{"id":2099,"date":"2020-03-03T13:53:03","date_gmt":"2020-03-03T12:53:03","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2099"},"modified":"2022-06-23T10:56:21","modified_gmt":"2022-06-23T08:56:21","slug":"application-du-produit-scalaire","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/03\/application-du-produit-scalaire\/","title":{"rendered":"Application du produit scalaire"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">En 1\u00e8re sp\u00e9cialit\u00e9 Math\u00e9matiques, les \u00e9l\u00e8ves abordent la notion de <em>produit scalaire<\/em> de deux vecteurs, notions plut\u00f4t abstraite au premier abord. \u00c0 l&#8217;aide du produit scalaire, on peut d\u00e9montrer des propri\u00e9t\u00e9s g\u00e9om\u00e9triques importantes, comme la <a rel=\"noreferrer noopener\" aria-label=\"loi des sinus (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Loi_des_sinus\" target=\"_blank\"><em>loi des sinus<\/em><\/a> ou encore le <em><a rel=\"noreferrer noopener\" aria-label=\"th\u00e9or\u00e8me d'Al-Kashi (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Loi_des_cosinus\" target=\"_blank\">th\u00e9or\u00e8me d&#8217;Al-Kashi<\/a><\/em>, aussi connu sous le nom de <em>loi des cosinus<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons voir dans cet article que ces deux r\u00e9sultats nous permettent de trouver la solution \u00e0 un probl\u00e8me pos\u00e9 lors d&#8217;un concours math\u00e9matique.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 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\/03\/03\/application-du-produit-scalaire\/#Le_probleme\" >Le probl\u00e8me<\/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\/03\/03\/application-du-produit-scalaire\/#Une_solution\" >Une solution<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/03\/application-du-produit-scalaire\/#Utilisation_de_la_loi_des_sinus\" >Utilisation de la loi des sinus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/03\/application-du-produit-scalaire\/#Utilisation_de_la_loi_des_cosinus\" >Utilisation de la loi des cosinus<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_probleme\"><\/span>Le probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image wp-duotone-000000-00a5ff-1\">\n<figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/produit-scalaire-1024x329.png\" alt=\"\" class=\"wp-image-2098\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Le point D appartient au c\u00f4t\u00e9 [BC] du triangle ABC, tel que AD = 10. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On sait que AB = 2AC.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Trouver la longueur BC.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_solution\"><\/span>Une solution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Utilisation_de_la_loi_des_sinus\"><\/span>Utilisation de la loi des sinus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons nous placer dans un premier temps dans le triangle DCA afin d&#8217;utiliser la loi des sinus pour \u00e9crire:$$\\frac{DC}{\\sin60^\\circ}=\\frac{AC}{\\sin\\widehat{CDA}}.$$Dans le triangle DBA, cette loi donne:$$\\frac{DB}{\\sin60^\\circ}=\\frac{AB}{\\sin(180-\\widehat{CDA})}$$c&#8217;est-\u00e0-dire: $$\\frac{DB}{\\sin60^\\circ}=\\frac{2AC}{\\sin(\\widehat{CDA})}$$car \\(\\sin(180-x)=\\sin(x)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut alors \u00e9crire:$$\\frac{BD}{DC}= \\frac{\\sin60^\\circ\\frac{2AC}{\\sin\\widehat{CDA}}}{\\sin60^\\circ\\frac{AC}{\\sin\\widehat{CDA}}}=2.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Utilisation_de_la_loi_des_cosinus\"><\/span>Utilisation de la loi des cosinus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, dans les triangles ADC et ADB, utilisons le th\u00e9or\u00e8me d&#8217;Al-Kashi:$$\\begin{cases}DC^2 &amp; = AC^2 + 100^2 &#8211; 200AC\\cos60^\\circ\\\\DB^2 &amp; =AB^2 + 100^2 &#8211; 200AB\\cos60^\\circ \\end{cases}$$c&#8217;est-\u00e0-dire: $$\\begin{cases}DC^2 &amp; = AC^2 + 10000 &#8211; 100AC\\\\4DC^2 &amp; =4AC^2 + 10000 &#8211; 200AC \\end{cases}$$ou encore, en multipliant par 4 la premi\u00e8re \u00e9galit\u00e9: $$\\begin{cases}4DC^2 &amp; = 4AC^2 + 40000 &#8211; 100AC\\\\4DC^2 &amp; =4AC^2 + 10000 &#8211; 200AC \\end{cases}$$  En soustrayant ces deux \u00e9galit\u00e9s, on obtient:$$AC=150$$et donc:$$DC^2=AC^2+10000-100AC=17500$$soit:$$AC=\\sqrt{17500}=50\\sqrt7.$$On en d\u00e9duit alors que:$$BC=3DC=150\\sqrt7.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">T\u00e9l\u00e9charger le fichier geogebra de la figure: <\/p>\n\n\n\n<center><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/produit-scalaire.ggb\">Ficher Geogebra<\/a><\/center>\n","protected":false},"excerpt":{"rendered":"<p>En 1\u00e8re sp\u00e9cialit\u00e9 Math\u00e9matiques, les \u00e9l\u00e8ves abordent la notion de produit scalaire de deux vecteurs, notions plut\u00f4t abstraite au premier abord. \u00c0 l&#8217;aide du produit scalaire, on peut d\u00e9montrer des propri\u00e9t\u00e9s g\u00e9om\u00e9triques importantes, comme la loi des sinus ou encore le th\u00e9or\u00e8me d&#8217;Al-Kashi, aussi connu sous le nom de loi [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":7690,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[49,27,182,181],"class_list":["post-2099","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-enseignement","category-mathematiques","tag-al-kashi","tag-geogebra","tag-loi-des-cosinus","tag-loi-des-sinus"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Application du 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\/03\/03\/application-du-produit-scalaire\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" 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