{"id":1914,"date":"2023-11-22T21:58:28","date_gmt":"2023-11-22T20:58:28","guid":{"rendered":"http:\/\/localhost:8080\/maxblog\/?p=1914"},"modified":"2023-11-22T23:46:58","modified_gmt":"2023-11-22T22:46:58","slug":"manipulation-des-vecteurs-dans-le-plan","status":"publish","type":"post","link":"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/","title":{"rendered":"Manipulation des vecteurs dans le plan"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1914\" class=\"elementor elementor-1914\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7d71a4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7d71a4a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-389cd7d\" data-id=\"389cd7d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a8169f1 elementor-widget elementor-widget-spacer\" data-id=\"a8169f1\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7643655 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7643655\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f297427\" data-id=\"f297427\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-361d80f elementor-widget elementor-widget-heading\" data-id=\"361d80f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-heading-title elementor-size-xl\">Manipulation des vecteurs dans le plan<\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c2bda19 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c2bda19\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d178bc4\" data-id=\"d178bc4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-de06178 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"de06178\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cc24190\" data-id=\"cc24190\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b7c611b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b7c611b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9a7e2c2\" data-id=\"9a7e2c2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fffe349 nc-justify-text elementor-widget elementor-widget-text-editor\" data-id=\"fffe349\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/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-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#Introduction\" >Introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#Cours\" >Cours<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#1_Vecteur_vecMM_associe_a_une_translation\" >1. Vecteur  \\vec{MM&#039;}  associ\u00e9 \u00e0 une translation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#2_Egalite_de_deux_vecteurs\" >2. \u00c9galit\u00e9 de deux vecteurs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#3_Vecteur_nul\" >3. Vecteur nul<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#4_Somme_de_deux_vecteurs_et_relation_de_Chasles\" >4. Somme de deux vecteurs et relation de Chasles<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#5_Base_orthonormee_et_coordonnees_dun_vecteur\" >5. Base orthonorm\u00e9e et coordonn\u00e9es d&rsquo;un vecteur<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#6_Expression_des_coordonnees_de_vecAB\" >6. Expression des coordonn\u00e9es de  \\vec{AB}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#7_Produit_dun_vecteur_par_un_nombre_reel_et_colinearite\" >7. Produit d&rsquo;un vecteur par un nombre r\u00e9el et colin\u00e9arit\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#8_Determinant_et_critere_de_colinearite\" >8. D\u00e9terminant et crit\u00e8re de colin\u00e9arit\u00e9<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#Methodes\" >M\u00e9thodes<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#1_Representer_geometriquement_des_vecteurs\" >1. Repr\u00e9senter g\u00e9om\u00e9triquement des vecteurs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#2_Construire_geometriquement_la_somme_de_deux_vecteurs\" >2. Construire g\u00e9om\u00e9triquement la somme de deux vecteurs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#3_Representer_et_lire_les_coordonnees_dun_vecteur\" >3. Repr\u00e9senter et lire les coordonn\u00e9es d\u2019un vecteur<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#4_Calculer_les_coordonnees_dune_somme_de_vecteurs_et_dun_produit_dun_vecteur_par_un_nombre_reel\" >4. Calculer les coordonn\u00e9es d&rsquo;une somme de vecteurs et d&rsquo;un produit d&rsquo;un\nvecteur par un nombre r\u00e9el<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#5_Calculer_la_distance_entre_deux_points_et_les_coordonnees_du_milieu_dun_segment\" >5. Calculer la distance entre deux points et les coordonn\u00e9es du milieu d\u2019un\nsegment<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#6_Caracteriser_alignement_et_parallelisme_par_la_colinearite_de_vecteurs\" >6. Caract\u00e9riser alignement et parall\u00e9lisme par la colin\u00e9arit\u00e9 de vecteurs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/www.maxdecours.com\/maxblog\/manipulation-des-vecteurs-dans-le-plan\/#7_Resoudre_des_problemes_en_utilisant_la_representation_la_plus_adaptee_des_vecteurs\" >7. R\u00e9soudre des probl\u00e8mes en utilisant la repr\u00e9sentation la plus adapt\u00e9e\ndes vecteurs<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"Introduction\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">Introduction<\/span><span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">La ma\u00eetrise de ces concepts permet de manipuler et de comprendre les\nvecteurs dans le plan, facilitant l&rsquo;analyse et la r\u00e9solution de probl\u00e8mes\ng\u00e9om\u00e9triques.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h1><span class=\"ez-toc-section\" id=\"Cours\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">Cours<\/span><span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"1_Vecteur_vecMM_associe_a_une_translation\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">1. Vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{MM&#039;} <\/span> associ\u00e9 \u00e0 une translation<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Le vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{MM&#039;} <\/span> est l&rsquo;entit\u00e9 g\u00e9om\u00e9trique qui\ncorrespond \u00e0 la translation qui transforme le point M en M&rsquo;. Ce vecteur poss\u00e8de\ntrois caract\u00e9ristiques :<br style=\"mso-special-character:line-break\">\n<br style=\"mso-special-character:line-break\">\n<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; <i>Direction<\/i> : La droite qui\npasse par les points M et M&rsquo; indique la direction du vecteur.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; <i>Sens<\/i> : Le sens du vecteur\nva de M \u00e0 M&rsquo;.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; <i>Norme<\/i> : La norme du\nvecteur, not\u00e9e <span class=\"katex-eq\" data-katex-display=\"false\"> \\lVert \\vec{MM&#039;} \\rVert <\/span>, est la distance entre\nles points M et M&rsquo;.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"2_Egalite_de_deux_vecteurs\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">2. \u00c9galit\u00e9 de deux vecteurs<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Deux vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} <\/span> sont\n\u00e9gaux si et seulement s&rsquo;ils ont la m\u00eame direction, le m\u00eame sens et la m\u00eame\nnorme. On \u00e9crit alors <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} = \\vec{v} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"3_Vecteur_nul\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">3. Vecteur nul<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Le vecteur nul, not\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{0} <\/span>, est un vecteur dont la norme\nest \u00e9gale \u00e0 z\u00e9ro. Il n&rsquo;a ni direction ni sens particulier.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"4_Somme_de_deux_vecteurs_et_relation_de_Chasles\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">4. Somme de deux vecteurs et relation de Chasles<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">La somme de deux vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} <\/span>\nest le vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{w} <\/span> obtenu en encha\u00eenant les translations\ncorrespondant \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} <\/span>. On note <span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{w} = \\vec{u} + \\vec{v} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>La relation de Chasles \u00e9tablit que\npour trois points A, B, et C dans le plan, <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{AC} = \\vec{AB} +\n\\vec{BC} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"5_Base_orthonormee_et_coordonnees_dun_vecteur\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">5. Base orthonorm\u00e9e et coordonn\u00e9es d&rsquo;un vecteur<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Une base orthonorm\u00e9e du plan est un couple de vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> (\\vec{i}, \\vec{j})\n<\/span> qui sont perpendiculaires et de norme 1. Les coordonn\u00e9es d&rsquo;un vecteur <span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{u} <\/span> sont les nombres r\u00e9els <span class=\"katex-eq\" data-katex-display=\"false\"> (x, y) <\/span> tels que <span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{u} = x\\vec{i} + y\\vec{j} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">La <i>norme<\/i> du vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> est donn\u00e9e par <span class=\"katex-eq\" data-katex-display=\"false\">\n\\lVert \\vec{u} \\rVert = \\sqrt{x^2 + y^2} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"6_Expression_des_coordonnees_de_vecAB\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">6. Expression des coordonn\u00e9es de <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{AB} <\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Les coordonn\u00e9es du vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{AB} <\/span>, avec A de coordonn\u00e9es\n<span class=\"katex-eq\" data-katex-display=\"false\"> (x_1, y_1) <\/span> et B de coordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\"> (x_2, y_2) <\/span>,\nsont donn\u00e9es par <span class=\"katex-eq\" data-katex-display=\"false\"> (x_2 &#8211; x_1, y_2 &#8211; y_1) <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"7_Produit_dun_vecteur_par_un_nombre_reel_et_colinearite\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">7. Produit d&rsquo;un vecteur par un nombre r\u00e9el et colin\u00e9arit\u00e9<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Le produit d&rsquo;un vecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> de coordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\"> (x,\ny) <\/span> par un nombre r\u00e9el <span class=\"katex-eq\" data-katex-display=\"false\"> \\lambda <\/span> donne un nouveau\nvecteur <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} = \\lambda \\vec{u} <\/span> de coordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\">\n(\\lambda x, \\lambda y) <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Deux vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} <\/span> sont dits\ncolin\u00e9aires s&rsquo;il existe un r\u00e9el <span class=\"katex-eq\" data-katex-display=\"false\"> \\lambda <\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{v} = \\lambda \\vec{u} <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"8_Determinant_et_critere_de_colinearite\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">8. D\u00e9terminant et crit\u00e8re de colin\u00e9arit\u00e9<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Le d\u00e9terminant de deux vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} = (x_1, y_1) <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{v} = (x_2, y_2) <\/span> dans une base orthonorm\u00e9e est donn\u00e9 par <span class=\"katex-eq\" data-katex-display=\"false\">\n\\text{det}(\\vec{u}, \\vec{v}) = x_1 y_2 &#8211; x_2 y_1 <\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">Les vecteurs <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{u} <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{v} <\/span> sont\ncolin\u00e9aires si et seulement si <span class=\"katex-eq\" data-katex-display=\"false\"> \\text{det}(\\vec{u}, \\vec{v}) = 0 <\/span>.\nCe crit\u00e8re est utilis\u00e9 pour v\u00e9rifier l&rsquo;alignement (trois points sont align\u00e9s si\nles vecteurs qu&rsquo;ils forment sont colin\u00e9aires) et le parall\u00e9lisme (deux vecteurs\nsont parall\u00e8les s&rsquo;ils sont colin\u00e9aires).<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h1><span class=\"ez-toc-section\" id=\"Methodes\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">M\u00e9thodes<\/span><span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"1_Representer_geometriquement_des_vecteurs\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">1. Repr\u00e9senter g\u00e9om\u00e9triquement des vecteurs<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Choisissez deux points <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span>\net <span class=\"katex-eq\" data-katex-display=\"false\">M&#039;<\/span> dans le plan.<br style=\"mso-special-character:line-break\">\n<br style=\"mso-special-character:line-break\">\n<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Tracez une fl\u00e8che allant de <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span>\n\u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">M&#039;<\/span>. La fl\u00e8che repr\u00e9sente le vecteur <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{MM&#039;}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Indiquez les caract\u00e9ristiques du\nvecteur : direction (ligne <span class=\"katex-eq\" data-katex-display=\"false\">MM&#039;<\/span>), sens (de <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span> \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">M&#039;<\/span>)\net norme (longueur de <span class=\"katex-eq\" data-katex-display=\"false\">MM&#039;<\/span>).<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"2_Construire_geometriquement_la_somme_de_deux_vecteurs\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">2. Construire g\u00e9om\u00e9triquement la somme de deux vecteurs<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Repr\u00e9sentez les vecteurs <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span>\net <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span> en les pla\u00e7ant bout \u00e0 bout : la queue de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span>\n\u00e0 la t\u00eate de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Tracez un nouveau vecteur <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{w}<\/span>\nallant de la queue de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span> \u00e0 la t\u00eate de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{w}<\/span> est alors la\nsomme des vecteurs <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span>, soit <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{w}\n= \\vec{u} + \\vec{v}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"3_Representer_et_lire_les_coordonnees_dun_vecteur\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">3. Repr\u00e9senter et lire les coordonn\u00e9es d\u2019un vecteur<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Pour repr\u00e9senter un vecteur <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span>\nde coordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\">(x, y)<\/span>, partez d&rsquo;un point <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>,\ntracez x unit\u00e9s horizontalement et y unit\u00e9s verticalement, puis marquez le\npoint <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>. <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span> est alors le vecteur <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{AB}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Pour lire les coordonn\u00e9es, mesurez\nles distances horizontale (x) et verticale (y) depuis le point de d\u00e9part du\nvecteur.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"4_Calculer_les_coordonnees_dune_somme_de_vecteurs_et_dun_produit_dun_vecteur_par_un_nombre_reel\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">4. Calculer les coordonn\u00e9es d&rsquo;une somme de vecteurs et d&rsquo;un produit d&rsquo;un\nvecteur par un nombre r\u00e9el<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Si <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span> a pour\ncoordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\">(x_1, y_1)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span> a pour\ncoordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\">(x_2, y_2)<\/span>, alors les coordonn\u00e9es de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}\n+ \\vec{v}<\/span> sont <span class=\"katex-eq\" data-katex-display=\"false\">(x_1 + x_2, y_1 + y_2)<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Pour un nombre r\u00e9el <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span>,\nles coordonn\u00e9es de <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\vec{u}<\/span> sont <span class=\"katex-eq\" data-katex-display=\"false\">(\\lambda x_1,\n\\lambda y_1)<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"5_Calculer_la_distance_entre_deux_points_et_les_coordonnees_du_milieu_dun_segment\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">5. Calculer la distance entre deux points et les coordonn\u00e9es du milieu d\u2019un\nsegment<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; La distance entre deux points <span class=\"katex-eq\" data-katex-display=\"false\">A(x_1,\ny_1)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B(x_2, y_2)<\/span> est <span class=\"katex-eq\" data-katex-display=\"false\">d = \\sqrt{(x_2 &#8211; x_1)^2 +\n(y_2 &#8211; y_1)^2}<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Les coordonn\u00e9es du milieu <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span>\ndu segment <span class=\"katex-eq\" data-katex-display=\"false\">[AB]<\/span> sont <span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1\n+ y_2}{2}\\right)<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"6_Caracteriser_alignement_et_parallelisme_par_la_colinearite_de_vecteurs\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">6. Caract\u00e9riser alignement et parall\u00e9lisme par la colin\u00e9arit\u00e9 de vecteurs<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Trois points A, B, et C sont\nalign\u00e9s si les vecteurs <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{AB}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{BC}<\/span> sont\ncolin\u00e9aires, c&rsquo;est-\u00e0-dire <span class=\"katex-eq\" data-katex-display=\"false\">det(\\vec{AB}, \\vec{BC}) = 0<\/span>.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Deux vecteurs <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{u}<\/span>\net <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}<\/span> sont parall\u00e8les s&rsquo;ils sont colin\u00e9aires.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"7_Resoudre_des_problemes_en_utilisant_la_representation_la_plus_adaptee_des_vecteurs\"><\/span><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\">7. R\u00e9soudre des probl\u00e8mes en utilisant la repr\u00e9sentation la plus adapt\u00e9e\ndes vecteurs<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Analysez le probl\u00e8me pour\nidentifier les concepts cl\u00e9s (somme de vecteurs, colin\u00e9arit\u00e9, etc.).<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; Choisissez la repr\u00e9sentation la plus\nadapt\u00e9e (graphique ou alg\u00e9brique) pour r\u00e9soudre le probl\u00e8me.<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"mso-fareast-font-family:&quot;Times New Roman&quot;;\nmso-bidi-font-family:Calibri;mso-bidi-theme-font:minor-latin;mso-fareast-language:\nFR\"><br>\n<span style=\"mso-spacerun:yes\">&nbsp;&nbsp; <\/span>&#8211; V\u00e9rifiez la coh\u00e9rence de votre\nsolution par rapport \u00e0 la situation initiale.<\/span><\/p>\n\n\n\n\n\n<style>@font-face\n\t{font-family:\"Cambria Math\";\n\tpanose-1:2 4 5 3 5 4 6 3 2 4;\n\tmso-font-charset:0;\n\tmso-generic-font-family:roman;\n\tmso-font-pitch:variable;\n\tmso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face\n\t{font-family:Calibri;\n\tpanose-1:2 15 5 2 2 2 4 3 2 4;\n\tmso-font-charset:0;\n\tmso-generic-font-family:swiss;\n\tmso-font-pitch:variable;\n\tmso-font-signature:-536859905 -1073732485 9 0 511 0;}@font-face\n\t{font-family:\"Calibri Light\";\n\tpanose-1:2 15 3 2 2 2 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class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8c43dc4\" data-id=\"8c43dc4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-aa0c527 elementor-widget elementor-widget-spacer\" data-id=\"aa0c527\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-789777a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"789777a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-dd93843\" data-id=\"dd93843\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b1d0d5c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b1d0d5c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7e963bc\" data-id=\"7e963bc\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-40909d1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"40909d1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3a01f62\" data-id=\"3a01f62\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Manipulation des vecteurs dans le plan Introduction &nbsp; La ma\u00eetrise de ces concepts permet de manipuler et de comprendre les vecteurs dans le plan, facilitant l&rsquo;analyse et la r\u00e9solution de probl\u00e8mes g\u00e9om\u00e9triques. &nbsp; Cours &nbsp; 1. Vecteur associ\u00e9 \u00e0 une translation &nbsp; Le vecteur est l&rsquo;entit\u00e9 g\u00e9om\u00e9trique qui correspond \u00e0 la translation qui transforme le [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_canvas","format":"standard","meta":{"footnotes":""},"categories":[25,26,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1914"}],"collection":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/comments?post=1914"}],"version-history":[{"count":7,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1914\/revisions"}],"predecessor-version":[{"id":2009,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1914\/revisions\/2009"}],"wp:attachment":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/media?parent=1914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/categories?post=1914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/tags?post=1914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}