{"id":167,"date":"2015-02-20T11:21:29","date_gmt":"2015-02-20T11:21:29","guid":{"rendered":"http:\/\/anthroponaute.fr\/blog-informatique\/?p=167"},"modified":"2020-03-26T12:17:37","modified_gmt":"2020-03-26T12:17:37","slug":"petit-cours-sur-les-vecteurs","status":"publish","type":"post","link":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/?p=167","title":{"rendered":"Les vecteurs"},"content":{"rendered":"<p><strong><a href=\"https:\/\/anthropoya.cluster014.ovh.net\/blog-informatique\/wp-content\/uploads\/2015\/02\/vector_2d_coordinates.png\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-medium wp-image-220\" src=\"https:\/\/anthropoya.cluster014.ovh.net\/blog-informatique\/wp-content\/uploads\/2015\/02\/vector_2d_coordinates-300x169.png\" alt=\"vector_2d_coordinates\" width=\"300\" height=\"169\" srcset=\"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/wp-content\/uploads\/2015\/02\/vector_2d_coordinates-300x169.png 300w, https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/wp-content\/uploads\/2015\/02\/vector_2d_coordinates.png 454w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/strong><\/p>\n<p><strong>Intro :<\/strong><\/p>\n<p>Ceci constitue un petit cours sur les vecteurs que vous avez vus normalement au lyc\u00e9e.<\/p>\n<p>Un vecteur est une entit\u00e9 math\u00e9matique qui d\u00e9signe aussi bien une <strong>position<\/strong> qu&rsquo;une <strong>direction<\/strong> dans un rep\u00e8re donn\u00e9.<br \/>\nUn vecteur peut servir \u00e0 repr\u00e9senter une force, une position, une vitesse, par exemple la direction \u00e0 laquelle le joueur regarde, etc&#8230;<\/p>\n<p>Dans cet article on notera un vecteur <img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv%7D&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{v}' title='{v}' class='latex' \/> plut\u00f4t que <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7Bv%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='\\vec{v} ' title='\\vec{v} ' class='latex' \/>.<\/p>\n<p><strong>Pr\u00e9requis :<\/strong><\/p>\n<p>&#8211; Avoir d\u00e9j\u00e0 lu et compris quelques rudiments sur les math\u00e9matiques des vecteurs.<\/p>\n<p><strong>Explications :<\/strong><\/p>\n<p>Un vecteur se note de cette fa\u00e7on : <img src='https:\/\/s0.wp.com\/latex.php?latex=v%3D%28%7Bx%2Cy%2Cz%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='v=({x,y,z})' title='v=({x,y,z})' class='latex' \/><\/p>\n<p><strong>Op\u00e9rations sur les vecteurs :<\/strong><\/p>\n<p>Obtenir la n\u00e9gation d&rsquo;un vecteur :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=-v%3D-%28%7Bx%2Cy%2Cz%7D%29+%3D+%28%7B-x%2C-y%2C-z%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='-v=-({x,y,z}) = ({-x,-y,-z})' title='-v=-({x,y,z}) = ({-x,-y,-z})' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>On peut ajouter deux vecteurs :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv+%2B+w%7D+%3D+%7B%28%7Bv_%7Bx%7D%2Cv_%7By%7D%2Cv_%7Bz%7D%7D%29+%2B+%28%7Bw_%7Bx%7D%2Cw_%7By%7D%2Cw_%7Bz%7D%7D%29%7D+%3D+%28%7Bv_%7Bx%7D+%2B+w_%7Bx%7D%2Cv_%7By%7D+%2B+w_%7By%7D%2Cv_%7Bz%7D+%2B+w_%7Bz%7D%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{v + w} = {({v_{x},v_{y},v_{z}}) + ({w_{x},w_{y},w_{z}})} = ({v_{x} + w_{x},v_{y} + w_{y},v_{z} + w_{z}})' title='{v + w} = {({v_{x},v_{y},v_{z}}) + ({w_{x},w_{y},w_{z}})} = ({v_{x} + w_{x},v_{y} + w_{y},v_{z} + w_{z}})' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Soustraire deux vecteurs :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv+-+w%7D+%3D+%7Bv+%2B+%28-w%29%7D+%3D+%7B%28%7Bv_%7Bx%7D%2Cv_%7By%7D%2Cv_%7Bz%7D%7D%29+%2B+%28%7B-w_%7Bx%7D%2C-w_%7By%7D%2C-w_%7Bz%7D%7D%29%7D+%3D+%28%7Bv_%7Bx%7D+-+w_%7Bx%7D%2Cv_%7By%7D+-+w_%7By%7D%2Cv_%7Bz%7D+-+w_%7Bz%7D%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{v - w} = {v + (-w)} = {({v_{x},v_{y},v_{z}}) + ({-w_{x},-w_{y},-w_{z}})} = ({v_{x} - w_{x},v_{y} - w_{y},v_{z} - w_{z}})' title='{v - w} = {v + (-w)} = {({v_{x},v_{y},v_{z}}) + ({-w_{x},-w_{y},-w_{z}})} = ({v_{x} - w_{x},v_{y} - w_{y},v_{z} - w_{z}})' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Multiplier un vecteur par un scalaire :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctextbf%7Bk%7Dv+%3D+%28%7B%5Ctextbf%7Bk%7Dv_%7Bx%7D%2C+%5Ctextbf%7Bk%7Dv_%7By%7D%2C+%5Ctextbf%7Bk%7Dv_%7Bz%7D%7D%29+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='\\textbf{k}v = ({\\textbf{k}v_{x}, \\textbf{k}v_{y}, \\textbf{k}v_{z}}) ' title='\\textbf{k}v = ({\\textbf{k}v_{x}, \\textbf{k}v_{y}, \\textbf{k}v_{z}}) ' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>On peut inverser la direction d&rsquo;un vecteur dans le m\u00eame genre :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7B%28%5Ctextbf%7B-1%7D%29%7Dv+%3D+-v+%3D+%28%7B-v_%7Bx%7D%2C+-v_%7By%7D%2C+-v_%7Bz%7D%7D%29+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{(\\textbf{-1})}v = -v = ({-v_{x}, -v_{y}, -v_{z}}) ' title='{(\\textbf{-1})}v = -v = ({-v_{x}, -v_{y}, -v_{z}}) ' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>On peut calculer la longueur d&rsquo;un vecteur (en utilisant le th\u00e9or\u00e8me de Pythagore) :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%5C%7Cv%5C%7C+%3D+%5Csqrt%7B%7Bv%7D_x%5E2%2B%7Bv%7D_y%5E2%2B%7Bv%7D_z%5E2%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='\\|v\\| = \\sqrt{{v}_x^2+{v}_y^2+{v}_z^2} ' title='\\|v\\| = \\sqrt{{v}_x^2+{v}_y^2+{v}_z^2} ' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>On peut calculer la distance entre deux vecteurs :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=distance%28%7Ba%7D%2C+%7Bb%7D%29%3D%5C%7C%7Bb%7D-%7Ba%7D%5C%7C%3D%5Csqrt%7B%28%7Bb%7D_x-%7Ba%7D_x%29%5E%7B2%7D%2B%28%7Bb%7D_y-%7Ba%7D_y%29%5E%7B2%7D%2B%28%7Bb%7D_z-%7Ba%7D_z%29%5E%7B2%7D%7D&#038;bg=ffffff&#038;fg=000000&#038;s=3' alt='distance({a}, {b})=\\|{b}-{a}\\|=\\sqrt{({b}_x-{a}_x)^{2}+({b}_y-{a}_y)^{2}+({b}_z-{a}_z)^{2}}' title='distance({a}, {b})=\\|{b}-{a}\\|=\\sqrt{({b}_x-{a}_x)^{2}+({b}_y-{a}_y)^{2}+({b}_z-{a}_z)^{2}}' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Normaliser un vecteur (rendre sa longueur de 1 unit\u00e9) :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv%7D_%7Bnorm%7D%3D%5Cfrac%7B%7Bv%7D%7D%7B%5C%7C%7Bv%7D%5C%7C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=3' alt='{v}_{norm}=\\frac{{v}}{\\|{v}\\|}' title='{v}_{norm}=\\frac{{v}}{\\|{v}\\|}' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Il existe aussi deux autres op\u00e9rations qui sont : <strong>le produit scalaire<\/strong> et <strong>le produit vectoriel<\/strong>.<\/p>\n<p>L&rsquo;op\u00e9ration du <strong>produit scalaire<\/strong> donne comme r\u00e9sultat un nombre (d&rsquo;o\u00f9 le nom \u00ab\u00a0scalaire\u00a0\u00bb) :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv%7D%5Ccdot%7Bw%7D%3D%7Bv%7D_%7Bx%7D%7Bw%7D_%7Bx%7D%2B%7Bv%7D_%7By%7D%7Bw%7D_%7By%7D%2B%7Bv%7D_%7Bz%7D%7Bw%7D_%7Bz%7D&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{v}\\cdot{w}={v}_{x}{w}_{x}+{v}_{y}{w}_{y}+{v}_{z}{w}_{z}' title='{v}\\cdot{w}={v}_{x}{w}_{x}+{v}_{y}{w}_{y}+{v}_{z}{w}_{z}' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Il s&rsquo;exprime aussi avec la fonction cosinus avec comme param\u00e8tre l&rsquo;angle entre les deux vecteurs :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bv%7D%5Ccdot%7Bw%7D%3D%5C%7C%7Bv%7D%5C%7C%5C%7C%7Bw%7D%5C%7C%5Ccos%28%5Ctheta%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{v}\\cdot{w}=\\|{v}\\|\\|{w}\\|\\cos(\\theta)' title='{v}\\cdot{w}=\\|{v}\\|\\|{w}\\|\\cos(\\theta)' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Le produit scalaire est une op\u00e9ration fondamentale dans le calcul des lumi\u00e8res dans le rendu graphique 3D.<\/p>\n<p>On peut trouver l&rsquo;angle entre les deux vecteurs ainsi :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta%3D%5Carccos%5Cleft%28%5Cfrac%7B%7Ba%7D%5Ccdot%7Bb%7D%7D%7B%5C%7C%7Ba%7D%5C%7C%5C%7C%7Bb%7D%5C%7C%7D%5Cright%29&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='\\theta=\\arccos\\left(\\frac{{a}\\cdot{b}}{\\|{a}\\|\\|{b}\\|}\\right)' title='\\theta=\\arccos\\left(\\frac{{a}\\cdot{b}}{\\|{a}\\|\\|{b}\\|}\\right)' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>La fonction arccos est tout simplement la fonction cos\u207b\u00b9 de votre calculatrice.<\/p>\n<p>L&rsquo;op\u00e9ration du <strong>produit vectoriel<\/strong> :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bw%7D%3D%7Bu%7D%5Ctimes%7Bv%7D+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{w}={u}\\times{v} ' title='{w}={u}\\times{v} ' class='latex' \/>\n<p>&nbsp;<\/p>\n<p>Le r\u00e9sultat du produit vectoriel est un vecteur perpendiculaire aux deux vecteurs (<strong>u<\/strong> et <strong>v<\/strong>) de l&rsquo;op\u00e9ration ; le <strong>sens<\/strong> de ce vecteur d\u00e9pend de <em>l&rsquo;ordre<\/em> des op\u00e9randes dans la multiplication.<\/p>\n<p>On le calcul d&rsquo;apr\u00e8s la formule suivante :<\/p>\n<img src='https:\/\/s0.wp.com\/latex.php?latex=%7Bw%7D%3D%7Bu%7D%5Ctimes%7Bv%7D%3D%28%28%7Bu%7D_%7By%7D%7Bv%7D_%7Bz%7D-%7Bu%7D_%7Bz%7D%7Bv%7D_%7By%7D%29%2C%28%7Bu%7D_%7Bz%7D%7Bv%7D_%7Bx%7D-%7Bu%7D_%7Bx%7D%7Bv%7D_%7Bz%7D%29%2C%28%7Bu%7D_%7Bx%7D+%7Bv%7D_%7By%7D-%7Bu%7D_%7By%7D%7Bv%7D_%7Bx%7D%29%29+&#038;bg=ffffff&#038;fg=000000&#038;s=2' alt='{w}={u}\\times{v}=(({u}_{y}{v}_{z}-{u}_{z}{v}_{y}),({u}_{z}{v}_{x}-{u}_{x}{v}_{z}),({u}_{x} {v}_{y}-{u}_{y}{v}_{x})) ' title='{w}={u}\\times{v}=(({u}_{y}{v}_{z}-{u}_{z}{v}_{y}),({u}_{z}{v}_{x}-{u}_{x}{v}_{z}),({u}_{x} {v}_{y}-{u}_{y}{v}_{x})) ' class='latex' \/>\n<p><strong><br \/>\nR\u00e9sum\u00e9 :<\/strong><\/p>\n<p>Nous avons vu la notion de vecteur ; objet math\u00e9matique fondamentale dans la r\u00e9alisation d&rsquo;un jeu vid\u00e9o en 2D ou 3D.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Intro : Ceci constitue un petit cours sur les vecteurs que vous avez vus normalement au lyc\u00e9e. Un vecteur est une entit\u00e9 math\u00e9matique qui d\u00e9signe aussi bien une position qu&rsquo;une direction dans un rep\u00e8re donn\u00e9. Un vecteur peut servir \u00e0 repr\u00e9senter une force, une position, une vitesse, par exemple la direction \u00e0 laquelle le joueur [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[6],"tags":[],"_links":{"self":[{"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/posts\/167"}],"collection":[{"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=167"}],"version-history":[{"count":182,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions"}],"predecessor-version":[{"id":4585,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions\/4585"}],"wp:attachment":[{"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.la-porte-des-nebuleuses.net\/blog-informatique\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}