{"id":2391,"date":"2020-05-15T18:48:54","date_gmt":"2020-05-15T16:48:54","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2391"},"modified":"2020-05-15T18:52:54","modified_gmt":"2020-05-15T16:52:54","slug":"calculer-la-valeur-dun-angle-avec-le-produit-scalaire","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/15\/calculer-la-valeur-dun-angle-avec-le-produit-scalaire\/","title":{"rendered":"Calculer la valeur d&rsquo;un angle avec le produit scalaire"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">En cette fin d&rsquo;ann\u00e9e, les \u00e9l\u00e8ves de 1\u00e8re abordent \u00e9ventuellement le produit scalaire. Nous allons en voir une application pour d\u00e9terminer la valeur d&rsquo;un angle.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/15\/calculer-la-valeur-dun-angle-avec-le-produit-scalaire\/#Un_peu_de_mathematiques\" >Un peu de math\u00e9matiques<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/15\/calculer-la-valeur-dun-angle-avec-le-produit-scalaire\/#En_Python\" >En Python<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_peu_de_mathematiques\"><\/span>Un peu de math\u00e9matiques<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Pla\u00e7ons-nous dans un rep\u00e8re orthonorm\u00e9, et consid\u00e9rons deux vecteurs \\(\\vec{u}\\) et \\(\\vec{v}\\) comme ci-dessous:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/produit-scalaire.png\" alt=\"\" class=\"wp-image-2392\" width=\"549\" height=\"510\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/produit-scalaire.png 549w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/05\/produit-scalaire-300x279.png 300w\" sizes=\"auto, (max-width: 549px) 100vw, 549px\" \/><figcaption>Deux vecteurs du plan<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Nous cherchons \u00e0 d\u00e9terminer la valeur de l&rsquo;angle \\(\\alpha\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour cela, nous allons d&rsquo;abord calculer le produit scalaire : $$\\vec{u}\\cdot\\vec{v} = xx&rsquo; + yy&rsquo; = 7\\times4 + 4\\times(-4) = 12.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, \\(\\vec{u}\\displaystyle\\binom{7}{4}\\) car il faut avancer de 7 unit\u00e9s en abscisse et de 4 unit\u00e9s en ordonn\u00e9es pour aller du point A au point B. De m\u00eame, \\(\\vec{v}\\displaystyle\\binom{4}{-4}\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or, nous savons aussi que:$$\\vec{u}\\cdot\\vec{v}=\\|\\vec{u}\\| \\times \\|\\vec{v}\\| \\times \\cos(\\vec{u},\\vec{v}).$$ Or, $$\\|\\vec{u}\\| = \\sqrt{x_{\\vec{u}}^2+y_{\\vec{u}}^2}=\\sqrt{7^2 + 4^2} = \\sqrt{65}$$ et $$\\|\\vec{v}\\| = \\sqrt{x_{\\vec{v}}^2+y_{\\vec{v}}^2}=\\sqrt{4^2 + (-4)^2} =4\\sqrt{2}.$$Donc:$$\\underbrace{\\vec{u}\\cdot\\vec{v}}_{=12}=\\sqrt{65}\\times4\\sqrt{2}\\times\\cos(\\vec{u},\\vec{v})$$soit:$$12=4\\sqrt{130}\\cos(\\vec{u},\\vec{v}).$$On en d\u00e9duit alors:$$\\cos(\\vec{u},\\vec{v})=\\frac{12}{4\\sqrt{130}}$$et donc :$$\\alpha=\\arccos\\left( \\frac{12}{4\\sqrt{130}}\\right)\\approx75^\\circ.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"En_Python\"><\/span>En Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Nous venons de voir \u00e0 l&rsquo;instant une m\u00e9thode que l&rsquo;on peut g\u00e9n\u00e9raliser pour \u00e9crire une fonction Python retournant une valeur approch\u00e9e de l&rsquo;angle en degr\u00e9s.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from numpy import arccos,sqrt,pi\n\ndef calcAngle(u,v): # u = (a,b) et v = (c,d)\n    prodscal = u[0] * v[0] + u[1] * v[1]\n    NormeU = sqrt(u[0]**2 + u[1]**2)\n    NormeV = sqrt(v[0]**2 + v[1]**2)\n    return arccos( prodscal \/ (NormeU * NormeV) ) * 180 \/ pi\n\nu = (7,4)\nv = (4,-4)\nprint(calcAngle(u,v))<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>En cette fin d&rsquo;ann\u00e9e, les \u00e9l\u00e8ves de 1\u00e8re abordent \u00e9ventuellement le produit scalaire. Nous allons en voir une application pour d\u00e9terminer la valeur d&rsquo;un angle.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[59,193],"class_list":["post-2391","post","type-post","status-publish","format-standard","hentry","category-mathematiques","category-python","tag-angles","tag-produit-scalaire"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Calculer la valeur d&#039;un angle avec le produit scalaire - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/05\/15\/calculer-la-valeur-dun-angle-avec-le-produit-scalaire\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calculer la valeur d&#039;un angle avec le produit scalaire - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"En cette fin d&rsquo;ann\u00e9e, les \u00e9l\u00e8ves de 1\u00e8re abordent \u00e9ventuellement le produit scalaire. 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