{"id":1556,"date":"2026-09-08T08:06:58","date_gmt":"2026-09-08T08:06:58","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1556"},"modified":"2026-09-08T08:06:58","modified_gmt":"2026-09-08T08:06:58","slug":"mesuer-projeter-orthogonaliser","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/mesuer-projeter-orthogonaliser\/","title":{"rendered":"Mesuer. Projeter. Orthogonaliser."},"content":{"rendered":"\n&#8220;`html\n<div id=\"mza-euclidiens-cpge\">\n\n<style>\n#mza-euclidiens-cpge,#mza-euclidiens-cpge *{box-sizing:border-box}\n#mza-euclidiens-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-euclidiens-cpge a{text-decoration:none;color:inherit}\n#mza-euclidiens-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-euclidiens-cpge section{padding:68px 0}\n#mza-euclidiens-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-euclidiens-cpge .badge{\n display:inline-block;padding:8px 13px;border-radius:999px;\n border:1px solid rgba(239,214,140,.4);color:var(--gold2);\n 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.method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n\n<span class=\"badge\">COURS 15 \u2022 ESPACES EUCLIDIENS<\/span>\n\n<h1>Mesurer.<br>Projeter. Orthogonaliser.<\/h1>\n\n<p class=\"lead\">\nLes espaces euclidiens ajoutent \u00e0 l&#8217;alg\u00e8bre lin\u00e9aire une notion de longueur,\nd&#8217;angle et d&#8217;orthogonalit\u00e9. Ils permettent de construire des bases orthonormales,\nd\u00e9terminer des projections, calculer des distances et simplifier profond\u00e9ment\nde nombreux probl\u00e8mes de g\u00e9om\u00e9trie et d&#8217;alg\u00e8bre.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. PRODUIT SCALAIRE<\/span>\n<h2>Mesurer l&#8217;angle entre deux vecteurs.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Dans \u211d\u207f<\/h3>\n\n<div class=\"formula\">\n\u27e8x,y\u27e9 = x\u2081y\u2081 + &#8230; + x\u2099y\u2099\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Sym\u00e9trie<\/h3>\n\n<div class=\"formula\">\n\u27e8x,y\u27e9 = \u27e8y,x\u27e9\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Lin\u00e9arit\u00e9<\/h3>\n\n<div class=\"formula\">\n\u27e8\u03bbx+\u03bcy,z\u27e9\n=\n\u03bb\u27e8x,z\u27e9+\u03bc\u27e8y,z\u27e9\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Positivit\u00e9<\/h3>\n\n<div class=\"formula\">\n\u27e8x,x\u27e9 \u2265 0\n<\/div>\n\n<p>avec \u00e9galit\u00e9 si et seulement si x=0.<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. NORME<\/span>\n<h2>Mesurer la longueur.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<div class=\"formula\">\n||x|| = \u221a\u27e8x,x\u27e9\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Homog\u00e9n\u00e9it\u00e9<\/h3>\n\n<div class=\"formula\">\n||\u03bbx|| = |\u03bb| ||x||\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>In\u00e9galit\u00e9 triangulaire<\/h3>\n\n<div class=\"formula\">\n||x+y|| \u2264 ||x||+||y||\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Distance<\/h3>\n\n<div class=\"formula\">\nd(x,y)=||x-y||\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. CAUCHY-SCHWARZ<\/span>\n<h2>L&#8217;in\u00e9galit\u00e9 fondamentale.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<div class=\"formula\">\n|\u27e8x,y\u27e9| \u2264 ||x|| ||y||\n<\/div>\n\n<p>\nCette in\u00e9galit\u00e9 permet notamment de d\u00e9finir l&#8217;angle entre deux vecteurs.\n<\/p>\n\n<div class=\"formula\">\ncos(\u03b8)=\u27e8x,y\u27e9 \/ (||x|| ||y||)\n<\/div>\n\n<\/div>\n\n<div class=\"tip\">\nL&#8217;\u00e9galit\u00e9 dans Cauchy-Schwarz appara\u00eet lorsque les vecteurs sont colin\u00e9aires.\n<\/div>\n\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. ORTHOGONALIT\u00c9<\/span>\n<h2>Quand le produit scalaire s&#8217;annule.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Vecteurs orthogonaux<\/h3>\n\n<div class=\"formula\">\nx \u27c2 y \u21d4 \u27e8x,y\u27e9=0\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Pythagore<\/h3>\n\n<p>Si x \u27c2 y :<\/p>\n\n<div class=\"formula\">\n||x+y||\u00b2=||x||\u00b2+||y||\u00b2\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Famille orthogonale<\/h3>\n\n<p>\nLes vecteurs sont deux \u00e0 deux orthogonaux.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Cons\u00e9quence<\/h3>\n\n<div class=\"formula\">\nfamille orthogonale de vecteurs non nuls \u21d2 famille libre\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA EUCLIDEAN ENGINE<\/span>\n<h2>Six r\u00e9flexes.<\/h2>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>PRODUIT SCALAIRE<\/span><\/div>\n<div><b>02<\/b><span>NORMES<\/span><\/div>\n<div><b>03<\/b><span>ORTHOGONALIT\u00c9<\/span><\/div>\n<div><b>04<\/b><span>PROJECTION<\/span><\/div>\n<div><b>05<\/b><span>DISTANCE<\/span><\/div>\n<div><b>06<\/b><span>BASE ORTHONORMALE<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. BASES ORTHONORMALES<\/span>\n<h2>Le rep\u00e8re id\u00e9al.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Orthonormale<\/h3>\n\n<p>\nUne famille (e\u2081,&#8230;,e\u2099) est orthonormale si :\n<\/p>\n\n<div class=\"formula\">\n\u27e8e\u1d62,e\u2c7c\u27e9=0 si i\u2260j\n<\/div>\n\n<div class=\"formula\">\n||e\u1d62||=1\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Coordonn\u00e9es<\/h3>\n\n<p>\nDans une base orthonormale :\n<\/p>\n\n<div class=\"formula\">\nx = \u03a3 \u27e8x,e\u1d62\u27e9 e\u1d62\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Norme<\/h3>\n\n<div class=\"formula\">\n||x||\u00b2 = \u03a3 \u27e8x,e\u1d62\u27e9\u00b2\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Pourquoi c&#8217;est puissant ?<\/h3>\n\n<p>\nLes coordonn\u00e9es s&#8217;obtiennent directement par produit scalaire.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. GRAM-SCHMIDT<\/span>\n<h2>Transformer une base en base orthonormale.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\n\u00c0 partir d&#8217;une famille libre (v\u2081,&#8230;,v\u2099), on construit une famille orthogonale puis orthonormale.\n<\/p>\n\n<div class=\"formula\">\nu\u2081 = v\u2081\n<\/div>\n\n<div class=\"formula\">\nu\u2082 = v\u2082 &#8211; proj<sub>u\u2081<\/sub>(v\u2082)\n<\/div>\n\n<div class=\"formula\">\nu\u2083 = v\u2083\n&#8211; proj<sub>u\u2081<\/sub>(v\u2083)\n&#8211; proj<sub>u\u2082<\/sub>(v\u2083)\n<\/div>\n\n<p>\nPuis :\n<\/p>\n\n<div class=\"formula\">\ne\u1d62 = u\u1d62 \/ ||u\u1d62||\n<\/div>\n\n<\/div>\n\n<div class=\"tip\">\nToujours orthogonaliser avant de normaliser.\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. PROJECTION ORTHOGONALE SUR UNE DROITE<\/span>\n<h2>Extraire la composante dans une direction.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Direction unitaire e<\/h3>\n\n<div class=\"formula\">\nproj<sub>e<\/sub>(x)=\u27e8x,e\u27e9e\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Direction u non unitaire<\/h3>\n\n<div class=\"formula\">\nproj<sub>u<\/sub>(x)\n=\n\u27e8x,u\u27e9 \/ \u27e8u,u\u27e9 \u00b7 u\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>D\u00e9composition<\/h3>\n\n<div class=\"formula\">\nx = proj<sub>F<\/sub>(x) + z\n<\/div>\n\n<p>avec z\u2208F\u22a5.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Caract\u00e9risation<\/h3>\n\n<div class=\"formula\">\nx-proj<sub>F<\/sub>(x) \u27c2 F\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. ORTHOGONAL D&#8217;UN SOUS-ESPACE<\/span>\n<h2>Les directions perpendiculaires \u00e0 tout F.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nPour un sous-espace F de E :\n<\/p>\n\n<div class=\"formula\">\nF\u22a5 = {x\u2208E | \u2200y\u2208F, \u27e8x,y\u27e9=0}\n<\/div>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Dimension finie<\/h3>\n\n<div class=\"formula\">\nE = F \u2295 F\u22a5\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Dimensions<\/h3>\n\n<div class=\"formula\">\ndim(F)+dim(F\u22a5)=dim(E)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. DISTANCE \u00c0 UN SOUS-ESPACE<\/span>\n<h2>Le point le plus proche.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi p<sub>F<\/sub>(x) est le projet\u00e9 orthogonal de x sur F :\n<\/p>\n\n<div class=\"formula\">\nd(x,F)=||x-p<sub>F<\/sub>(x)||\n<\/div>\n\n<p>\nLe projet\u00e9 orthogonal est le point de F le plus proche de x.\n<\/p>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">10. EXERCICES PROGRESSIFS<\/span>\n<h2>Du produit scalaire \u00e0 la projection.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n\n<h3>Exercice 1 \u2014 Orthogonalit\u00e9<\/h3>\n\n<p>\nDans \u211d\u00b3, d\u00e9terminer si :\n<\/p>\n\n<div class=\"formula\">\nu=(1,1,0)<br>\nv=(1,-1,2)\n<\/div>\n\n<p>sont orthogonaux.<\/p>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclHint('euclHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"euclHint1\" class=\"hint\">\nCalculez simplement le produit scalaire u\u00b7v.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n\n<h3>Exercice 2 \u2014 Normalisation<\/h3>\n\n<p>\nNormaliser :\n<\/p>\n\n<div class=\"formula\">\nu=(3,4)\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclHint('euclHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"euclHint2\" class=\"hint\">\nCommencez par calculer ||u||, puis divisez le vecteur par sa norme.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n\n<h3>Exercice 3 \u2014 Projection<\/h3>\n\n<p>\nProjeter x=(2,3) sur la droite engendr\u00e9e par :\n<\/p>\n\n<div class=\"formula\">\nu=(1,1)\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclHint('euclHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"euclHint3\" class=\"hint\">\nUtilisez proj_u(x)=\u27e8x,u\u27e9\/\u27e8u,u\u27e9 \u00d7 u.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n\n<h3>Exercice 4 \u2014 Orthogonal<\/h3>\n\n<p>\nDans \u211d\u00b3, d\u00e9terminer F\u22a5 pour :\n<\/p>\n\n<div class=\"formula\">\nF=Vect((1,1,0),(0,1,1))\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclHint('euclHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"euclHint4\" class=\"hint\">\nCherchez x=(a,b,c) orthogonal aux deux vecteurs g\u00e9n\u00e9rateurs de F.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">11. PROBLEM LAB<\/span>\n<h2>Mission : Gram-Schmidt complet.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Mission CPGE<\/h3>\n\n<p>\nDans \u211d\u00b3 muni du produit scalaire canonique, on consid\u00e8re :\n<\/p>\n\n<div class=\"formula\">\nv\u2081=(1,1,0)<br>\nv\u2082=(1,0,1)<br>\nv\u2083=(0,1,1)\n<\/div>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>v\u00e9rifier que la famille est libre ;<\/li>\n<li>poser u\u2081=v\u2081 ;<\/li>\n<li>construire u\u2082 orthogonal \u00e0 u\u2081 ;<\/li>\n<li>construire u\u2083 orthogonal \u00e0 u\u2081 et u\u2082 ;<\/li>\n<li>normaliser les trois vecteurs ;<\/li>\n<li>obtenir une base orthonormale ;<\/li>\n<li>exprimer un vecteur x=(2,1,3) dans cette nouvelle base ;<\/li>\n<li>v\u00e9rifier la norme par les coordonn\u00e9es obtenues.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclHint('euclMission1')\">\ud83d\udca1 INDICE<\/button>\n<button type=\"button\" class=\"btn\" style=\"background:#06172c;color:#fff\" onclick=\"mzaEuclHint('euclMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"euclMission1\" class=\"hint\">\nPour construire u\u2082, retranchez \u00e0 v\u2082 sa projection sur u\u2081.\n<\/div>\n\n<div id=\"euclMission2\" class=\"hint\">\nPour u\u2083, retirez successivement les projections de v\u2083 sur u\u2081 puis u\u2082. Normalisez uniquement \u00e0 la fin.\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA ORTHOGONAL LAB<\/span>\n<h2>Orthogonaliser pour simplifier.<\/h2>\n\n<p>\nUne base orthonormale transforme de nombreux calculs en simples produits scalaires.\n<\/p>\n<\/div>\n\n<div class=\"method\">\n<div><b>A<\/b><span>PRODUIT<\/span><\/div>\n<div><b>B<\/b><span>ORTHOGONAL ?<\/span><\/div>\n<div><b>C<\/b><span>PROJETER<\/span><\/div>\n<div><b>D<\/b><span>SOUSTRAIRE<\/span><\/div>\n<div><b>E<\/b><span>NORMALISER<\/span><\/div>\n<div><b>F<\/b><span>CONTR\u00d4LER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section id=\"qcm\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">12. QCM DE VALIDATION<\/span>\n<h2>Diagnostic Espaces euclidiens.<\/h2>\n\n<p>Le score est affich\u00e9 sans r\u00e9v\u00e9ler les r\u00e9ponses correctes.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n\n<form id=\"euclQuiz\">\n\n<div class=\"question\">\n<strong>1. Deux vecteurs x et y sont orthogonaux si :<\/strong>\n<label><input type=\"radio\" name=\"e1\" value=\"1\"> \u27e8x,y\u27e9=0<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> ||x||=||y||<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> x=y<\/label>\n<label><input type=\"radio\" name=\"e1\" value=\"0\"> x+y=0 n\u00e9cessairement<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. La norme associ\u00e9e au produit scalaire est :<\/strong>\n<label><input type=\"radio\" name=\"e2\" value=\"1\"> ||x||=\u221a\u27e8x,x\u27e9<\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> ||x||=\u27e8x,x\u27e9\u00b2<\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> ||x||=\u27e8x,0\u27e9<\/label>\n<label><input type=\"radio\" name=\"e2\" value=\"0\"> ||x||=0 pour tout x<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. Une famille orthogonale de vecteurs non nuls est :<\/strong>\n<label><input type=\"radio\" name=\"e3\" value=\"1\"> libre<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> toujours li\u00e9e<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> forc\u00e9ment nulle<\/label>\n<label><input type=\"radio\" name=\"e3\" value=\"0\"> jamais g\u00e9n\u00e9ratrice<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. Une base orthonormale est :<\/strong>\n<label><input type=\"radio\" name=\"e4\" value=\"1\"> orthogonale et compos\u00e9e de vecteurs unitaires<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> seulement g\u00e9n\u00e9ratrice<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> forc\u00e9ment canonique<\/label>\n<label><input type=\"radio\" name=\"e4\" value=\"0\"> constitu\u00e9e de vecteurs de norme 0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. Pour projeter x sur une direction u non nulle :<\/strong>\n<label><input type=\"radio\" name=\"e5\" value=\"1\"> proj_u(x)=\u27e8x,u\u27e9\/\u27e8u,u\u27e9 \u00b7 u<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> proj_u(x)=x+u<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> proj_u(x)=0 toujours<\/label>\n<label><input type=\"radio\" name=\"e5\" value=\"0\"> proj_u(x)=||x||u<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. En dimension finie :<\/strong>\n<label><input type=\"radio\" name=\"e6\" value=\"1\"> E=F\u2295F\u22a5<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> F=F\u22a5 toujours<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> F\u22a5 est toujours nul<\/label>\n<label><input type=\"radio\" name=\"e6\" value=\"0\"> dim(F\u22a5)=dim(F) toujours<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaEuclScore()\">\nVALIDER MON QCM\n<\/button>\n\n<\/form>\n\n<div id=\"euclResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n\n<div style=\"font-size:43px\">x = p<sub>F<\/sub>(x) + z<\/div>\n\n<h2>Orthogonaliser, c&#8217;est rendre la g\u00e9om\u00e9trie calculable.<\/h2>\n\n<p>\nAvec un produit scalaire et une base orthonormale,\nles angles, les distances, les projections et les coordonn\u00e9es\nse ram\u00e8nent \u00e0 des calculs structur\u00e9s et rapides.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 REFAIRE LE QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaEuclHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaEuclScore(){\n const form=document.getElementById(\"euclQuiz\");\n const result=document.getElementById(\"euclResult\");\n\n let score=0;\n let complete=true;\n\n [\"e1\",\"e2\",\"e3\",\"e4\",\"e5\",\"e6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n\n   if(!answer){\n     complete=false;\n   }else{\n     score+=Number(answer.value);\n   }\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n\n let level=\"Espaces euclidiens \u00e0 consolider\";\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes et la correction d\u00e9taill\u00e9e restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html COURS 15 \u2022 ESPACES EUCLIDIENS Mesurer.Projeter. Orthogonaliser. Les espaces euclidiens ajoutent \u00e0 l&#8217;alg\u00e8bre lin\u00e9aire une notion de longueur, d&#8217;angle et d&#8217;orthogonalit\u00e9. Ils permettent de construire des bases orthonormales, d\u00e9terminer des projections, calculer des distances et simplifier profond\u00e9ment de nombreux probl\u00e8mes de g\u00e9om\u00e9trie et d&#8217;alg\u00e8bre. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM 1. PRODUIT [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1556","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1556"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1556\/revisions"}],"predecessor-version":[{"id":1558,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1556\/revisions\/1558"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}