{"id":39107,"date":"2024-06-26T10:02:27","date_gmt":"2024-06-26T08:02:27","guid":{"rendered":"https:\/\/www.swisslearn.org\/?p=39107"},"modified":"2024-06-28T09:09:04","modified_gmt":"2024-06-28T07:09:04","slug":"representation-vectorielle-dune-droite-passant-par-deux-points-en-3d-exemple-2","status":"publish","type":"post","link":"https:\/\/www.swisslearn.org\/?p=39107","title":{"rendered":"Repr\u00e9sentation vectorielle d&rsquo;une droite passant par deux points en 3D :  exemple 2"},"content":{"rendered":"\n<p><a href=\"https:\/\/www.swisslearn.org\/?p=39141\">Exemples<\/a> n\u00b0 <a href=\"https:\/\/www.swisslearn.org\/?p=39065\">1<\/a>, 2, <a href=\"https:\/\/www.swisslearn.org\/?p=39122\">3<\/a><\/p>\n<p><strong>Enonc\u00e9<\/strong><\/p>\n<p>Donner une repr\u00e9sentation vectorielle d&rsquo;une droite passant par les deux points P(2; 12; -4) et Q(2; 3; 2).<\/p>\n<div id=\"objectives\" class=\"objectives\">\n<p><strong>Rappels de la th\u00e9orie<\/strong><\/p>\n<p>La repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;une droite dans l&rsquo;espace est de type :\u00a0<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\left\\{\\begin{matrix}&amp;space;x=x_0+kx_v&amp;space;\\\\&amp;space;y=y_0+ky_v&amp;space;\\\\&amp;space;z=z_0+kz_v&amp;space;\\end{matrix}\\right.\" alt=\"\\left\\{\\begin{matrix} x=x_0+kx_v \\\\ y=y_0+ky_v \\\\ z=z_0+kz_v \\end{matrix}\\right.\" align=\"absmiddle\" \/><\/p>\n<p>On peut aussi l&rsquo;\u00e9crire sous une autre forme :<\/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&amp;space;\\\\&amp;space;y_v&amp;space;\\\\&amp;space;z_v&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 \\\\ y_v \\\\ z_v \\end{pmatrix}\" align=\"absmiddle\" \/><\/p>\n<p>Dans cette repr\u00e9sentation <em>x<sub>0<\/sub><\/em>, <em>y<sub>0<\/sub><\/em> et <em>z<sub>0<\/sub><\/em> sont les coordonn\u00e9es d&rsquo;un point appartenant \u00e0 la droite et <em>x<sub>v<\/sub><\/em>, <em>y<sub>v<\/sub><\/em> et <em>z<sub>v<\/sub> <\/em>sont les composants d&rsquo;un vecteur parall\u00e8le \u00e0 la droite (<em>vecteur directeur<\/em>). Il existe une infinit\u00e9 de points sur une droite et une infinit\u00e9 de vecteurs parall\u00e8le \u00e0 une droite. Ainsi, selon le point et le vecteur choisis, il existe une infinit\u00e9 de repr\u00e9sentations vectorielles d&rsquo;une m\u00eame droite.\u00a0<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<p><strong>Corrections<\/strong><\/p>\n<p>En pratique, pour donner la repr\u00e9sentation vectorielle d&rsquo;une droite passant par deux points P et Q on a le choix entre les coordonn\u00e9es d&rsquo;un de ces deux points. Il est pratique de choisir, si possible, le point qui a des coordonn\u00e9es nulles.<\/p>\n<p>Dans notre exemple avec les points on choisira le point P :<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?P\\in&amp;space;d\\Rightarrow&amp;space;x_0=2\" alt=\"P\\in d\\Rightarrow x_0=2\" align=\"absmiddle\" \/>\u00a0 \u00a0\u00a0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?y_0=12\" alt=\"y_0=12\" align=\"absmiddle\" \/>\u00a0 \u00a0 \u00a0<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?z_0=-4\" alt=\"z_0=-4\" align=\"absmiddle\" \/><\/p>\n<p>Il faut encore d\u00e9terminer les composants d&rsquo;un vecteur parall\u00e8le \u00e0 la droite. Si la droite passe par les points P et Q alors elle est parall\u00e8le au vecteur <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{PQ}\" alt=\"\\inline \\small \\overrightarrow{PQ}\" align=\"absmiddle\" \/> ou <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{QP}\" alt=\"\\inline \\small \\overrightarrow{QP}\" align=\"absmiddle\" \/>.<\/p>\n<p>Ici on choisit le vecteur <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{PQ}\" alt=\"\\inline \\small \\overrightarrow{PQ}\" align=\"absmiddle\" \/>\u00a0:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\overrightarrow{PQ}=&amp;space;\\begin{pmatrix}&amp;space;x_Q&amp;space;-&amp;space;x_P&amp;space;\\\\&amp;space;y_Q&amp;space;-&amp;space;y_P&amp;space;\\\\&amp;space;z_Q&amp;space;-&amp;space;z_P&amp;space;\\end{pmatrix}&amp;space;=&amp;space;\\begin{pmatrix}&amp;space;2-2&amp;space;\\\\&amp;space;3-12&amp;space;\\\\&amp;space;2-(-4)&amp;space;\\end{pmatrix}&amp;space;=&amp;space;\\begin{pmatrix}&amp;space;0&amp;space;\\\\&amp;space;15&amp;space;\\\\&amp;space;-9&amp;space;\\end{pmatrix}\" alt=\"\\overrightarrow{PQ}= \\begin{pmatrix} x_Q - x_P \\\\ y_Q - y_P \\\\ z_Q - z_P \\end{pmatrix} = \\begin{pmatrix} 2-2 \\\\ 3-12 \\\\ 2-(-4) \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 15 \\\\ -9 \\end{pmatrix}\" align=\"absmiddle\" \/><\/p>\n<p>On peut utiliser sans autre les composante de ce vecteur pour la repr\u00e9sentation param\u00e9trique de notre droite.\u00a0 Mais on note que les trois composantes sont divisibles par 3. Il serait plus \u00e9l\u00e9gant d&rsquo;utiliser le vecteur <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\overrightarrow{v}\" alt=\"\\inline \\small \\overrightarrow{v}\" align=\"absmiddle\" \/> qui est \u00e9gale \u00e0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\inline&amp;space;\\small&amp;space;\\frac{\\overrightarrow{PQ}}{3}\" alt=\"\\inline \\small \\frac{\\overrightarrow{PQ}}{3}\" align=\"absmiddle\" \/>\u00a0. Ce vecteur est aussi parall\u00e8le \u00e0 la droite <em>d<\/em> :<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\overrightarrow{v}=\\begin{pmatrix}&amp;space;0&amp;space;\\\\&amp;space;5&amp;space;\\\\&amp;space;-2&amp;space;\\end{pmatrix}\\parallel&amp;space;d&amp;space;\\Rightarrow&amp;space;x_v=0\" alt=\"\\overrightarrow{v}=\\begin{pmatrix} 0 \\\\ 5 \\\\ -2 \\end{pmatrix}\\parallel d \\Rightarrow x_v=0\" align=\"absmiddle\" \/> \u00a0 <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?y_v=5\" alt=\"y_v=5\" align=\"absmiddle\" \/> <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?z_v=-2\" alt=\"z_v=-2\" align=\"absmiddle\" \/><\/p>\n<p>On peut maintenant donner la repr\u00e9sentation param\u00e9trique de notre droite :<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\left\\{\\begin{matrix}&amp;space;x=2&amp;space;\\\\&amp;space;y=12+5k&amp;space;\\\\&amp;space;z=-4-2k&amp;space;\\end{matrix}\\right.\" alt=\"\\left\\{\\begin{matrix} x=2 \\\\ y=12+5k \\\\ z=-4-2k \\end{matrix}\\right.\" align=\"absmiddle\" \/> \u00a0ou <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;2&amp;space;\\\\&amp;space;12&amp;space;\\\\&amp;space;-4&amp;space;\\end{pmatrix}&amp;space;+k&amp;space;\\begin{pmatrix}&amp;space;0&amp;space;\\\\&amp;space;5&amp;space;\\\\&amp;space;-2&amp;space;\\end{pmatrix}\" alt=\"\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 12 \\\\ -4 \\end{pmatrix} +k \\begin{pmatrix} 0 \\\\ 5 \\\\ -2 \\end{pmatrix}\" align=\"absmiddle\" \/><\/p>\n<p>\u00a0<\/p>\n<p>Pour savoir comment d\u00e9terminer la partie visible de cette droite, <a href=\"https:\/\/www.swisslearn.org\/?p=38836\">cliquez ici<\/a>.<\/p>\n<p><iframe loading=\"lazy\" style=\"border: 0px;\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/nb4u44yk\/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","protected":false},"excerpt":{"rendered":"<p>Exemples n\u00b0 1, 2, 3 Enonc\u00e9 Donner une repr\u00e9sentation vectorielle d&rsquo;une droite passant par les deux points P(2; 12; -4) et Q(2; 3; 2). Rappels de la th\u00e9orie La repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;une droite dans l&rsquo;espace est de type :\u00a0 On peut aussi l&rsquo;\u00e9crire sous une autre forme : Dans cette repr\u00e9sentation x0, y0 [&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":[40,10,58],"tags":[],"class_list":["post-39107","post","type-post","status-publish","format-standard","hentry","category-exercice","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=40\" rel=\"category\">Exercice<\/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":"Exemples n\u00b0 1, 2, 3 Enonc\u00e9 Donner une repr\u00e9sentation vectorielle d&rsquo;une droite passant par les deux points P(2; 12; -4) et Q(2; 3; 2). Rappels de la th\u00e9orie La repr\u00e9sentation vectorielle ou param\u00e9trique d&rsquo;une droite dans l&rsquo;espace est de type :\u00a0 On peut aussi l&rsquo;\u00e9crire sous une autre forme : Dans cette repr\u00e9sentation x0, y0\u2026","_links":{"self":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39107"}],"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=39107"}],"version-history":[{"count":10,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39107\/revisions"}],"predecessor-version":[{"id":39197,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=\/wp\/v2\/posts\/39107\/revisions\/39197"}],"wp:attachment":[{"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=39107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=39107"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.swisslearn.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=39107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}