{"id":2156,"date":"2020-03-16T16:17:07","date_gmt":"2020-03-16T15:17:07","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2156"},"modified":"2020-10-02T14:58:29","modified_gmt":"2020-10-02T12:58:29","slug":"utiliser-le-produit-scalaire-et-les-fonctions","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/16\/utiliser-le-produit-scalaire-et-les-fonctions\/","title":{"rendered":"Utiliser le produit scalaire et les fonctions"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Le probl\u00e8me est le suivant: \u00e9tant donn\u00e9 un carr\u00e9 ABCD de c\u00f4t\u00e9 10, on choisit deux points E et F respectivement sur [AD] et [BC] tels que ED = BF = x. On note G le point d&#8217;intersection de [BE] et [AF]. L&#8217;angle \\(\\widehat{AGB}\\) est-il constant ? Si oui, que vaut sa mesure ? Si non, entre quelles valeurs varie sa mesure ?<\/p>\n\n\n\n<figure class=\"wp-block-video aligncenter\"><video height=\"532\" style=\"aspect-ratio: 530 \/ 532;\" width=\"530\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/prodscal.mp4\"><\/video><\/figure>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">Rapportons le carr\u00e9 au rep\u00e8re orthonorm\u00e9 \\(A;\\vec{AB},\\vec{AD}\\). Dans ce rep\u00e8re, on a :$$\\begin{align}\\vec{FA}\\cdot\\vec{EB}&amp;=\\binom{-10}{-x}\\cdot\\binom{10}{x-10}\\\\&amp;=-100-x(x-10)\\\\&amp;=-x^2+10x-100. \\quad(1)\\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, utilisons la formule:$$\\vec{u}\\cdot\\vec{v}=\\|\\vec{u}\\|\\times\\|\\vec{v}\\|\\times\\cos\\left(\\vec{u},\\vec{v}\\right)$$pour calculer le m\u00eame produit scalaire: $$\\begin{align}\\vec{FA}\\cdot\\vec{EB}&amp;=FA \\times EB \\times \\cos\\alpha\\\\&amp;=\\sqrt{x^2+100}\\times\\sqrt{100+(10-x)^2}\\times\\cos\\alpha\\\\&amp;=\\sqrt{(x^2+100)(x^2-20x+200)} \\times \\cos\\alpha.\\quad (2) \\end{align}$$ <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Des r\u00e9sultats (1) et (2), on peut \u00e9tablir l&#8217;\u00e9galit\u00e9:$$ -x^2+10x-100 =  \\sqrt{(x^2+100)(x^2-20x+200)} \\times \\cos\\alpha$$et en d\u00e9duire que:$$\\cos\\alpha=\\frac{-x^2+10x-100}{ \\sqrt{(x^2+100)(x^2-20x+200)}}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut ainsi r\u00e9pondre \u00e0 la premi\u00e8re question: l&#8217;angle \\(\\alpha\\) n&#8217;est pas constant car son cosinus ne l&#8217;est pas.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">R\u00e9pondre \u00e0 la seconde est bien plus compliqu\u00e9e si on veut le faire alg\u00e9briquement. En effet, en d\u00e9rivant la fonction en <em>x<\/em> obtenue pour le cosinus de \\(\\alpha\\), on obtient une fonction qui peut nous freiner (m\u00eame si au final, on peut trouver son signe, voir remarque en fin d&#8217;article). Ci-apr\u00e8s, la capture d&#8217;\u00e9cran Xcas donnant cette d\u00e9riv\u00e9e:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"856\" height=\"331\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal.png\" alt=\"\" class=\"wp-image-2161\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal.png 856w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal-300x116.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal-600x232.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal-768x297.png 768w\" sizes=\"auto, (max-width: 856px) 100vw, 856px\" \/><figcaption>Capture d&#8217;\u00e9cran Xcas<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">On peut alors avoir envie de tracer la courbe repr\u00e9sentative de la fonction \u00e0 l&#8217;aide de GeoGebra:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"782\" height=\"164\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/geogebra-prodscal.png\" alt=\"\" class=\"wp-image-2162\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/geogebra-prodscal.png 782w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/geogebra-prodscal-300x63.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/geogebra-prodscal-600x126.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/geogebra-prodscal-768x161.png 768w\" sizes=\"auto, (max-width: 782px) 100vw, 782px\" \/><figcaption>Courbe repr\u00e9sentative du cosinus de \\(\\alpha\\) sous GeoGebra<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">M\u00eame en effectuant des supers m\u00e9ga zooms, difficile d&#8217;\u00eatre pr\u00e9cis: on peut juste voir que le cosinus ne varie pas beaucoup quand <em>x<\/em> varie de 0 \u00e0 10. On peut alors s&#8217;aider d&#8217;un programme Python pour d\u00e9finir le minimum et le maximum de la fonction. Mais pour des raisons de sym\u00e9trie, la courbe est n\u00e9cessairement sym\u00e9trique par rapport \u00e0 <em>x<\/em>=5. On peut alors dire que <em>f<\/em>(0) est la valeur minimale et <em>f<\/em>(5) la valeur maximale. Or,$$f(0)\\approx-0,707\\quad\\text{et}\\quad f(5)=-0,6.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, $$127^\\circ &lt; \\alpha \\leq 135^\\circ.$$En effet, \\(\\arccos(-0,6)=135^\\circ\\) et \\(\\arccos(-0,707)\\approx 126,86989765\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Une remarque sur la d\u00e9riv\u00e9e<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Le passage conjecturant que le minimum et le maximum sont atteints pour x=0 et x=5 n&#8217;est pas farfelu; en effet, si on regarde la fonction d\u00e9riv\u00e9e donn\u00e9e par Xcas (sous la forme factoris\u00e9e), on voit qu&#8217;un facteur est \u00e9gal \u00e0 (5-<em>x<\/em>) (il y a un signe &#8220;-&#8221; avant la fraction donc -(x-5) se transforme en 5-<em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">De plus, les autres facteurs du d\u00e9nominateurs sont positifs sur [0;10]. Il ne reste plus que les deux facteurs du num\u00e9rateurs de la forme:$$\\sqrt{u(x)}-v(x)$$et$$\\sqrt{u(x)}+v(x)$$en posant$$u(x)=x^4-20x^3+300x^2-2000x+20000\\text{ et }v(x)=x^2-10x+100$$qui, en d\u00e9veloppant leur produit, donne:$$x^4-20x^3+300x^2-2000x+20000-(x^2-10x+100)^2$$c&#8217;est-\u00e0-dire:$$400x^2-4000x+10000\\geq0.$$On peut donc montrer que <em>f<\/em>&#8216;(<em>x<\/em>) est positive sur [0;5] et donc que <em>f<\/em> est croissante sur cet intervalle.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Le probl\u00e8me est le suivant: \u00e9tant donn\u00e9 un carr\u00e9 ABCD de c\u00f4t\u00e9 10, on choisit deux points E et F respectivement sur [AD] et [BC] tels que ED = BF = x. On note G le point d&#8217;intersection de [BE] et [AF]. L&#8217;angle \\(\\widehat{AGB}\\) est-il constant ? Si oui, que [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[178,186,27,193,143],"class_list":["post-2156","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-cosinus","tag-fonction","tag-geogebra","tag-produit-scalaire","tag-xcas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Utiliser le produit scalaire et les fonctions - 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\/16\/utiliser-le-produit-scalaire-et-les-fonctions\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Utiliser le produit scalaire et les fonctions - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Le probl\u00e8me est le suivant: \u00e9tant donn\u00e9 un carr\u00e9 ABCD de c\u00f4t\u00e9 10, on choisit deux points E et F respectivement sur [AD] et [BC] tels que ED = BF = x. On note G le point d&#8217;intersection de [BE] et [AF]. L&#8217;angle (widehat{AGB}) est-il constant ? Si oui, que [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/16\/utiliser-le-produit-scalaire-et-les-fonctions\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2020-03-16T15:17:07+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-10-02T12:58:29+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/xcas-prodscal.png\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Utiliser le produit scalaire et les fonctions\",\"datePublished\":\"2020-03-16T15:17:07+00:00\",\"dateModified\":\"2020-10-02T12:58:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/\"},\"wordCount\":600,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/03\\\/xcas-prodscal.png\",\"keywords\":[\"cosinus\",\"fonction\",\"geogebra\",\"produit scalaire\",\"xcas\"],\"articleSection\":[\"Enseignement\",\"Math\u00e9matiques\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/03\\\/16\\\/utiliser-le-produit-scalaire-et-les-fonctions\\\/\",\"name\":\"Utiliser le produit scalaire et les fonctions - 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On note G le point d&#8217;intersection de [BE] et [AF]. L&#8217;angle (widehat{AGB}) est-il constant ? 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