a rel="nofollow" class="external autonumber" href="http://www.euclides.org/menu/elements_esp/01/definicioneslibro1.htm">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090306065647/http://www.euclides.org/menu/elements_esp/01/definicioneslibro1.htm">Arxivat</a> 2009-03-06 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-a-5"><span class="mw-cite-backlink"><a href="#cite_ref-a_5-0">↑</a></span> <span class="reference-text"><span class="citació mathworld" id="Referència-Mathworld-Ray"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Rayo.html">«Ray»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></span> </li> <li id="cite_note-b-6"><span class="mw-cite-backlink"><a href="#cite_ref-b_6-0">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <i>Pequeña enciclopedia de matemáticas</i>.  Pagoulatos, 1981.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Peque%C3%B1a+enciclopedia+de+matem%C3%A1ticas&rft.date=1981&rft.pub=Pagoulatos"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://dle.rae.es/?w=semirrecta">semirrecta</a>». <i><a href="/wiki/Diccionario_de_la_lengua_espa%C3%B1ola" title="Diccionario de la lengua española">Diccionario de la lengua española</a></i>.  <a href="/wiki/Real_Academia_Espa%C3%B1ola" class="mw-redirect" title="Real Academia Española">Real Academia Española</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r33711417" /><span class="languageicon" title="En castellà">(castellà)</span>.</span></span> </li> <li id="cite_note-c-8"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-c_8-0">8,0</a></sup> <sup><a href="#cite_ref-c_8-1">8,1</a></sup> <sup><a href="#cite_ref-c_8-2">8,2</a></sup></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> Real Academia de Ciencias Exactas, Física y Naturales. <i>Diccionario esencial de las ciencias</i>.  Espsa, 1999. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/84-239-7921-0" title="Especial:Fonts bibliogràfiques/84-239-7921-0">ISBN 84-239-7921-0</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Diccionario+esencial+de+las+ciencias&rft.date=1999&rft.pub=Espsa&rft.isbn=84-239-7921-0"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-d-9"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-d_9-0">9,0</a></sup> <sup><a href="#cite_ref-d_9-1">9,1</a></sup></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <i>Diccionario de matematicas</i>.  Akal Editores, 1979.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Diccionario+de+matematicas&rft.date=1979&rft.pub=Akal+Editores"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <i>Docta guia educativa</i>.  Carroggio,s.a..</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Docta+guia+educativa&rft.pub=Carroggio%2Cs.a."><span style="display: none;"> </span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <i>Enciclopedia didáctica de matemáticas</i>.  Oceano.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enciclopedia+did%C3%A1ctica+de+matem%C3%A1ticas&rft.pub=Oceano"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <i>Léxico de matemáticas</i>.  Akal Editores.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=L%C3%A9xico+de+matem%C3%A1ticas&rft.pub=Akal+Editores"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.superprof.es/diccionario/matematicas/geometria/rectas.html/">rectas</a>» (en castellà). [Consulta: 21 gener 2022].</span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml">Equations of a Straight Line</a>». [Consulta: 29 gener 2022].</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Recta&action=edit&section=20" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Lines" class="extiw" title="commons:Category:Lines">Recta</a></b></i></div></div> </div> <ul><li><a href="/wiki/Axiomes_de_la_geometria" title="Axiomes de la geometria">Axiomes de la geometria</a></li> <li><a href="/wiki/Recta_num%C3%A8rica" title="Recta numèrica">Recta numèrica</a></li> <li><a href="/wiki/Segment_lineal" title="Segment lineal">Segment lineal</a></li></ul> <p><span style="display: none;" class="interProject"><a href="https://ca.wiktionary.org/wiki/recta" class="extiw" title="wikt:recta">Viccionari</a></span> </p> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> (<a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4156780-8">1</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bases d'informació</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Encyclop%C3%A6dia_Britannica_Online" title="Encyclopædia Britannica Online">Britannica</a> (<a rel="nofollow" class="external text" href="https://www.britannica.com/topic/line-mathematics">1</a>)</li> <li><a href="/wiki/Enciclop%C3%A8dia_Larousse" title="Enciclopèdia Larousse">Larousse</a> (<a rel="nofollow" class="external text" href="https://www.larousse.fr/encyclopedie/divers/droite/186112">1</a>)</li> <li><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=4342&url_prefix=https://snl.no/&id=linje">SNL</a></li> <li><a href="/wiki/Treccani" title="Treccani">Treccani</a> (<a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/retta">1</a>)</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; 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