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-1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 295 22 "Laboratoire 14 - Plans " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):wi th(student):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 305 51 "Principales com mandes utilis\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "vector, evalm, norm, dotprod, crossprod, solve, subs, au gment, det, genmatrix, gausselim, backsub, isolate et equate de la bib lioth\350que student, GramSchmidt, normalize." }}{PARA 0 "" 0 "" {TEXT -1 104 "Commandes graphiques : implicitplot3d et plot3d avec wit h(plots), sphere avec with(plottools), display, " }}{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utiliser l'aide pour plu s d'information" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 307 43 "Plans et co mbinaisons lin\351aires de vecteurs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 "Que repr\351sente l'ensemble M des combinaisons lin\351aires d es vecteurs u et v pour les vecteurs u, v suivants? Repr\351sentons d 'abord 5 combinaisons lin\351aires des vecteurs u et v, ru + sv, avec \+ -1<= r <=1 et -1<= s <=1 choisis al\351atoirement" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 624 "u:=vector([1,2,-1]):v:=vector([-1,4,2]):\nr andomize():alea:=rand(-50..50)/50:\nCombLin:=[seq(evalm(alea()*u+alea( )*v),i=1..5)]:\nwith(plots):with(plottools,arrow):\ncl1:=arrow([0,0,0] ,CombLin[1],0.15,0.4,0.2,color=red):\ncl2:=arrow([0,0,0],CombLin[2],0. 15,0.4,0.2,color=red):\ncl3:=arrow([0,0,0],CombLin[3],0.15,0.4,0.2,col or=red):\ncl4:=arrow([0,0,0],CombLin[4],0.15,0.4,0.2,color=red):\ncl5: =arrow([0,0,0],CombLin[5],0.15,0.4,0.2,color=red):\nvecteuru1:=arrow([ 0,0,0],u,0.15,0.4,0.2,color=blue):\nvecteuru2:=arrow([0,0,0],v,0.15,0. 4,0.2,color=blue):\ndisplay(cl1,cl2,cl3,cl4,cl5,vecteuru1,vecteuru2,ax es=normal,orientation=[-25,55]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 26 "Faire tourner le graphique" }{TEXT -1 47 " pour voir que les vecte urs sont dans un plan. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Place z le curseur sur la ligne CombLin et faites des retours pour obtenir 5 autres combinaisons lin\351aires (en rouge) de u et v (en bleu)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "En convenant que chaque vecteur O P repr\351sente le point P (son extr\351mit\351) alors les 5 combinais ons lin\351aires (en rouge) repr\351sente 5 points dans un m\352me pla n. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Repr\351sentons 1000 point s avec -1<= r <=1 et -1<= s <=1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pts:=\{seq(evalm(alea()*u+alea()*v),i=1..1000)\}:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "with(plottools,arrow):\nve cteuru1:=arrow([0,0,0],u,0.1,0.3,0.2,color=red):\nvecteuru2:=arrow([0, 0,0],v,0.1,0.3,0.2,color=blue):\ngrPts:=pointplot3d(pts,axes=normal,co lor=black):\ndisplay(vecteuru1,vecteuru2,grPts,orientation=[0,45]);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 309 26 "Faire tourner le graphique" } {TEXT -1 49 " pour voir que les 1000 points sont dans un plan." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "Chaque combinaison lin\351aire r* u + s*v o\371 -1<= r <=1 et -1<= s <=1 repr\351sente un point du paral l\351logramme ci-haut contenant l'origine. \{r*u + s*v| r, s sont des \+ r\351els\}repr\351sente le plan contenant le parall\351logramme et don c contenant l'origine." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Si le \+ plan ne passe pas par l'origine et A est un point du plan alors OA + r *u + s*v repr\351sente un point P du plan. " }}{PARA 0 "" 0 "" {TEXT -1 24 "On a OP = OA + r*u + s*v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 37 "\311quations vector ielles, param\351triques" }{TEXT 257 26 " et cart\351siennes d'un plan " }}{SECT 1 {PARA 4 "" 0 "" {TEXT 292 7 "Rappels" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Les 3 situations suivantes d\351finissent un plan" } }{PARA 0 "" 0 "" {TEXT -1 46 "a) 3 points A,B,C non align\351s(non col in\351aires)" }}{PARA 0 "" 0 "" {TEXT -1 49 "b) un point et 2 vecteurs v1 et v2 non parall\350les" }}{PARA 0 "" 0 "" {TEXT -1 109 "c) Un poi nt et un vecteur normal N au plan i.e. un vecteur perpendiculire \340 \+ chaque vecteur parall\350le au plan " }}{PARA 0 "" 0 "" {TEXT -1 91 "L e cas a) peut \352tre ramen\351 au cas b) en cr\351ant par exemple, le s vecteurs v1 = AB et v2 = AC" }}{PARA 0 "" 0 "" {TEXT -1 89 "De m\352 me le cas b) peut \352tre ramen\351 au cas c) en prenant N = v1xv2 (x \+ : produit vectoriel)" }}{PARA 0 "" 0 "" {TEXT 258 52 " \311quations ve ctorielles, param\351triques d'un plan P\n" }{TEXT -1 99 "Soit un poi nt A = (a1,a2,a3) du plan P et 2 vecteurs // au plan P, u = (u1,u2,u3) et v = (v1,v2,v3)" }}{PARA 0 "" 0 "" {TEXT -1 65 "Soit P(x,y,z) un po int quelconque du plan P et O = (0,0,0) alors " }}{PARA 0 "" 0 "" {TEXT 259 59 "L'\351quation vectorielle du plan P est : OP = OA + r*u \+ + s*v " }{TEXT -1 33 "o\371 r et s sont des r\351els. On a : " }} {PARA 0 "" 0 "" {TEXT 261 51 "(x,y,z) = (a1,a2,a3) + r*(u1,u2,u3) + s* (v1,v2,v3) " }{TEXT -1 21 "(1) On tire de (1) : " }}}{EXCHG {PARA 0 " " 0 "" {TEXT 260 101 "Les \351quations param\351triques du plan P sont : x = a1+r*u1+s*v1, y = a2+r*u2+s*v2 et z = a3+r*u3+s*v3" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "Remarque : Les \351qu ations x = f(r,s), y = g(r,s) et z = h(r,s) d\351finissent une surface dans l'espace. Si les fonctions f(r,s), g(r,s) et h(r,s) sont lin\351 aires alors la surface est un plan" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Soit un plan P passant par un point A(a1,a2,a3) et ayant comme \+ vecteur normal N = (a,b,c) : " }}{PARA 0 "" 0 "" {TEXT 262 54 "\311qua tion cart\351sienne d'un plan : ax + by + cz + d = 0 " }{TEXT -1 12 "o \371 d = -OAoN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "Soit un plan P passant par un point A(x1,y1,z1) et ayant comme vecteurs directeurs v 1= (a1,b1,c1) et v2 = (a2,b2,c2) alors l'\351quation cart\351sienne pe ut \352tre obtenue en calculant le d\351terminant : " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "det(matrix([[x-x1, a1, a2], [y-y1, b1, b2], [ z-z1, c1, c2]])) = 0;" "6#/-%$detG6#-%'matrixG6#7%7%,&%\"xG\"\"\"%#x1G !\"\"%#a1G%#a2G7%,&%\"yGF.%#y1GF0%#b1G%#b2G7%,&%\"zGF.%#z1GF0%#c1G%#c2 G\"\"!" }{TEXT -1 0 "" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 11 "Expli cation" }{TEXT 294 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Soit M( x,y,z) un point de R^3 alors AM = (x-x1,y-y1,z-z1) et AMo(v1xv2) = (+/ - volume du parall\351lipip\350de de c\364t\351s AM, v1 et v2) = " } {XPPEDIT 18 0 "det(matrix([[x-x1, a1, a2], [y-y1, b1, b2], [z-z1, c1, \+ c2]]))" "6#-%$detG6#-%'matrixG6#7%7%,&%\"xG\"\"\"%#x1G!\"\"%#a1G%#a2G7 %,&%\"yGF-%#y1GF/%#b1G%#b2G7%,&%\"zGF-%#z1GF/%#c1G%#c2G" }{TEXT -1 131 ". M appartient au plan si et seulement si volume du parall\351lip ip\350de = 0 donc M appartient au plan si et seulement si AMo(v1xv2) = 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{EXCHG {PARA 0 "" 0 " " {TEXT 277 12 "Exemple 1 : " }{TEXT -1 134 "Trouver les \351quations \+ vectorielles, param\351triques et cart\351siennes du plan passant par \+ les 3 points A(2,0,-5), B(5,-10,7) et C(-3,7,11)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=[2,0,-5];B:=[5,-10,7];C:=[-3,7,11];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "OA:=evalm(A):OB:=evalm(B):OC :=evalm(C):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "2 vecteurs parall \350les au plan" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u:=evalm(OB-OA); v:=evalm(OC-OA);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "L'\351quation vectorielle du plan P est:OP = OA + r*u + s*v" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " [x,y,z]=A+r*evalm(u)+s*evalm(v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Les \351quatons param\351triques du plan" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "equate(lhs(%),rhs(%));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Premi\350re solution : Pour l'\351quation cart\351sienne du plan, prenons comme vecteur normal N = uxv" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "N:=crossprod(u,v);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "L'\351quation du plan v\351rifie" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqplan:=N[1]*x+N[2]*y+N[3]*z+d=0; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " Trouvons d = -OAoN" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d:=-dotprod(A,N);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "eqplan;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 31 "Autre m\351thode pour d\351terminer d" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "d:='d':#efface la valeur de d" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Le point A(2,0,-5) v\351rifie l' \351quation du plan" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs (\{x=2,y=0,z=-5\},\{eqplan\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,d);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Deuxi\350 me solution : avec le point A = (2,0,-5), et les vecteurs directeurs u = (3,-10,12) et v = (-5,7,16) avec : " }{XPPEDIT 18 0 "det(matrix([[x -x1, a1, a2], [y-y1, b1, b2], [z-z1, c1, c2]])) = 0" "6#/-%$detG6#-%'m atrixG6#7%7%,&%\"xG\"\"\"%#x1G!\"\"%#a1G%#a2G7%,&%\"yGF.%#y1GF0%#b1G%# b2G7%,&%\"zGF.%#z1GF0%#c1G%#c2G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "M:=augment(vector([x-2,y,z+5]),u,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "det(M)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 263 55 "Plans p arall\350les, perpendiculaires, angle entre 2 plans" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT 279 7 "Rappels" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "a) \+ Deux plans sont parall\350les si leurs vecteurs normaux sont parall \350les, N1 // N2, i.e. N1 = k*N2" }}{PARA 0 "" 0 "" {TEXT -1 97 "b) D eux plans sont perpendiculaires si leurs vecteurs normaux sont perpend iculaires i.e.N1oN2 = 0" }}{PARA 0 "" 0 "" {TEXT -1 3 "c) " }{TEXT 286 24 "L'angle t entre 2 plans " }{TEXT -1 83 "P1 et P2 est l'angle e ntre les 2 droites portant les vecteurs normaux N1 et N2 i.e." }} {PARA 0 "" 0 "" {TEXT -1 9 "cos(t) = " }{TEXT 280 14 "valeur absolue" }{TEXT -1 36 "(N1oN2) / ||N1||*||N2|| et 0<= t <= " }{TEXT 281 6 "Pi / 2" }}{PARA 0 "" 0 "" {TEXT -1 7 "Note : " }{TEXT 282 27 "L'angle a en tre 2 vecteurs " }{TEXT -1 10 "N1 et N2 :" }}{PARA 0 "" 0 "" {TEXT -1 49 "a) v\351rifie cos(a) = (N1oN2) / (||N1|| ||N2||) o\371" }{TEXT 283 13 " 0<= a <= Pi " }{TEXT -1 9 "radians. " }}{PARA 0 "" 0 "" {TEXT -1 61 "b) est donn\351 en radians, par la commande Maple : angle (N1,N2)" }}{PARA 0 "" 0 "" {TEXT -1 60 "Remarque 1: L'angle t entre P 1 et P2 \351gale angle(v1,v2) si " }{TEXT 284 15 "0<= t <= Pi/2 " } {TEXT -1 55 "et l'angle t entre P1 et P2 \351gale Pi -angle(P1,P2) si " }{TEXT 285 14 "Pi/2< t <= Pi" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 12 "Exemple 2 : " }{TEXT -1 151 "Soit P1 le plan d'\351quation: 3x + 8y +13z - 15 = 0 et P2 le pla n passant par le point A(3,5,7) et parall\350le aux 2 vecteurs v1= (-2 ,5,9) et v2 = (4,6,-1)" }}{PARA 0 "" 0 "" {TEXT -1 63 "P1 et P2 sont-i ls parall\350les? N1 = (3,8,13), Prenons N2 = v1xv2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "N1:=vector([3,8,13]);N2:=crossprod(vector ([-2,5,9]),vector([4,6,-1]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 " N1 n'est pas // \340 N2 donc P1 n'est pas // \340 P2. P1 est-il perpen diculaire \340 P2?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dotpr od(N1,N2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Les plans P1 et P2 \+ ne sont pas perpendiculaires. " }}{PARA 0 "" 0 "" {TEXT -1 83 "Trouvon s l'angle t entre P1 et P2 o\371 cos(t) = valeur absolue(N1oN2) / ||N1 ||*||N2||" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "t:=arccos(abs( dotprod(N1,N2))/(norm(N1,2)*norm(N2,2)));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "evalf(%);#en radians" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 273 1 "I" }{TEXT 276 20 "ntersection de plans" }}{EXCHG {PARA 0 "" 0 "" {TEXT 271 46 "C as o\371 l'intersection des plans est un point :" }{TEXT 287 1 " " } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 12 "Exemple 3 : " }{TEXT -1 61 "Trouver l'intersection des plans donn\351s par leurs \351quations :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eq1:=-x+y+z=2;eq2:=-10*x+ y-11*z=-8;eq3:=4*x+7*y-3*z=3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 " La solution " }{TEXT 306 16 "comme \340 la main " }{TEXT -1 50 ": On r \351soud avec la m\351thode d'\351limination de Gauss" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "genmatrix([eq1,eq2,eq3],[x,y,z],fla g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gausselim(%);backsub (%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "La solution avec Maple : \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,eq2,eq3\}, \{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "La solution repr \351sente le point d'intersection des 3 plans. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 80 "implicitplot3d(\{eq1,eq2,eq3\},x=-5..5,y=-5..5 ,z=-5..5,axes=boxed,numpoints=1000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "On peut tracer, avec plot3d, les plans comme des fonctions z = \+ f(x,y): \nIsolons z dans eq1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "solve(eq1,z);# on aurait pu utiliser isolate " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(plottools):#permet d'utiliser sphere " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "display(\{plot3d(solve (eq1,z),x=-5..5,y=-5..5,color=red),\nplot3d(solve(eq2,z),x=-5..5,y=-5. .5,color=blue),\nplot3d(solve(eq3,z),x=-5..5,y=-5..5,color=green),\nsp here([ -16/111, 203/222, 209/222],1/2)\},\naxes =boxed,view=[-5..5,-5. .5,-5..5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 66 "Cas o\371 l'intersection des plans conti ent une infinit\351 de points : " }}{PARA 0 "" 0 "" {TEXT 289 11 "Exem ple 4 :" }{TEXT -1 107 " Trouvons les points d'intersections des plan s d'\351quations : eq1 : 2*x+5*y+3*z = 2;eq2: 3*x- 8*y+2*z = 10;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(plots):with(lin alg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq1:=2*x+5*y+3*z = 2;eq2:=3*x-8*y+2*z= 10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "intersection :=solve(\{eq1,eq2\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Il y a une infinit\351 de solutions car on a une" }{TEXT 270 15 " variable libre" }{TEXT -1 9 ". Posant " }{TEXT 290 8 "y = t," }{TEXT -1 79 " on obtient les \+ \351quations param\351triques de la droite d'intersection des 2 plans " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(\{y=t\} union int ersection,\{x,y,z\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "D roiteIntersection:=spacecurve([26/5+34/5*t,t,-14/5-31/5*t,t=-2..1],col or=red,thickness = 3):\nPlans:=implicitplot3d(\{eq1,eq2\},x=-10..10,y= -10..10,z=-10..10,axes=boxed):\ndisplay(DroiteIntersection,Plans,orien tation=[85,45]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Chaque point \+ de la droite d'intersection (en rouge) est une solution du syst\350me \+ d'\351quations " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 65 "Cas o\371 l'i ntersection des plans est vide : Maple ne retourne rien" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eq3:=2 *x+5*y+3*z = 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{ eq1,eq2,eq3\},\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Map le ne retourne rien car il n'y a pas de points communs aux 3 plans d' \351quations eq1,eq2,eq3. Le plan 1 est parall\350le au plan 3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "Remarque : 3 plans peuvent n'avoi r aucun point commun aux 3 plans, m\352me s'il n'y a pas de plans para ll\350les. Par exemple, les plans y - z = 0, y + z = 0, z = 9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "display(implicitplot3d(\{y-z =0,y+z=0,z=9\},x=-10..10,y=-10..10,z=-10..10,axes=boxed));\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT 274 29 "Distance d'un point \340 un plan" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 " Trouver la distance d d'un point \+ A \340 un plan P : " }}{PARA 0 "" 0 "" {TEXT -1 149 "Soit B un point d u plan P alors la distance d d'un point A \340 un plan P est la norme \+ de la projection du vecteur AB sur le vecteur normal N du plan : " } {TEXT 275 20 "d = ||proj(AB / N)||" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 11 "Exemple 5 :" }{TEXT -1 68 " Trouver la distance du point A(5,- 7,10) au plan 7x - 8y +12z -9 = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A:=[5,-7,10];N:=vector([7,-8,12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Trouvons un point B du plan: Posons x = 0, y = \+ 0 alors z = 9/12" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B:=[0,0 ,9/12];AB:=evalm(B-A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "p roj('AB/N'):=(dotprod(AB,N)/dotprod(N,N))*N;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "d=norm(proj('AB/N'),2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Autre solution : la distance d d'un point A(x0,y0,z0) au plan ax + by + cz + d = 0 est :" }}{PARA 0 "" 0 "" {TEXT 293 63 "distance = valeur absolue(ax0+by0+cz0+d) / ||N|| o\371 N = (a,b,c)" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Trouver la distance du point A(5,- 7,10) au plan 7x - 8y +12z -9 = 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "distance=abs(7*5-8*(-7)+12*10-9)/norm(N,2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 264 40 "Plans para ll\350les aux axes de coordonn\351es" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 329 "Dans R^3, 3x-5y = 20 repr\351sente un plan dont le vecteur N = (3,-5,0) est un vecteur normal. Or tout vecteur perpendiculaire \340 \+ N est parall\350le au plan. Le vecteur k = (0,0,1) est perpendiculaire au vecteur N car koN = 0, donc le vecteur k est parall\350le au plan, et par cons\351quent, l'axe des z (portant k) est parall\350le au pla n.. " }}{PARA 0 "" 0 "" {TEXT -1 151 "Le plan 3x-5y =20 est un plan p arall\350le \340 l'axe des z (z n'apparait pas dans son \351quation) e t son intersection avec le plan XY est la droite 3x-5y = 20" }}{PARA 0 "" 0 "" {TEXT 291 159 "Principe : Si une variable est abscente dans \+ l'\351quation cart\351sienne cart\351sienne d'un plan, alors le plan e st parall\350le \340 l'axe ayant cette variable comme nom. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "implicitplot3d(3*x-5*y=20,x=-5..5,y =-5..5,z=-5..5,axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "L e plan 3x-5y = 20 est parall\350le \340 l'axe des z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "implicitplot3d(x=4,x=-5..5,y=-5..5,z=-5.. 5,axes=normal,orientation=[-74,71]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Le plan x = 2 est parall\350le, \340 la fois \340 l'axe des y \+ et \340 l'axe des z et donc au plan de coordonn\351e YZ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Remarque : L'\351quation x = 2 repr\351s ente " }}{PARA 0 "" 0 "" {TEXT -1 77 "a) dans R, un point sur la droit e des r\351els, situ\351 \340 2 unit\351s de l'origine 0 " }}{PARA 0 " " 0 "" {TEXT -1 93 "b) dans R^2, une droite dans le plan XY, parall \350le \340 l'axe des y et \340 2 unit\351s de l'axe des y" }}{PARA 0 "" 0 "" {TEXT -1 116 "c) dans R^3, un plan dans R^3, parall\350le aux \+ 2 axes, y et z (donc au plan YZ) et \340 2 unit\351s du plan de coordo nn\351e YZ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 265 38 "Bases d'un plan, passant par l'origine" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Un plan P passant par l'origine \+ a comme \351quation cart\351sienne est ax + by + cz = 0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 14 "Exemple 6 : a)" }{TEXT -1 42 " Trouver u ne base du plan 7x - 3y + 4z = 0" }}{PARA 0 "" 0 "" {TEXT 300 2 "b)" } {TEXT -1 54 " Trouver une base orthogonale du plan 7x - 3y + 4z = 0" } }{PARA 0 "" 0 "" {TEXT 301 2 "c)" }{TEXT -1 55 " Trouver une base orth onormale du plan 7x - 3y + 4z = 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "L'origine (0,0,0) est un point du plan. Si un point P(x,y,z) appa rtient au plan alors le vecteur OP = (x,y,z) est parall\350le au plan. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(linalg):w ith(student):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 2 "a)" }{TEXT -1 34 " Trouvons deux vecteurs // au plan" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Un point A appartenant au plan fournit le vecteur OA // a u plan. Prenons x = 1, y = 1, alors z = -1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "v1:=vector([1,1,-1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Le point (0,4,3) appartient au plan" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v2:=vector([0,4,3]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "v1 et v2 ne sont pas // donc \{v1,v2\}est une base du \+ plan" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 41 "Bases orthogonales et ba ses orthonormales" }{TEXT -1 1 " " }{TEXT 299 31 "d'un plan passant pa r l'origine" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 303 2 "b)" }{TEXT -1 51 " Trouvons une base orthogonale du plan 7x-3y+4z = 0" }}{PARA 0 "" 0 " " {TEXT -1 6 "Note: " }{TEXT 266 74 "une base orthogonale est une base dont les vecteurs sont orthogonaux 2 \340 2" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Le proc\351d\351 de GramSchmidt re nd une base orthogonale" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " GramSchmidt([v1,v2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Comme \+ \340 la main : " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 109 "Si on soustr ait d'un vecteur v, sa projection sur un vecteur u, on obtient un vect eur orthogonal au vecteur u" }{TEXT -1 118 ". Ceci est la base de la \+ m\351thode de Gram-Schmidt pour orthogonaliser une base. Voir GramSchm idt dans l'aide de Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Le v ecteur u - proj(u/v) est un vecteur perpendiculaire \340 v et dans le m\352me plan que u et v. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "P enons u = v1 et v = v2-proj(v2/v1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "u:=evalm(v1);proj(v2/v1):=(dotprod(v2,v1)/dotprod(v1, v1))*v1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v:=evalm(v2-pro j(v2/v1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Une base orthogonal e du plan est < u,v>. V\351rifions que uov = 0" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "dotprod(u,v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 2 "c)" }{TEXT -1 51 " Trouver une base orthonormale du plan \+ 7x-3y+4z = 0" }}{PARA 0 "" 0 "" {TEXT -1 6 "Note: " }{TEXT 268 78 "une base orthogonale est une base orthogonale dont les vecteurs sont unit aires" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Rendons u et v unitaires" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "u2:=evalm(u/norm(u,2));v2:=evalm(v/norm(v,2));#ou nor malize(u)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 310 9 "Exercices" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 329 5 "No 1)" }{TEXT -1 155 " Trouver l'\351quation vectorielle, les \351quations param\351triques et l'\351quation alg\351brique du p lan P passant par les 3 points A(2,-3,5), B(-2,5,12) et C(-3,0,5)" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT 332 1 " " }{TEXT 331 31 "Espace de trava il de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 311 5 "No 2)" }{TEXT -1 80 " Plan PI passe par le point A(3,-19, 7) et est // v1 = (9,8,-5) et v2 = ( 7,0,12)" }}{PARA 0 "" 0 "" {TEXT -1 98 "a) Trouver l'\351quation vectorielle, les \351quations param \351triques et l'\351quation alg\351brique du plan P1" }}{PARA 0 "" 0 "" {TEXT -1 48 "b) Trouver un point du plan autre que la point A" }} {PARA 0 "" 0 "" {TEXT -1 43 "c) Le point B(5,2,8) appartient-il au pla n?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 334 1 " " }{TEXT 333 31 "Espace d e travail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 5 "No 3)" }{TEXT -1 28 " Soit P1,P2 et P3, 3 plans :" }} {PARA 0 "" 0 "" {TEXT -1 25 "P1 : 12x - 20y + 17z = 9" }}{PARA 0 "" 0 "" {TEXT -1 61 "P2 : x = 4 + 180k - 41w, y = 5 +92k -95w, z = 10 +10 k + 44 w " }}{PARA 0 "" 0 "" {TEXT -1 51 "P3 : (x,y,z) = (8,8,-2) + r( 10,-2,5) + s(-7,15,-12)" }}{PARA 0 "" 0 "" {TEXT -1 57 "Trouver a) 2 p lans parall\350les b) 2 plans perpendiculaires" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 336 1 " " }{TEXT 335 31 "Espace de travail de l'\351tudiant " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 5 "No 4)" } {TEXT -1 143 " Trouver a pour que le point (a,2,10) appartienne au pla n P passant par les points B(3,6,-8) et C(2,0,11) et parall\350le au v ecteur v = ( 1,7,-3)" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 338 1 " " } {TEXT 337 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 314 5 "No 5)" }{TEXT -1 36 " Soit P1 et P 2 les plans suivants : " }}{PARA 0 "" 0 "" {TEXT -1 22 "P1 : 9x -14y + 7z = 31" }}{PARA 0 "" 0 "" {TEXT -1 17 "P2 : -7y + 5z = 5" }}{PARA 0 "" 0 "" {TEXT -1 33 "a) Trouver l'angle entre P1 et P2" }}{PARA 0 "" 0 "" {TEXT -1 93 "b) Trouver un plan P perpendiculaire aux 2 plans P1 \+ et P2 et passant par le point A(8,-10,11)" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 340 1 " " }{TEXT 339 31 "Espace de travail de l'\351tudiant" } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 315 5 "No 6)" } {TEXT -1 79 " Trouver l'\351quation alg\351brique du plan P contenant \+ les 2 deux droites D1 et D2" }}{PARA 0 "" 0 "" {TEXT -1 42 "D1 : x = \+ 3 + 5t, y = -4 - 9t, z = 12 + 11t" }}{PARA 0 "" 0 "" {TEXT -1 39 "D2 : (x-5)/3 = y + (186/43) = (z-12)/-2" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 342 1 " " }{TEXT 341 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 5 "No 7)" }{TEXT -1 38 " S oit P le plan : 7x - 12y + 35z = 140" }}{PARA 0 "" 0 "" {TEXT -1 52 "a ) Trouver la distance du point A( 6,7,-5) au plan P" }}{PARA 0 "" 0 " " {TEXT -1 52 "b) Trouver la distance du point B(10,0,2 ) au plan P" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT 344 1 " " }{TEXT 343 31 "Espace de tra vail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 6 "No 8) " }{TEXT -1 55 "Soit P1 et P2, 2 plans parall\350le s et un point A(3,-2,1)" }}{PARA 0 "" 0 "" {TEXT -1 23 "P1 : 4x + 7y + 12 z = 18" }}{PARA 0 "" 0 "" {TEXT -1 25 "P2 : 12x + 21y + 36z = 9 " } }{PARA 0 "" 0 "" {TEXT 325 2 "a)" }{TEXT -1 42 " Trouver la distance d u point A au plan P1" }}{PARA 0 "" 0 "" {TEXT 326 3 "b) " }{TEXT -1 41 "Trouver la distance du point A au plan P2" }}{PARA 0 "" 0 "" {TEXT 327 2 "c)" }{TEXT -1 38 " Trouver la distance entre les 2 plans " }}{PARA 0 "" 0 "" {TEXT 328 2 "d)" }{TEXT -1 49 " Le point A est-il \+ entre les 2 plans parall\350les ?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 346 1 " " }{TEXT 345 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 5 "No 9)" }{TEXT -1 95 " T rouver l'\351quation alg\351brique du plan P passant par le point A(5, 5,5) et contenant la droite D" }}{PARA 0 "" 0 "" {TEXT -1 34 "D : x = \+ 7 - 8k, y = 9 + 11k, z = 3" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 348 1 " \+ " }{TEXT 347 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 319 6 "No 10)" }{TEXT -1 56 " Trouver une base orthonormale du plan 5x + 7y - 12z = 4" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 350 1 " " }{TEXT 349 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 320 6 "No 11)" } {TEXT -1 105 " Soit P le plan passant par le point A(7,-7,12) et paral l\350le aux vecteurs v1 = ( 4,5,6) et v2 = ( 5,-3,7)" }}{PARA 0 "" 0 " " {TEXT -1 71 "Trouver un vecteur v parall\350le au plan et perpendicu laire au vecteur v1" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 352 1 " " } {TEXT 351 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 321 6 "No 12)" }{TEXT -1 29 " Soit P1 et \+ P2 les 2 plans : " }}{PARA 0 "" 0 "" {TEXT -1 17 "P1 : 3y - 4z = -6" } }{PARA 0 "" 0 "" {TEXT -1 17 "P2 : 5y + 2z = 16" }}{PARA 0 "" 0 "" {TEXT -1 62 "a) D\351crire g\351om\351triquement l'intersection des 2 \+ plans P1 et P2" }}{PARA 0 "" 0 "" {TEXT -1 61 "b) Tracer les 2 plans P 1 et P2 dans la m\352me fen\352tre graphique" }}{PARA 0 "" 0 "" {TEXT -1 88 "c) Trouver l'\351quation de la droite d'intersection des 2 plan s et ajoutez la au graphique" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 354 1 " " }{TEXT 353 31 "Espace de travail de l'\351tudiant" }{TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 322 6 "No 13)" }{TEXT -1 84 " Soit 3 plan s : P1 : x + 2y + az = 4, P2 : 2x - y + 3z = b, P3 : 3x - 4y + 2z = -3 ; " }}{PARA 0 "" 0 "" {TEXT -1 79 "Trouver les valeurs des param\350tr es a et b pour que l'intersection des 3 plans :" }}{PARA 0 "" 0 "" {TEXT -1 16 "a) soit un point" }}{PARA 0 "" 0 "" {TEXT -1 18 "b) soit \+ une droite" }}{PARA 0 "" 0 "" {TEXT -1 12 "c) soit vide" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 356 1 " " }{TEXT 355 31 "Espace de travail de l' \351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 323 6 "No 14 )" }{TEXT -1 49 " Soit A(-3,7), B(2,5), et C(5,-7), trois points. " }} {PARA 0 "" 0 "" {TEXT -1 22 "a) Trouver la parabole" }{TEXT 324 21 " y = a*x^2 + b*x + c." }{TEXT -1 34 " passe par les 3 points A, B et C. " }}{PARA 0 "" 0 "" {TEXT -1 77 "b) Trouver les \351quations de 3 plan s dont l'intersection est le point (a,b,c) " }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 358 1 " " }{TEXT 357 31 "Espace de travail de l'\351tudiant" } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 330 6 "No 15)" } {TEXT -1 98 " D\351crire g\351om\351triquement l'ensemble M des points de R^3 : M = \{(3x-5y, 4x+7y, 10x-2y)| x, y r\351els\}" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 360 1 " " }{TEXT 359 31 "Espace de travail de l' \351tudiant" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }