{"id":39172,"date":"2024-06-27T15:47:58","date_gmt":"2024-06-27T13:47:58","guid":{"rendered":"https:\/\/www.swisslearn.org\/?p=39172"},"modified":"2024-06-28T08:59:11","modified_gmt":"2024-06-28T06:59:11","slug":"equation-et-partie-visible-dun-plan-passant-par-trois-points","status":"publish","type":"post","link":"https:\/\/www.swisslearn.org\/?p=39172","title":{"rendered":"Repr\u00e9sentation vectorielle d&rsquo;un plan passant par trois points"},"content":{"rendered":"\n<p><strong>Repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;un plan<\/strong><\/p>\n<p>L&rsquo;\u00e9quation vectorielle d&rsquo;un plan est compos\u00e9e des coordonn\u00e9es d&rsquo;un point et des composants de deux vecteurs. Elle peut \u00eatre \u00e9crite de la mani\u00e8re suivante\u00a0 :<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\begin{pmatrix}&amp;space;x&amp;space;\\\\&amp;space;y&amp;space;\\\\&amp;space;z&amp;space;\\end{pmatrix}&amp;space;=&amp;space;\\begin{pmatrix}&amp;space;x_0&amp;space;\\\\&amp;space;y_0&amp;space;\\\\&amp;space;z_0&amp;space;\\end{pmatrix}&amp;space;+k&amp;space;\\begin{pmatrix}&amp;space;x_{v_1}&amp;space;\\\\&amp;space;y_{v_1}&amp;space;\\\\&amp;space;z_{v_1}&amp;space;\\end{pmatrix}+l\\begin{pmatrix}&amp;space;x_{v_2}\\\\&amp;space;y_{v_2}\\\\&amp;space;z_{v_2}&amp;space;\\end{pmatrix}\" alt=\"\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} x_0 \\\\ y_0 \\\\ z_0 \\end{pmatrix} +k \\begin{pmatrix} x_{v_1} \\\\ y_{v_1} \\\\ z_{v_1} \\end{pmatrix}+l\\begin{pmatrix} x_{v_2}\\\\ y_{v_2}\\\\ z_{v_2} \\end{pmatrix}\" align=\"absmiddle\" \/>\u00a0 \u00a0 \u00a0<\/p>\n<p>ou\u00a0 \u00a0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\left\\{\\begin{matrix}&amp;space;x=x_0+kx_{v_1}+lx_{v_2}&amp;space;\\\\&amp;space;y=y_0+ky_{v_1}+ly_{v_2}&amp;space;\\\\&amp;space;z=z_0+kz_{v_1}+ly_{v_2}&amp;space;\\end{matrix}\\right.\" alt=\"\\left\\{\\begin{matrix} x=x_0+kx_{v_1}+lx_{v_2} \\\\ y=y_0+ky_{v_1}+ly_{v_2} \\\\ z=z_0+kz_{v_1}+ly_{v_2} \\end{matrix}\\right.\" align=\"absmiddle\" \/>\u00a0 \u00a0 avec\u00a0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?k\\in\\mathbb{R}\" alt=\"k\\in\\mathbb{R}\" align=\"absmiddle\" \/>\u00a0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?l\\in\\mathbb{R}\" alt=\"l\\in\\mathbb{R}\" align=\"absmiddle\" \/><\/p>\n<p>Les termes <em>x<sub>0<\/sub><\/em>, <em>y<sub>0<\/sub><\/em> et <em>z<sub>0<\/sub><\/em> sont les coordonn\u00e9es de n&rsquo;importe quel point qui appartient au plan.<\/p>\n<p>Les termes <em>x<sub>v1<\/sub><\/em>, <em>y<sub>v1<\/sub><\/em>, <em>z<sub>v1<\/sub><\/em> et <em>x<sub>v2<\/sub><\/em>, <em>y<sub>v2<\/sub><\/em>, <em>z<sub>v2<\/sub><\/em> sont les composantes de deux vecteurs parall\u00e8les au plan (vecteurs directeurs. Attention : ces deux vecteurs ne doivent pas \u00eatre colin\u00e9aires (parall\u00e8les) entre eux.<\/p>\n<p>Les lettres <em>k<\/em> et <em>l<\/em> sont deux param\u00e8tres repr\u00e9sentant un nombre r\u00e9el.<\/p>\n<p>Il existe une infinit\u00e9 de points sur un plan et une infinit\u00e9 de vecteurs parall\u00e8les \u00e0 un plan. Il existe donc une infinit\u00e9 de repr\u00e9sentations vectorielle d&rsquo;un m\u00eame plan.<\/p>\n<p>\u00a0<\/p>\n<p><strong>D\u00e9terminer la repr\u00e9sentation vectorielle d&rsquo;un plan passant par trois points A, B et C<\/strong><\/p>\n<p>Pour<em> x<sub>0<\/sub><\/em>, <em>y<sub>0<\/sub><\/em> et <em>z<sub>0<\/sub><\/em> et on choisira les coordonn\u00e9es de n&rsquo;importe quel de ces trois points. Si possible il est pr\u00e9f\u00e9rable de choisir le point qui a des coordonn\u00e9es nulles.<\/p>\n<p>Pour <em>x<sub>v1<\/sub><\/em>, <em>y<sub>v1<\/sub><\/em>, <em>z<sub>v1<\/sub><\/em> et <em>x<sub>v2<\/sub><\/em>, <em>y<sub>v2<\/sub><\/em>, <em>z<sub>v2<\/sub><\/em>\u00a0 on peut choisir n&rsquo;importe quel vecteur qui relie deux de ces points comme les vecteurs <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{AB}\" alt=\"\\inline \\small \\overrightarrow{AB}\" align=\"absmiddle\" \/>, <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{AC}\" alt=\"\\inline \\small \\overrightarrow{AC}\" align=\"absmiddle\" \/>, <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{BC}\" alt=\"\\inline \\small \\overrightarrow{BC}\" align=\"absmiddle\" \/>, <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{BA}\" alt=\"\\inline \\small \\overrightarrow{BA}\" align=\"absmiddle\" \/>\u00a0etc.<\/p>\n<p>\u00a0<\/p>\n<p>Le simulateur dessous utilise le point A et les vecteurs <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{AB}\" alt=\"\\inline \\small \\overrightarrow{AB}\" align=\"absmiddle\" \/> et <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{AC}\" alt=\"\\inline \\small \\overrightarrow{AC}\" align=\"absmiddle\" \/> pour donner une repr\u00e9sentation vectorielle du plan <em>\u03b1<\/em>.<\/p>\n<p><iframe loading=\"lazy\" style=\"border: 0px;\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/zfwtsjvp\/width\/1200\/height\/715\/border\/888888\/rc\/true\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/true\/ld\/false\/sri\/false\/sfsb\/true\" width=\"1600px\" height=\"715px\" scrolling=\"no\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p>Pour voir comment on d\u00e9termine les traces d&rsquo;un plan <a href=\"https:\/\/www.swisslearn.org\/?p=38934\">cliquez ici<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;un plan L&rsquo;\u00e9quation vectorielle d&rsquo;un plan est compos\u00e9e des coordonn\u00e9es d&rsquo;un point et des composants de deux vecteurs. Elle peut \u00eatre \u00e9crite de la mani\u00e8re suivante\u00a0 : \u00a0 \u00a0 \u00a0 ou\u00a0 \u00a0 \u00a0 \u00a0 avec\u00a0 \u00a0 Les termes x0, y0 et z0 sont les coordonn\u00e9es de n&rsquo;importe quel point qui appartient [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"off","_et_pb_old_content":"","_et_gb_content_width":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[38,10,58],"tags":[],"class_list":["post-39172","post","type-post","status-publish","format-standard","hentry","category-cours","category-geometrie","category-geometrie-de-lespace"],"blocksy_meta":"","rttpg_featured_image_url":null,"rttpg_author":{"display_name":"Bahram Zaerpour","author_link":"https:\/\/www.swisslearn.org\/?author=1"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/www.swisslearn.org\/?cat=38\" rel=\"category\">Cours<\/a> <a href=\"https:\/\/www.swisslearn.org\/?cat=10\" rel=\"category\">G\u00e9om\u00e9trie<\/a> <a href=\"https:\/\/www.swisslearn.org\/?cat=58\" rel=\"category\">G\u00e9om\u00e9trie de l'espace<\/a>","rttpg_excerpt":"Repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;un plan L&rsquo;\u00e9quation vectorielle d&rsquo;un plan est compos\u00e9e des coordonn\u00e9es d&rsquo;un point et des composants de deux vecteurs. Elle peut \u00eatre \u00e9crite de la mani\u00e8re suivante\u00a0 : \u00a0 \u00a0 \u00a0 ou\u00a0 \u00a0 \u00a0 \u00a0 avec\u00a0 \u00a0 Les termes x0, y0 et z0 sont les coordonn\u00e9es de n&rsquo;importe quel point qui appartient\u2026","_links":{"self":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39172"}],"collection":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=39172"}],"version-history":[{"count":6,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39172\/revisions"}],"predecessor-version":[{"id":39191,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39172\/revisions\/39191"}],"wp:attachment":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=39172"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=39172"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=39172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}