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</div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Plan (geometri)</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="Gå till en artikel på ett annat språk. Tillgänglig på 88 språk" > <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 språk</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 – afrikaans" lang="af" hreflang="af" data-title="Vlak" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik) – schweizertyska" lang="gsw" hreflang="gsw" data-title="Ebene (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="schweizertyska" 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) – arabiska" lang="ar" hreflang="ar" data-title="مستو (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabiska" 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) – asturiska" lang="ast" hreflang="ast" data-title="Planu (xeometría)" data-language-autonym="Asturianu" data-language-local-name="asturiska" 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 – azerbajdzjanska" lang="az" hreflang="az" data-title="Müstəvi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzjanska" class="interlanguage-link-target"><span>Azərbaycanca</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 – bengali" lang="bn" hreflang="bn" data-title="সমতল" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</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-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 – basjkiriska" lang="ba" hreflang="ba" data-title="Яҫылыҡ" data-language-autonym="Башҡортса" data-language-local-name="basjkiriska" 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 – belarusiska" lang="be" hreflang="be" data-title="Плоскасць" data-language-autonym="Беларуская" data-language-local-name="belarusiska" 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 – belarusiska (tarasjkevitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Роўніца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusiska (tarasjkevitsa)" 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) – bulgariska" lang="bg" hreflang="bg" data-title="Равнина (математика)" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ravan" title="Ravan – bosniska" lang="bs" hreflang="bs" data-title="Ravan" data-language-autonym="Bosanski" data-language-local-name="bosniska" 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 – katalanska" lang="ca" hreflang="ca" data-title="Pla" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D1%82%D0%B0%D0%BA" title="Лаптак – tjuvasjiska" lang="cv" hreflang="cv" data-title="Лаптак" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rovina" title="Rovina – tjeckiska" lang="cs" hreflang="cs" data-title="Rovina" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</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-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Pl%C3%A2n_geometraidd" title="Plân geometraidd – walesiska" lang="cy" hreflang="cy" data-title="Plân geometraidd" data-language-autonym="Cymraeg" data-language-local-name="walesiska" 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) – danska" lang="da" hreflang="da" data-title="Plan (matematik)" data-language-autonym="Dansk" data-language-local-name="danska" 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) – tyska" lang="de" hreflang="de" data-title="Ebene (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tasand" title="Tasand – estniska" lang="et" hreflang="et" data-title="Tasand" data-language-autonym="Eesti" data-language-local-name="estniska" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%AF%CF%80%CE%B5%CE%B4%CE%BF" title="Επίπεδο – grekiska" lang="el" hreflang="el" data-title="Επίπεδο" data-language-autonym="Ελληνικά" data-language-local-name="grekiska" 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 – engelska" lang="en" hreflang="en" data-title="Euclidean plane" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</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) – spanska" lang="es" hreflang="es" data-title="Plano (geometría)" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Plano" title="Plano – baskiska" lang="eu" hreflang="eu" data-title="Plano" data-language-autonym="Euskara" data-language-local-name="baskiska" 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) – persiska" lang="fa" hreflang="fa" data-title="صفحه (هندسه)" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Plan_(math%C3%A9matiques)" title="Plan (mathématiques) – franska" lang="fr" hreflang="fr" data-title="Plan (mathématiques)" data-language-autonym="Français" data-language-local-name="franska" 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) – iriska" lang="ga" hreflang="ga" data-title="Plána (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="iriska" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Plano_(xeometr%C3%ADa)" title="Plano (xeometría) – galiciska" lang="gl" hreflang="gl" data-title="Plano (xeometría)" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EB%A9%B4" title="평면 – koreanska" lang="ko" hreflang="ko" data-title="평면" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</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 – armeniska" lang="hy" hreflang="hy" data-title="Հարթություն" data-language-autonym="Հայերեն" data-language-local-name="armeniska" class="interlanguage-link-target"><span>Հայերեն</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 – kroatiska" lang="hr" hreflang="hr" data-title="Ravnina" data-language-autonym="Hrvatski" data-language-local-name="kroatiska" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bidang_(geometri)" title="Bidang (geometri) – indonesiska" lang="id" hreflang="id" data-title="Bidang (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Plano" title="Plano – interlingua" lang="ia" hreflang="ia" data-title="Plano" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</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) – isländska" lang="is" hreflang="is" data-title="Slétta (rúmfræði)" data-language-autonym="Íslenska" data-language-local-name="isländska" 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) – italienska" lang="it" hreflang="it" data-title="Piano (geometria)" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%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) – hebreiska" lang="he" hreflang="he" data-title="מישור (גאומטריה)" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</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 – georgiska" lang="ka" hreflang="ka" data-title="სიბრტყე" data-language-autonym="ქართული" data-language-local-name="georgiska" class="interlanguage-link-target"><span>ქართული</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) – kazakiska" lang="kk" hreflang="kk" data-title="Жазықтық (геометрия)" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</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-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 – kirgiziska" lang="ky" hreflang="ky" data-title="Тегиздик" data-language-autonym="Кыргызча" data-language-local-name="kirgiziska" 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) – latin" lang="la" hreflang="la" data-title="Planum (geometria)" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Plakne" title="Plakne – lettiska" lang="lv" hreflang="lv" data-title="Plakne" data-language-autonym="Latviešu" data-language-local-name="lettiska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Plok%C5%A1tuma" title="Plokštuma – litauiska" lang="lt" hreflang="lt" data-title="Plokštuma" data-language-autonym="Lietuvių" data-language-local-name="litauiska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/S%C3%ADk_(geometria)" title="Sík (geometria) – ungerska" lang="hu" hreflang="hu" data-title="Sík (geometria)" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%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) – makedonska" lang="mk" hreflang="mk" data-title="Рамнина (математика)" data-language-autonym="Македонски" data-language-local-name="makedonska" class="interlanguage-link-target"><span>Македонски</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) – malagassiska" lang="mg" hreflang="mg" data-title="Lemaka (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malagassiska" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Satah" title="Satah – malajiska" lang="ms" hreflang="ms" data-title="Satah" data-language-autonym="Bahasa Melayu" data-language-local-name="malajiska" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vlak_(meetkunde)" title="Vlak (meetkunde) – nederländska" lang="nl" hreflang="nl" data-title="Vlak (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2" title="平面 – japanska" lang="ja" hreflang="ja" data-title="平面" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Plan_(matematikk)" title="Plan (matematikk) – norskt bokmål" lang="nb" hreflang="nb" data-title="Plan (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – nynorska" lang="nn" hreflang="nn" data-title="Plan i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorska" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tekislik" title="Tekislik – uzbekiska" lang="uz" hreflang="uz" data-title="Tekislik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekiska" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Plien_(matimatix)" title="Plien (matimatix) – jamaikansk engelsk kreol" lang="jam" hreflang="jam" data-title="Plien (matimatix)" data-language-autonym="Patois" data-language-local-name="jamaikansk engelsk kreol" class="interlanguage-link-target"><span>Patois</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 – kambodjanska" lang="km" hreflang="km" data-title="ប្លង់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kambodjanska" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Flach_(Mathematik)" title="Flach (Mathematik) – lågtyska" lang="nds" hreflang="nds" data-title="Flach (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="lågtyska" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C5%82aszczyzna" title="Płaszczyzna – polska" lang="pl" hreflang="pl" data-title="Płaszczyzna" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Plano_(geometria)" title="Plano (geometria) – portugisiska" lang="pt" hreflang="pt" data-title="Plano (geometria)" data-language-autonym="Português" data-language-local-name="portugisiska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Plan_(geometrie)" title="Plan (geometrie) – rumänska" lang="ro" hreflang="ro" data-title="Plan (geometrie)" data-language-autonym="Română" data-language-local-name="rumänska" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27allta" title="P'allta – quechua" lang="qu" hreflang="qu" data-title="P'allta" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</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 – ryska" lang="ru" hreflang="ru" data-title="Плоскость" data-language-autonym="Русский" data-language-local-name="ryska" 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 – jakutiska" lang="sah" hreflang="sah" data-title="Хаптал" data-language-autonym="Саха тыла" data-language-local-name="jakutiska" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rrafshi_Euklidian" title="Rrafshi Euklidian – albanska" lang="sq" hreflang="sq" data-title="Rrafshi Euklidian" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</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) – slovakiska" lang="sk" hreflang="sk" data-title="Rovina (geometria)" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" 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 – slovenska" lang="sl" hreflang="sl" data-title="Ravnina" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D9%88%D9%88%D8%AA%DB%95%D8%AE%D8%AA" title="ڕووتەخت – centralkurdiska" lang="ckb" hreflang="ckb" data-title="ڕووتەخت" data-language-autonym="کوردی" data-language-local-name="centralkurdiska" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%B0%D0%BD" title="Раван – serbiska" lang="sr" hreflang="sr" data-title="Раван" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ravan" title="Ravan – serbokroatiska" lang="sh" hreflang="sh" data-title="Ravan" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatiska" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Taso" title="Taso – finska" lang="fi" hreflang="fi" data-title="Taso" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Plano_(heometriya)" title="Plano (heometriya) – tagalog" lang="tl" hreflang="tl" data-title="Plano (heometriya)" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%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) – tamil" lang="ta" hreflang="ta" data-title="தளம் (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Agni_(tanzeggit)" title="Agni (tanzeggit) – kabyliska" lang="kab" hreflang="kab" data-title="Agni (tanzeggit)" data-language-autonym="Taqbaylit" data-language-local-name="kabyliska" class="interlanguage-link-target"><span>Taqbaylit</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) – tatariska" lang="tt" hreflang="tt" data-title="Яссылык (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="tatariska" class="interlanguage-link-target"><span>Татарча / tatarça</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 – thailändska" lang="th" hreflang="th" data-title="ระนาบ" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</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 – cherokesiska" lang="chr" hreflang="chr" data-title="ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="cherokesiska" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/D%C3%BCzlem" title="Düzlem – turkiska" lang="tr" hreflang="tr" data-title="Düzlem" data-language-autonym="Türkçe" data-language-local-name="turkiska" 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%D0%BB%D0%BE%D1%89%D0%B8%D0%BD%D0%B0" title="Площина – ukrainska" lang="uk" hreflang="uk" data-title="Площина" data-language-autonym="Українська" data-language-local-name="ukrainska" 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-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) – vietnamesiska" lang="vi" hreflang="vi" data-title="Mặt phẳng (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" 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 – jiddisch" lang="yi" hreflang="yi" data-title="פלאך" data-language-autonym="ייִדיש" data-language-local-name="jiddisch" class="interlanguage-link-target"><span>ייִדיש</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) – kantonesiska" lang="yue" hreflang="yue" data-title="面 (幾何)" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学) – kinesiska" lang="zh" hreflang="zh" data-title="平面 (数学)" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q17285#sitelinks-wikipedia" 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#9f9fff;background-color:#efefff;text-align:center}.mw-parser-output .messagebox.standard-talk{border:1px solid #c0c090;background-color:#f8eaba}.mw-parser-output .messagebox.nested-talk{border:1px solid #c0c090;background-color:#f8eaba;width:100%;margin:2px 4px}</style><table class="metadata ambox ambox-content" style=""> <tbody><tr> <td class="ambox-image"> <div style="width:52px;"><span typeof="mw:File"><a href="/wiki/Fil:Question_book-4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/40px-Question_book-4.svg.png" decoding="async" width="40" height="31" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/60px-Question_book-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/80px-Question_book-4.svg.png 2x" data-file-width="262" data-file-height="204" /></a></span></div></td> <td class="ambox-text">Den här artikeln <b>behöver <a href="/wiki/Wikipedia:K%C3%A4llh%C3%A4nvisningar" title="Wikipedia:Källhänvisningar">källhänvisningar</a> för att kunna <a href="/wiki/Wikipedia:Verifierbarhet" title="Wikipedia:Verifierbarhet">verifieras</a>.</b> <small>(2020-03)</small> <br /><small>Åtgärda genom att lägga till pålitliga källor (<a href="/wiki/Wikipedia:Introduktion_till_k%C3%A4llh%C3%A4nvisningar" title="Wikipedia:Introduktion till källhänvisningar">gärna som fotnoter</a>). Uppgifter utan källhänvisning kan <a href="/wiki/Mall:K%C3%A4lla_beh%C3%B6vs" title="Mall:Källa behövs">ifrågasättas</a> och tas bort utan att det behöver diskuteras på <a href="/w/index.php?title=Diskussion:Plan_(geometri)&action=edit&redlink=1" class="new" title="Diskussion:Plan (geometri) [inte skriven än]">diskussionssidan</a>.</small></td> </tr> </tbody></table> <p>Inom matematiken är ett <b>plan</b> en flat, tvådimensionell yta med oändlig utsträckning. Plan kan förekomma som underrum till rum av godtyckliga dimensioner eller kan ha en oberoende existens som i fallet <a href="/wiki/Euklidisk_geometri" title="Euklidisk geometri">euklidisk geometri</a>. <a href="/wiki/Plangeometri" title="Plangeometri">Plangeometri</a> är läran om geometriska figurer i planet. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Plan_i_R3">Plan i R<sup>3</sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plan_(geometri)&veaction=edit&section=1" title="Redigera avsnitt: Plan i R3" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plan_(geometri)&action=edit&section=1" title="Redigera avsnitts källkod: Plan i R3"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:Plane-definition-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Plane-definition-2.svg/300px-Plane-definition-2.svg.png" decoding="async" width="300" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Plane-definition-2.svg/450px-Plane-definition-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Plane-definition-2.svg/600px-Plane-definition-2.svg.png 2x" data-file-width="334" data-file-height="220" /></a><figcaption></figcaption></figure> <p>Om (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>z</i><sub>0</sub>) är en <a href="/wiki/Ortsvektor" title="Ortsvektor">ortsvektor</a> till en punkt i planet och (<i>A</i>, <i>B</i>, <i>C</i>) är en <a href="/wiki/Normalvektor" title="Normalvektor">normalvektor</a> till planet, kan planets ekvation skrivas som <a href="/wiki/Skal%C3%A4rprodukt" title="Skalärprodukt">skalärprodukten</a> av en normalvektor och <a href="/wiki/Vektor" title="Vektor">vektorn</a> (<i>x</i> - <i>x</i><sub>0</sub>, <i>y</i> - <i>y</i><sub>0</sub>, <i>z</i> - <i>z</i><sub>0</sub>): </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 (A,\ B,\ C)(x\,-\,x_{0},\ y\,-\,y_{0},\ z\,-\,z_{0})=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mtext> </mtext> <mi>B</mi> <mo>,</mo> <mtext> </mtext> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>z</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,\ B,\ C)(x\,-\,x_{0},\ y\,-\,y_{0},\ z\,-\,z_{0})=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68c4bf3d8fcf491a92e8266b80bd90a301a627d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.128ex; height:2.843ex;" alt="{\displaystyle (A,\ B,\ C)(x\,-\,x_{0},\ y\,-\,y_{0},\ z\,-\,z_{0})=0\,}"></span></dd></dl> <p>vilket ger den <i>allmänna formen</i> av <b>planets ekvation</b> som </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 Ax\,+\,By\,+\,Cz\,+\,D\,=\,0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>B</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>C</mi> <mi>z</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>D</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax\,+\,By\,+\,Cz\,+\,D\,=\,0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9026e74eaca0ea11c94e05e21f2fe7d1c8f84be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.65ex; height:2.509ex;" alt="{\displaystyle Ax\,+\,By\,+\,Cz\,+\,D\,=\,0}"></span></dd></dl> <p>där <i>D</i> är </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 -(Ax_{0}\,+\,By_{0}\,+\,Cz_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>B</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>C</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(Ax_{0}\,+\,By_{0}\,+\,Cz_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9a15df3fab981b802d96f34fdc0db844ba1554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.833ex; height:2.843ex;" alt="{\displaystyle -(Ax_{0}\,+\,By_{0}\,+\,Cz_{0})}"></span></dd></dl> <p>En ekvation av första graden representerar alltid ett plan. För planets normal är <b>riktningscosinerna</b> (cosinus för de vinklar som normalvektorn bildar med koordinataxlarna) </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 {\frac {A}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {B}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {C}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {B}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {C}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817cbfe9a48c9365a462b8c62bbbca1f5ef79377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.421ex; height:6.676ex;" alt="{\displaystyle {\frac {A}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {B}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}},\quad {\frac {C}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}"></span></dd></dl> <p>Tecknet framför roten väljs så att </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 {\frac {D}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5edb40c80470b4ded171f7b1b706cbd1917a3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.569ex; height:6.509ex;" alt="{\displaystyle {\frac {D}{\pm {\sqrt {A^{2}\,+\,B^{2}\,+\,C^{2})}}}}}"></span> alltid är positiv. Därigenom är normalen riktad mot planets "positiva" sida.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Normalform">Normalform</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plan_(geometri)&veaction=edit&section=2" title="Redigera avsnitt: Normalform" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plan_(geometri)&action=edit&section=2" title="Redigera avsnitts källkod: Normalform"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Genom division med </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 \pm {\sqrt {A^{2}+B^{2}+C^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <mo>+</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {A^{2}+B^{2}+C^{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0458fdf0cb4da2a0830f9b9000061ec446c2e2b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.185ex; height:4.843ex;" alt="{\displaystyle \pm {\sqrt {A^{2}+B^{2}+C^{2})}}}"></span></dd></dl> <p>erhålls planets ekvation på <i>normalform</i> </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\cos \alpha +y\cos \beta +z\cos \gamma =p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>+</mo> <mi>y</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> <mo>+</mo> <mi>z</mi> <mi>cos</mi> <mo>⁡</mo> <mi>γ</mi> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cos \alpha +y\cos \beta +z\cos \gamma =p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de3a27263ab085f575987afb040ce6ef575dbde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.26ex; height:2.676ex;" alt="{\displaystyle x\cos \alpha +y\cos \beta +z\cos \gamma =p}"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\,\beta ,\,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>β</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\,\beta ,\,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7863da8eadaf36524d46892201f182ab39a1c873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.924ex; height:2.676ex;" alt="{\displaystyle \alpha ,\,\beta ,\,\gamma }"></span> är de vinklar som planets normal bildar med koordinataxlarna och <i>p</i> är längden av normalen från origo till planet. </p> <div class="mw-heading mw-heading3"><h3 id="Vektorform">Vektorform</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plan_(geometri)&veaction=edit&section=3" title="Redigera avsnitt: Vektorform" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plan_(geometri)&action=edit&section=3" title="Redigera avsnitts källkod: Vektorform"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:Plane-definition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Plane-definition.svg/300px-Plane-definition.svg.png" decoding="async" width="300" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Plane-definition.svg/450px-Plane-definition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Plane-definition.svg/600px-Plane-definition.svg.png 2x" data-file-width="334" data-file-height="220" /></a><figcaption></figcaption></figure> <p>Ekvationen för ett plan med normalvektorn <b>n</b>, en given punkt <b>r</b><sub>0</sub> och med <b>r</b> som ortsvektor för en godtycklig punkt <i>(x, y, z)</i> i planet är </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {r} -\mathbf {r} _{0})\mathbf {n} =0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {r} -\mathbf {r} _{0})\mathbf {n} =0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee455a4ea27a34be4ee277f5de016ebefbd3162e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.041ex; height:2.843ex;" alt="{\displaystyle (\mathbf {r} -\mathbf {r} _{0})\mathbf {n} =0\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Se_även"><span id="Se_.C3.A4ven"></span>Se även</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plan_(geometri)&veaction=edit&section=4" title="Redigera avsnitt: Se även" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | 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