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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">Início</div> </a> </li> <li id="toc-Planos_no_espaço" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Planos_no_espaço"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Planos no espaço</span> </div> </a> <ul id="toc-Planos_no_espaço-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometria_Euclidiana" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometria_Euclidiana"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geometria Euclidiana</span> </div> </a> <button aria-controls="toc-Geometria_Euclidiana-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>Alternar a subsecção Geometria Euclidiana</span> </button> <ul id="toc-Geometria_Euclidiana-sublist" class="vector-toc-list"> <li id="toc-Orientação" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientação"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Orientação</span> </div> </a> <ul id="toc-Orientação-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Planos_embutidos_no_R_3" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Planos_embutidos_no_R_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Planos embutidos no R <sup> 3 </sup></span> </div> </a> <button aria-controls="toc-Planos_embutidos_no_R_3-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>Alternar a subsecção Planos embutidos no R <sup> 3 </sup></span> </button> <ul id="toc-Planos_embutidos_no_R_3-sublist" class="vector-toc-list"> <li id="toc-Propriedades" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriedades"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Propriedades</span> </div> </a> <ul id="toc-Propriedades-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-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="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <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">Alternar o índice</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">Plano (geometria)</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="Ir para um artigo noutra língua. Disponível em 88 línguas" > <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-88" 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">88 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vlak" title="Vlak — africanês" lang="af" hreflang="af" data-title="Vlak" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik) — alemão suíço" lang="gsw" hreflang="gsw" data-title="Ebene (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%88_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مستو (رياضيات) — árabe" lang="ar" hreflang="ar" data-title="مستو (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Planu_(xeometr%C3%ADa)" title="Planu (xeometría) — asturiano" lang="ast" hreflang="ast" data-title="Planu (xeometría)" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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/M%C3%BCst%C9%99vi" title="Müstəvi — azerbaijano" lang="az" hreflang="az" data-title="Müstəvi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AF%D2%AB%D1%8B%D0%BB%D1%8B%D2%A1" title="Яҫылыҡ — bashkir" lang="ba" hreflang="ba" data-title="Яҫылыҡ" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%81%D0%BA%D0%B0%D1%81%D1%86%D1%8C" title="Плоскасць — bielorrusso" lang="be" hreflang="be" data-title="Плоскасць" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" 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%A0%D0%BE%D1%9E%D0%BD%D1%96%D1%86%D0%B0" title="Роўніца — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Роўніца" 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%A0%D0%B0%D0%B2%D0%BD%D0%B8%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Равнина (математика) — búlgaro" lang="bg" hreflang="bg" data-title="Равнина (математика)" data-language-autonym="Български" data-language-local-name="búlgaro" 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%B8%E0%A6%AE%E0%A6%A4%E0%A6%B2" title="সমতল — bengalês" lang="bn" hreflang="bn" data-title="সমতল" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ravan" title="Ravan — bósnio" lang="bs" hreflang="bs" data-title="Ravan" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Pla" title="Pla — catalão" lang="ca" hreflang="ca" data-title="Pla" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%AD%E1%8F%AB%E1%8E%BE%E1%8F%97%E1%8F%A2_%E1%8F%97%E1%8F%8E%E1%8F%8D%E1%8F%97_%E1%8E%A4%E1%8E%AC%E1%8F%A9%E1%8E%B5" title="ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ — cherokee" lang="chr" hreflang="chr" data-title="ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%AA%DB%95%D8%AE%D8%AA" title="ڕووتەخت — curdo central" lang="ckb" hreflang="ckb" data-title="ڕووتەخت" data-language-autonym="کوردی" data-language-local-name="curdo 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/Rovina" title="Rovina — checo" lang="cs" hreflang="cs" data-title="Rovina" data-language-autonym="Čeština" data-language-local-name="checo" 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%9B%D0%B0%D0%BF%D1%82%D0%B0%D0%BA" title="Лаптак — chuvash" lang="cv" hreflang="cv" data-title="Лаптак" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" 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/Pl%C3%A2n_geometraidd" title="Plân geometraidd — galês" lang="cy" hreflang="cy" data-title="Plân geometraidd" data-language-autonym="Cymraeg" data-language-local-name="galês" 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/Plan_(matematik)" title="Plan (matematik) — dinamarquês" lang="da" hreflang="da" data-title="Plan (matematik)" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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/Ebene_(Mathematik)" title="Ebene (Mathematik) — alemão" lang="de" hreflang="de" data-title="Ebene (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemão" 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%80%CE%AF%CF%80%CE%B5%CE%B4%CE%BF" title="Επίπεδο — grego" lang="el" hreflang="el" data-title="Επίπεδο" data-language-autonym="Ελληνικά" data-language-local-name="grego" 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/Euclidean_plane" title="Euclidean plane — inglês" lang="en" hreflang="en" data-title="Euclidean plane" data-language-autonym="English" data-language-local-name="inglês" 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/Ebeno_(matematiko)" title="Ebeno (matematiko) — esperanto" lang="eo" hreflang="eo" data-title="Ebeno (matematiko)" 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/Plano_(geometr%C3%ADa)" title="Plano (geometría) — espanhol" lang="es" hreflang="es" data-title="Plano (geometría)" data-language-autonym="Español" data-language-local-name="espanhol" 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/Tasand" title="Tasand — estónio" lang="et" hreflang="et" data-title="Tasand" data-language-autonym="Eesti" data-language-local-name="estónio" 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/Plano" title="Plano — basco" lang="eu" hreflang="eu" data-title="Plano" data-language-autonym="Euskara" data-language-local-name="basco" 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%B5%D9%81%D8%AD%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="صفحه (هندسه) — persa" lang="fa" hreflang="fa" data-title="صفحه (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/Taso" title="Taso — finlandês" lang="fi" hreflang="fi" data-title="Taso" data-language-autonym="Suomi" data-language-local-name="finlandês" 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/Plan_(math%C3%A9matiques)" title="Plan (mathématiques) — francês" lang="fr" hreflang="fr" data-title="Plan (mathématiques)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Pl%C3%A1na_(matamaitic)" title="Plána (matamaitic) — irlandês" lang="ga" hreflang="ga" data-title="Plána (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandês" 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/%E5%B9%B3%E9%9D%A2" title="平面 — gan" lang="gan" hreflang="gan" data-title="平面" data-language-autonym="贛語" data-language-local-name="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/Plan_(mat%C3%A9matik)" title="Plan (matématik) — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Plan (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-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Plano_(xeometr%C3%ADa)" title="Plano (xeometría) — galego" lang="gl" hreflang="gl" data-title="Plano (xeometría)" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%99%D7%A9%D7%95%D7%A8_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="מישור (גאומטריה) — hebraico" lang="he" hreflang="he" data-title="מישור (גאומטריה)" data-language-autonym="עברית" data-language-local-name="hebraico" 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%AE%E0%A4%A4%E0%A4%B2_(%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF)" title="समतल (ज्यामिति) — hindi" lang="hi" hreflang="hi" data-title="समतल (ज्यामिति)" 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/Ravnina" title="Ravnina — croata" lang="hr" hreflang="hr" data-title="Ravnina" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/S%C3%ADk_(geometria)" title="Sík (geometria) — húngaro" lang="hu" hreflang="hu" data-title="Sík (geometria)" data-language-autonym="Magyar" data-language-local-name="húngaro" 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%80%D5%A1%D6%80%D5%A9%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հարթություն — arménio" lang="hy" hreflang="hy" data-title="Հարթություն" data-language-autonym="Հայերեն" data-language-local-name="arménio" 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/Plano" title="Plano — interlíngua" lang="ia" hreflang="ia" data-title="Plano" data-language-autonym="Interlingua" data-language-local-name="interlíngua" 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/Bidang_(geometri)" title="Bidang (geometri) — indonésio" lang="id" hreflang="id" data-title="Bidang (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" 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/Plano" title="Plano — ido" lang="io" hreflang="io" data-title="Plano" 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/Sl%C3%A9tta_(r%C3%BAmfr%C3%A6%C3%B0i)" title="Slétta (rúmfræði) — islandês" lang="is" hreflang="is" data-title="Slétta (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="islandês" 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/Piano_(geometria)" title="Piano (geometria) — italiano" lang="it" hreflang="it" data-title="Piano (geometria)" data-language-autonym="Italiano" data-language-local-name="italiano" 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/%E5%B9%B3%E9%9D%A2" title="平面 — japonês" lang="ja" hreflang="ja" data-title="平面" data-language-autonym="日本語" data-language-local-name="japonês" 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/Plien_(matimatix)" title="Plien (matimatix) — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Plien (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%91%E1%83%A0%E1%83%A2%E1%83%A7%E1%83%94" title="სიბრტყე — georgiano" lang="ka" hreflang="ka" data-title="სიბრტყე" data-language-autonym="ქართული" data-language-local-name="georgiano" 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/Agni_(tanzeggit)" title="Agni (tanzeggit) — kabyle" lang="kab" hreflang="kab" data-title="Agni (tanzeggit)" data-language-autonym="Taqbaylit" data-language-local-name="kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%B7%D1%8B%D2%9B%D1%82%D1%8B%D2%9B_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Жазықтық (геометрия) — cazaque" lang="kk" hreflang="kk" data-title="Жазықтық (геометрия)" data-language-autonym="Қазақша" data-language-local-name="cazaque" 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%9F%92%E1%9E%9B%E1%9E%84%E1%9F%8B" title="ប្លង់ — khmer" lang="km" hreflang="km" data-title="ប្លង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EB%A9%B4" title="평면 — coreano" lang="ko" hreflang="ko" data-title="평면" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B3%D0%B8%D0%B7%D0%B4%D0%B8%D0%BA" title="Тегиздик — quirguiz" lang="ky" hreflang="ky" data-title="Тегиздик" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Planum_(geometria)" title="Planum (geometria) — latim" lang="la" hreflang="la" data-title="Planum (geometria)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Plok%C5%A1tuma" title="Plokštuma — lituano" lang="lt" hreflang="lt" data-title="Plokštuma" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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/Plakne" title="Plakne — letão" lang="lv" hreflang="lv" data-title="Plakne" data-language-autonym="Latviešu" data-language-local-name="letão" 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/Lemaka_(je%C3%B4metria)" title="Lemaka (jeômetria) — malgaxe" lang="mg" hreflang="mg" data-title="Lemaka (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgaxe" 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%A0%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Рамнина (математика) — macedónio" lang="mk" hreflang="mk" data-title="Рамнина (математика)" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Satah" title="Satah — malaio" lang="ms" hreflang="ms" data-title="Satah" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Flach_(Mathematik)" title="Flach (Mathematik) — baixo-alemão" lang="nds" hreflang="nds" data-title="Flach (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="baixo-alemão" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vlak_(meetkunde)" title="Vlak (meetkunde) — neerlandês" lang="nl" hreflang="nl" data-title="Vlak (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" 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/Plan_i_matematikk" title="Plan i matematikk — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Plan i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês 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/Plan_(matematikk)" title="Plan (matematikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Plan (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C5%82aszczyzna" title="Płaszczyzna — polaco" lang="pl" hreflang="pl" data-title="Płaszczyzna" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta" title="P'allta — quíchua" lang="qu" hreflang="qu" data-title="P'allta" data-language-autonym="Runa Simi" data-language-local-name="quíchua" 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/Plan_(geometrie)" title="Plan (geometrie) — romeno" lang="ro" hreflang="ro" data-title="Plan (geometrie)" data-language-autonym="Română" data-language-local-name="romeno" 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%D0%BB%D0%BE%D1%81%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Плоскость — russo" lang="ru" hreflang="ru" data-title="Плоскость" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A5%D0%B0%D0%BF%D1%82%D0%B0%D0%BB" title="Хаптал — sakha" lang="sah" hreflang="sah" data-title="Хаптал" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ravan" title="Ravan — servo-croata" lang="sh" hreflang="sh" data-title="Ravan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" 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/Plane_(mathematics)" title="Plane (mathematics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Plane (mathematics)" 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/Rovina_(geometria)" title="Rovina (geometria) — eslovaco" lang="sk" hreflang="sk" data-title="Rovina (geometria)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Ravnina" title="Ravnina — esloveno" lang="sl" hreflang="sl" data-title="Ravnina" data-language-autonym="Slovenščina" data-language-local-name="esloveno" 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/Mutsendo" title="Mutsendo — shona" lang="sn" hreflang="sn" data-title="Mutsendo" 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/Rrafshi_Euklidian" title="Rrafshi Euklidian — albanês" lang="sq" hreflang="sq" data-title="Rrafshi Euklidian" data-language-autonym="Shqip" data-language-local-name="albanês" 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%A0%D0%B0%D0%B2%D0%B0%D0%BD" title="Раван — sérvio" lang="sr" hreflang="sr" data-title="Раван" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" 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/Plan_(geometri)" title="Plan (geometri) — sueco" lang="sv" hreflang="sv" data-title="Plan (geometri)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%B3%E0%AE%AE%E0%AF%8D_(%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="தளம் (வடிவவியல்) — tâmil" lang="ta" hreflang="ta" data-title="தளம் (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%99%E0%B8%B2%E0%B8%9A" title="ระนาบ — tailandês" lang="th" hreflang="th" data-title="ระนาบ" data-language-autonym="ไทย" data-language-local-name="tailandês" 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/Plano_(heometriya)" title="Plano (heometriya) — tagalo" lang="tl" hreflang="tl" data-title="Plano (heometriya)" data-language-autonym="Tagalog" data-language-local-name="tagalo" 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/D%C3%BCzlem" title="Düzlem — turco" lang="tr" hreflang="tr" data-title="Düzlem" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%AF%D1%81%D1%81%D1%8B%D0%BB%D1%8B%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Яссылык (математика) — tatar" lang="tt" hreflang="tt" data-title="Яссылык (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%BE%D1%89%D0%B8%D0%BD%D0%B0" title="Площина — ucraniano" lang="uk" hreflang="uk" data-title="Площина" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%88%DB%8C_(%DB%81%D9%86%D8%AF%D8%B3%DB%81)" title="مستوی (ہندسہ) — urdu" lang="ur" hreflang="ur" data-title="مستوی (ہندسہ)" data-language-autonym="اردو" data-language-local-name="urdu" 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/Tekislik" title="Tekislik — usbeque" lang="uz" hreflang="uz" data-title="Tekislik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" 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/M%E1%BA%B7t_ph%E1%BA%B3ng_(to%C3%A1n_h%E1%BB%8Dc)" title="Mặt phẳng (toán học) — vietnamita" lang="vi" hreflang="vi" data-title="Mặt phẳng (toán học)" 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/Plano_(heyometriya)" title="Plano (heyometriya) — waray" lang="war" hreflang="war" data-title="Plano (heyometriya)" 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/%E5%B9%B3%E9%9D%A2" title="平面 — wu" lang="wuu" hreflang="wuu" data-title="平面" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%9C%D7%90%D7%9A" title="פלאך — iídiche" lang="yi" hreflang="yi" data-title="פלאך" data-language-autonym="ייִדיש" data-language-local-name="iídiche" 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/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学) — chinês" lang="zh" hreflang="zh" data-title="平面 (数学)" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/P%C3%AE%E2%81%BF-b%C4%ABn" title="Pîⁿ-bīn — min nan" lang="nan" hreflang="nan" data-title="Pîⁿ-bīn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" 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/%E9%9D%A2_(%E5%B9%BE%E4%BD%95)" title="面 (幾何) — cantonês" lang="yue" hreflang="yue" data-title="面 (幾何)" data-language-autonym="粵語" data-language-local-name="cantonês" 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/Q17285#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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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">Aspeto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</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">Origem: Wikipédia, a enciclopédia livre.</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="pt" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:PlaneIntersection.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/220px-PlaneIntersection.png" decoding="async" width="220" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/330px-PlaneIntersection.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/PlaneIntersection.png/440px-PlaneIntersection.png 2x" data-file-width="497" data-file-height="539" /></a><figcaption>Dois planos não <a href="/wiki/Paralelismo" title="Paralelismo">paralelos</a> no espaço</figcaption></figure> <p>Na <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a>, um <b>plano</b> é um ente primitivo <a href="/wiki/Geom%C3%A9trico" class="mw-redirect" title="Geométrico">geométrico</a> <a href="/wiki/Infinito" title="Infinito">infinito</a> a duas <a href="/wiki/Dimens%C3%A3o_(matem%C3%A1tica)" class="mw-redirect" title="Dimensão (matemática)">dimensões</a>. Nos <a href="/wiki/Elementos_de_Euclides" class="mw-redirect" title="Elementos de Euclides">Elementos de Euclides</a>, não possui definição enquanto conceito genérico. Mas um plano qualquer é definido, ou determinado, de várias formas equivalentes. </p><p>Em uma geometria <a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a>, é possível definir de forma genérica um plano como um conjunto infinito de retas, onde todas são perpendiculares a um mesmo vetor (<a href="/wiki/Vetor_normal" class="mw-redirect" title="Vetor normal">vetor normal</a> do plano). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Planos_no_espaço"><span id="Planos_no_espa.C3.A7o"></span>Planos no espaço</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=1" title="Editar secção: Planos no espaço" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=1" title="Editar código-fonte da secção: Planos no espaço"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um plano em <a href="/wiki/Produto_cartesiano" title="Produto cartesiano">ℝ<sup>3</sup></a> pode ser definido </p> <ul><li>dando um ponto do plano e um <a href="/wiki/Vetor_normal" class="mw-redirect" title="Vetor normal">vetor normal</a> a esse plano;</li> <li>dando um ponto do plano e dois <a href="/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática)">vetores</a> do plano;</li> <li>dando uma <a href="/wiki/Reta" title="Reta">reta</a> do plano e um ponto do plano exterior a esta reta;</li> <li>dando duas retas do plano;</li> <li>dando três pontos não colineares</li></ul> <div class="mw-heading mw-heading2"><h2 id="Geometria_Euclidiana">Geometria Euclidiana</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=2" title="Editar secção: Geometria Euclidiana" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=2" title="Editar código-fonte da secção: Geometria Euclidiana"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>No Espaço euclidiano, um plano é uma superfície tal que, dada a quaisquer pontos na superfície, a superfície também contém a única linha reta que passa pelos pontos. A estrutura fundamental de dois planos sempre será a mesma. Na matemática isto pode ser determinado como "homomorfismo". Informalmente, isso ocorre quando dois planos parecem o mesmo. Um plano pode ser unicamente determinado por um destes objetos: </p> <ul><li>três pontos não-colineares (não estão numa mesma reta)</li> <li>uma reta e um ponto fora desta reta</li> <li>duas retas concorrentes (duas retas que se cruzam num único ponto)</li> <li>duas retas paralelas distintas</li></ul> <div class="mw-heading mw-heading3"><h3 id="Orientação"><span id="Orienta.C3.A7.C3.A3o"></span>Orientação</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=3" title="Editar secção: Orientação" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=3" title="Editar código-fonte da secção: Orientação"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Planes_parallel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Planes_parallel.svg/150px-Planes_parallel.svg.png" decoding="async" width="150" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Planes_parallel.svg/225px-Planes_parallel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Planes_parallel.svg/300px-Planes_parallel.svg.png 2x" data-file-width="550" data-file-height="289" /></a><figcaption>Três planos paralelos.</figcaption></figure> <p>Como as linhas, os planos podem ser paralelos ou concorrentes. Diferentemente das linhas, os planos não podem ser deformes.Linhas traçadas em dois planos paralelos podem ser paralelas ou deformes, mas nunca concorrentes. Planos concorrentes podem ser perpendiculares, ou podem formar outros ângulos. </p> <div class="mw-heading mw-heading2"><h2 id="Planos_embutidos_no_R_3">Planos embutidos no R <sup> 3 </sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=4" title="Editar secção: Planos embutidos no R 3" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=4" title="Editar código-fonte da secção: Planos embutidos no R 3"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Essa seção fala de planos com três dimensões : especificamente, no <a href="/wiki/Produto_cartesiano" title="Produto cartesiano">ℝ<sup>3</sup></a>. </p> <div class="mw-heading mw-heading3"><h3 id="Propriedades">Propriedades</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=5" title="Editar secção: Propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=5" title="Editar código-fonte da secção: Propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>No espaço euclidiano <a href="/wiki/Tridimensional" class="mw-redirect" title="Tridimensional">tridimensional</a>, nós podemos explorar os seguintes factos que não detêm, em dimensões superiores: </p> <ul><li>Dois planos são ou paralelos ou concorrentes.</li> <li>Uma linha ou é paralela ou é concorrente num único ponto ou está contida no plano.</li> <li>Duas linhas <a href="/wiki/Perpendicular" class="mw-redirect" title="Perpendicular">perpendiculares</a> para o mesmo plano devem ser paralelas entre si.</li> <li>Dois planos perpendiculares para a mesma linha devem ser paralelos entre si.</li> <li>O plano é uma <a href="/wiki/Superf%C3%ADcie_m%C3%ADnima" title="Superfície mínima">superfície mínima</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=6" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=6" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">Geometria euclidiana</a></li> <li><a href="/wiki/Ponto_(matem%C3%A1tica)" title="Ponto (matemática)">Ponto</a></li> <li><a href="/wiki/Reta" title="Reta">Reta</a></li> <li><a href="/wiki/Superf%C3%ADcie" title="Superfície">Superfície</a></li> <li><a href="/wiki/Face_(matem%C3%A1tica)" class="mw-redirect" title="Face (matemática)">Face</a></li> <li><a href="/wiki/Entes_geom%C3%A9tricos_fundamentais" title="Entes geométricos fundamentais">Entes geométricos fundamentais</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plano_(geometria)&veaction=edit&section=7" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plano_(geometria)&action=edit&section=7" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.geometriaanalitica.com.br/livros/av.pdf">Livro Álgebra Vetorial e Geometria Analítica</a>: Livro do Prof. Jacir J. Venturi, de 242 páginas, disponível na íntegra para acesso gratuito.</li> <li><a rel="nofollow" class="external text" href="http://www.wdl.org/pt/item/2850">"Facilitando a Dificuldade da Aritmética e da Geometria Planar"</a> é um manuscrito árabe, do século 15, que serve como um tutorial sobre geometria plana e aritmética</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Plano_(geometria)&oldid=63608660">https://pt.wikipedia.org/w/index.php?title=Plano_(geometria)&oldid=63608660</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:Geometria" title="Categoria:Geometria">Geometria</a></li><li><a href="/wiki/Categoria:Desenho_geom%C3%A9trico" title="Categoria:Desenho geométrico">Desenho geométrico</a></li><li><a href="/wiki/Categoria:Conceitos_matem%C3%A1ticos" title="Categoria:Conceitos matemáticos">Conceitos matemáticos</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 14h35min de 17 de maio de 2022.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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