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0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 291 59 "Laboratoire 11-Produit \+ vectoriel et produit mixte, dans R^3" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire, claude.sthilaire@videotron.ca" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 294 51 "Principales commandes utilis\351 es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "vector, \+ crossprod, dotprod, norm, evalm, matrix." }}{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utiliser l'aide pour plus d'infor mation" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 26 "Produit vectoriel da ns R^3" }{TEXT -1 0 "" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 40 "Produit vectoriel de 2 vecteurs u et v: " }{TEXT 257 15 "crossprod(u,v);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "u:=vector([u1,u2,u3]);v:=vec tor([v1,v2,v3]);crossprod(u,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 7 "Rappels" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Le produit vectoriel des vecteurs u et v \+ not\351 " }{TEXT 258 25 "uxv est un vecteur dont :" }}{PARA 0 "" 0 "" {TEXT -1 3 "a) " }{TEXT 259 42 "la direction est perpendiculaire \340 \+ u et v " }}{PARA 0 "" 0 "" {TEXT -1 3 "b) " }{TEXT 260 31 "le sens sui t la r\350gle de la vis" }{TEXT -1 76 " (si une vis tourne de u vers v alors uxv suit le sens de la vis) ou encore " }{TEXT 265 26 "la r\350 gle de la main droite" }{TEXT -1 110 " (on place le bord de la main dr oite sur u, on ferme les doigts vers v alors le pouce indique le sens \+ de uxv) " }}{PARA 0 "" 0 "" {TEXT -1 121 "ou on place le majeur perpen diculaire \340 la main et l'index selon u, le majeur selon v alors le \+ pouce donne le sens de uxv" }}{PARA 0 "" 0 "" {TEXT -1 3 "c) " }{TEXT 261 38 "la norme ||uxv|| = ||u||*||v||*sin(t) " }{TEXT -1 42 "o\371 t \+ est l'angle entre u et v, 0<= t <= Pi" }}{PARA 0 "" 0 "" {TEXT 266 71 "Le module ||uxv|| repr\351sente l'aire du parall\351logramme de c\364 t\351s u et v." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Le produit vect oriel de " }{TEXT 292 14 "u = (u1,u2,u3)" }{TEXT -1 3 " et" }{TEXT 293 15 " v = (v1,v2,v3)" }{TEXT -1 16 " est le vecteur " }{TEXT 263 39 "uxv = (u2v3-v2u3, v1u3-u1v3, u1v2-v1u2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Remarque : on peut calculer uxv en calculant le \"pseudo d\351terminant\" de la \+ matrice " }{XPPEDIT 18 0 "A := matrix([[i, j, k], [u1, u2, u3], [v1, v 2, v3]]);" "6#>%\"AG-%'matrixG6#7%7%%\"iG%\"jG%\"kG7%%#u1G%#u2G%#u3G7% %#v1G%#v2G%#v3G" }{TEXT -1 79 " o\371 i, j, k repr\351sentent les vect eurs i = (1,0,0), j = (0,1,0) et k = (0,0,1). " }}{PARA 0 "" 0 "" {TEXT -1 89 "On a: uxv = i(u2v3-v2u3) -j (u1v3-v1u3) + k(u1v2-v1u2) = (u2v3-v2u3,v1u3-u1v3,u1v2-v1u2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:=matrix(3,3,[i,j,k,u1,u2,u 3,v1,v2,v3]);uxv=det(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "collect(%,[i,j,k]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT 267 24 "Repr\351sentation graphique" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "u:=vector([2,1,5]);v:=vecto r([-3,1,2]);uxv:=crossprod(u,v);\n#u:=vector(-[2,1,5]);v:=vector([-3,1 ,2]);uxv:=crossprod(u,v);\n#u:=vector([2,1,5]);v:=vector(-[-3,1,2]);ux v:=crossprod(u,v);\n#u:=vector(-[2,1,5]);v:=vector(-[-3,1,2]);uxv:=cro ssprod(u,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "with(plots ):with(plottools):\nvecteuru:=arrow([0,0,0],u,.3,.5,.1,color=red):\nve cteurv:=arrow([0,0,0],v,.3,.5,.1,color=blue):\nvecteuruxv:=arrow([0,0, 0],uxv,.3,.5,.1,color=black):\nnomu:=textplot3d([u[1],u[2],u[3],'u'],c olor=black):\nnomv:=textplot3d([v[1],v[2],v[3],'v'],color=black):\nnom uxv:=textplot3d([uxv[1],uxv[2],uxv[3],'uxv'],color=black):\nplanuv:=po lygon([[0,0,0],[u[1],u[2],u[3]],[u[1]+v[1],u[2]+v[2],u[3]+v[3]],[v[1], v[2],v[3]],[0,0,0]],color=green):\ndisplay(vecteuru,vecteurv,vecteurux v,nomu,nomv,nomuxv,planuv,orientation =[36,83],axes=normal);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Faire tourner la figure et pratiqu er la r\350gle de la vis ou la r\350gle de la main droite" }}{PARA 0 " " 0 "" {TEXT -1 114 "Les vecteurs pr\351c\351d\351s de # sont inactifs . Changez la position des # pour visualiser d'autres produits vectorie ls. " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 11 "Exemple 1 :" }{TEXT -1 94 " Trouver l'aire du \+ triangle dont les sommets sont les points A(2,5,7), B(-3,6,12) et C(4, -4,0)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "L'aire du triangle ABC \+ est la moiti\351 de l'aire du parall\351logramme de c\364t\351s AB et \+ AC donc l'aire du triangle ABC = (1/2)||ABxAC)|| " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "OA:=vector([2,5,7]);OB:=vector([-3,6,12]);O C:=vector([4,-4,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "AB: =evalm(OB-OA);AC:=evalm(OC-OA);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "#aire du triangle ABC = (aire du parall\351logramme de c\364t \351s AB et AC)/2 =\n(1/2)*norm(crossprod(AB,AC),2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 11 "Exemple 2 :" }{TEXT -1 35 " Soit u = (3,5,9) e t v = (-7,-12,8)" }}{PARA 0 "" 0 "" {TEXT 275 3 "a) " }{TEXT -1 57 "Tr ouver l'angle entre u et v \340 l'aide du produit scalaire" }}{PARA 0 "" 0 "" {TEXT 276 2 "b)" }{TEXT -1 59 " Trouver l'angle entre u et v \+ \340 l'aide du produit vectoriel" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "a) u*v = ||u||*||v||*cos(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "u:='u':v:='v':u:=vector([3,5,9]);v:=vector([-7,-12,8]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "t=arccos(dotprod(u,v)/(norm( u,2)*norm(v,2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(% );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "b) ||uxv|| = ||u||*||v||*si n(t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "t=arcsin((norm(cros sprod(u,v),2)/(norm(u,2)*norm(v,2))));evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "On n'a pas la m\352me r\351ponse qu'en a). Cela est d\373 au fait que sin(t) = ||uxv|| / ( ||u||*||v|| ) a 2 solutions p our 0<= t<= Pi" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sin(t)=ev alf(norm(crossprod(u,v),2)/(norm(u,2)*norm(v,2)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Il y a 2 solutions pour 0 " 0 "" {MPLTEXT 1 0 57 "t1=arcsi n(.9986287340);\nt2=evalf(Pi-arcsin(.9986287340));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "La valeur est t2 radians. Il est plus simple de \+ calculer l'angle entre 2 vecteurs avec le produit scalaire. " }}} {EXCHG {PARA 257 "" 0 "" {TEXT 295 11 "Exemple 3 :" }{TEXT -1 34 " Pro pri\351t\351s du produit vectoriel :" }{TEXT 278 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 2 "1)" }{TEXT -1 15 " V\351rifions que " }{TEXT 274 63 "uxv est perpendiculaire \340 u et \340 v : uo(uxv) = 0 et vo(u xv) = 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "uxv:=crossprod(u ,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dotprod(u,uxv);dotp rod(v,uxv);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 17 "2) uxu = (0,0,0) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crossprod(u,u);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 281 2 "3)" }{TEXT -1 43 " Le produit vect oriel est anticommutatif : " }{TEXT 269 11 "uxv = - vxu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "uxv:=crossprod(u,v);vxu:=crossprod( v,u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 2 "4)" }{TEXT -1 44 " Le pr oduit vectoriel n'est pas associatif. " }{TEXT 270 29 "En g\351n\351ra l ux(vxw) <> (uxv)xw" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "w:= vector([2,-2,7]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "ux(vxw) :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "crossprod(u,crossprod(v,w));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "(uxv)xw :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "crossprod(crossprod(u,v),w);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 2 "5)" }{TEXT -1 67 " Le produit vectoriel est distributi f sur l'addition de vecteurs : " }{TEXT 271 21 "ux(v + w) = uxv + uxw " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "w:=vector([2,4,9]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ux(v+w)=crossprod(u,v+w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "uxv+uxw=evalm(crossprod(u ,v)+crossprod(u,w));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "C'est la \+ distributivit\351 \340 gauche. On a aussi la distributivit\351 \340 dr oite : " }{TEXT 286 21 "(v + w)xu = vxu + wxu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 2 "6)" }{TEXT -1 1 " " }{TEXT 272 34 "k(uxv) = (ku)xv et \+ k(uxv) = ux(kv)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "k*crossp rod(u,v);\ncrossprod(k*u,v);\ncrossprod(u,k*v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 2 "7)" }{TEXT -1 46 " Produit vectoriel triple. Formule de Gibbs : " }{TEXT 273 25 "ux(vxw) = (uow)v - (uov)w" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "crossprod(u,crossprod(v,w));\nevalm (dotprod(u,w)*v-dotprod(u,v)*w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 260 "" 0 "" {TEXT 305 8 "Exercice " }{TEXT -1 6 "s A : " }{TEXT 324 26 "Produit vectoriel dans R^3" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 308 5 "No 1)" }{TEXT -1 70 " Trouver un v ecteur perpendiculaire \340 u = ( 4,-8,12) et v = (-10, 2 9)" }}{PARA 0 "" 0 "" {TEXT 320 2 "a)" }{TEXT -1 34 " en utilisant le produit vect oriel" }}{PARA 0 "" 0 "" {TEXT 321 2 "b)" }{TEXT -1 33 " en utilisant \+ le produit scalaire" }}{PARA 0 "" 0 "" {TEXT 323 2 "c)" }{TEXT -1 61 " Montrer que les vecteurs obtenus en a) et b) sont parall\350les" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 326 1 " " }{TEXT 325 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 313 5 "No 2)" }{TEXT -1 54 " Soit v1 = (3,-5,2), v2 = (5,-7,9) e t v3 = (62,34,-8) " }}{PARA 0 "" 0 "" {TEXT 315 2 "a)" }{TEXT -1 47 " \+ Montrer que v3 est perpendiculaire \340 v1 et v2 " }}{PARA 0 "" 0 "" {TEXT 314 2 "b)" }{TEXT -1 35 " Trouver k tel que v3 = k(v1 x v2) " }} }{SECT 1 {PARA 3 "" 0 "" {TEXT 328 1 " " }{TEXT 327 31 "Espace de trav ail 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 309 5 "No 3)" }{TEXT -1 36 " Soit v1 = (3,5,9) et v2 = (-7,12,8) " }}{PARA 0 "" 0 "" {TEXT 310 3 "a) " }{TEXT -1 52 "Trouver l'aire du \+ parall\351logramme de c\364t\351s v1 et v2" }}{PARA 0 "" 0 "" {TEXT 311 2 "b)" }{TEXT -1 58 " Trouver l'aire du triangle dont 2 des c\364t \351s sont v1 et v2" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 330 1 " " } {TEXT 329 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 312 5 "No 4)" }{TEXT -1 95 " Trouver l'ai re du triangle dont les sommets sont les points A(2,5,7) , B(-6,7,12) \+ et C(2,-5,8)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 332 1 " " }{TEXT 331 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 5)" }{TEXT -1 48 " Voici 2 vecteurs v1 = (4, 1,-2), v2 = (-3,0,9) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 1 " " } {TEXT -1 152 " \300 l'aide du produit vectoriel, trouver un vecteur v3 est perpendiculaire \340 v1 et qui est une combinaison lin\351aire de v1 et v2 (dans le plan de v1, v2) " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 334 1 " " }{TEXT 333 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 306 5 "No 6)" }{TEXT 319 1 " \+ " }{TEXT 322 92 "Soit u = (2,-5,6) et v = (-3,8,3). Trouver l'angle t \+ entre u et v en utilisant les formules " }}{PARA 0 "" 0 "" {TEXT 307 11 "a) uov = |" }{TEXT -1 17 "|u||*||v||*cos(t)" }}{PARA 0 "" 0 "" {TEXT -1 32 "b) ||uxv|| = ||u||*||v||*sin(t) " }}}{SECT 1 {PARA 3 "" 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 316 5 "No 7)" } {TEXT -1 139 " Trouver l'aire totale du t\351tra\350dre (l'aire des 4 \+ faces triangulaires) dont les 4 sommets sont A(2,2,-5), B(-3,7,1), C(5 ,8,-12) et E(6,0,7)" }}}{SECT 1 {PARA 3 "" 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 "" }}}} }{SECT 1 {PARA 260 "" 0 "" {TEXT 262 30 "Produit mixte uo(vxw) dans R^ 3" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 288 7 "Rappels" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 90 "u produit scalaire(v produit vectoriel W), not\351 uo(vxw), est appel\351 produit mixte de u,v,w" }}{PARA 0 "" 0 "" {TEXT -1 158 "Si u = (u1,u2,3), v = (v1,v2,v3) et w = (w1,w2,w3) alors uo(vxw) est \351gal au d\351terminant de la matrice ayant comme ligne s (ou colonnes) u,v,w : uo(vxw) = det(" }{XPPEDIT 18 0 "matrix([[u1, u 2, u3], [v1, v2, v3], [w1, w2, w3]]);" "6#-%'matrixG6#7%7%%#u1G%#u2G%# u3G7%%#v1G%#v2G%#v3G7%%#w1G%#w2G%#w3G" }{TEXT -1 2 ") " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "u:=vector([4,0,9]);v:=vector([5,12, 77]);w:=vector([-5,-7,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "uovxw:= dotprod(u,crossprod(v,w));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=matrix([u,v,w]);det(A);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 91 "Le volume du parall\351l\351pip\350de de c\364t\351s u, v,w est la valeur absolue du produit mixte uo(vxw) " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 303 41 "Repr\351sentation graphique du produit m ixte" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(linalg) :with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=-[1,3,0 ]:B:=[-2,3,0]:C:=[1,2,6]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "OA:=evalm(A);OB:=evalm(B);OC:=evalm(C);OAxOB:=crossprod(OA,OB);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Le produit mixte " }{TEXT 304 37 "OCo(OAxOB) = ||OC||*||OAxOB|| cos(t) " }{TEXT -1 51 "o\371 t est l 'angle entre OC et OAxOB et 0 <= t <= Pi." }}{PARA 0 "" 0 "" {TEXT 301 2 "1)" }{TEXT -1 56 " ||OAxOB|| = l'aire du parall\351logramme de \+ c\364t\351s OA et OB" }}{PARA 0 "" 0 "" {TEXT 302 2 "2)" }{TEXT -1 97 " ||OC|| cos(t) = mesure alg\351brique de la projection du vecteur OC \+ sur (la droite portant) OAxOB. " }}{PARA 0 "" 0 "" {TEXT -1 74 "Si 0 < = t <= Pi/2 alors cos(t) >= 0 et si Pi/2 < t <= Pi alors cos(t) <= 0" }}{PARA 0 "" 0 "" {TEXT -1 221 "On a ||OC|| cos(t) = ||proj(OC/OAxOB)| | si 0<= t <= Pi/2 et ||OC|| cos(t) = - ||proj(OC/OAxOB)|| si Pi/2 <= \+ t <= Pi donc le produit mixte OCo(OAxOB) = (l'aire du parall\351logra mme de c\364t\351s OA et OB)* +/- ||proj(OC/OAxOB)||" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 967 "with(plottools):\nvectOA:=arrow([0,0,0], OA,0.2,0.4,0.1,color=red):\nvectOB:=arrow([0,0,0],OB,0.2,0.4,0.1,color =red):\nvectOC:=arrow([0,0,0],OC,0.2,0.4,0.1,color=green):\nvectOAxOB: =arrow([0,0,0],OAxOB,0.2,0.6,0.1,color=blue):\nproj(OC/OAxOB):=evalm(( dotprod(OC,OAxOB)/dotprod(OAxOB,OAxOB))*OAxOB):\nvectproj(OC/OAxOB):=a rrow([0,0,0],proj(OC/OAxOB),0.1,0.6,0.1,color=black):projection:=curve ([C,convert(proj(OC/OAxOB),list)]):\n#noms:=textplot3d([[A[1],A[2],A[3 ],`A`],[B[1],B[2],B[3],`B`],[C[1],C[2],C[3],`C`],[OAxOB[1],OAxOB[2],OA xOB[3],`OAxOB`],[proj(OC/OAxOB)[1],proj(OC/OAxOB)[2],proj(OC/OAxOB)[3] ,`proj(OC/OAxOB)`]]):\nnoms:=textplot3d([[A[1],A[2],A[3],`OA`],[B[1],B [2],B[3],`OB`],[C[1],C[2],C[3],`OC`],[OAxOB[1],OAxOB[2],OAxOB[3],`OAxO B`],[proj(OC/OAxOB)[1],proj(OC/OAxOB)[2],proj(OC/OAxOB)[3],`proj(OC/OA xOB)`]],color=black):\nbase:=polygon([[0,0,0],A,A+B,B,[0,0,0]]):\ndisp lay(vectOA,vectOB,vectOC,vectOAxOB,vectproj(OC/OAxOB),projection,noms, base,orientation =[-2,66]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "L e produit mixte OCo(OAxOB) = (l'aire du parall\351logramme de c\364t \351s OA et OB)* +/- ||proj(OC/OAxOB)||." }}{PARA 0 "" 0 "" {TEXT -1 103 "On a + si les vecteurs OAxOB et proj(OC/OAxOB) sont dans le m\352 me sens et - s'ils sont de sens contraire" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Construisons le parall\351l\351pi\350de de c\364t\351s OA , OB, OC " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "parallelepipe de:=curve([[0,0,0],A,A+B,B,[0,0,0],C,C+B,B,A+B,A+B+C,C+B,C,A+C,A+B+C,A +C,A]):\ndisplay(vectOA,vectOB,vectOC,vectOAxOB,vectproj(OC/OAxOB),pro jection,noms,base,parallelepipede,orientation =[-2,66]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "||proj(OC/OAxOB)|| repr\351sente la hauteur du parall\351l\351p ip\350de. On a donc :" }}{PARA 0 "" 0 "" {TEXT -1 111 "le produit mixt e OCo(OAxOB) = (l'aire du parall\351logramme de c\364t\351s OA et OB) * +/- la hauteur du parall\351l\351pip\350de" }}{PARA 0 "" 0 "" {TEXT -1 101 "le produit mixte OCo(OAxOB) = l'aire de la base du parall\351l \351pip\350de* +/- la hauteur du parall\351l\351pip\350de" }}{PARA 0 " " 0 "" {TEXT -1 63 "le produit mixte OCo(OAxOB) = +/- le volume du pa rall\351l\351pip\350de" }{TEXT 298 1 " " }{TEXT 299 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 300 79 "Le volume du parall\351l\351pip\350de de c\364t\351s OA, OB, OC = la valeur absolue OCo(OAxOB)" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Changer le point A = (1,3,0) pour A = (-1,-3,0) pour visualiser un cas o\371 OAxOB et proj(OC/OAxO B) sont de sens contraire." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 7 "Exemple" }{TEXT -1 1 " " }{TEXT 296 1 "4" }{TEXT -1 130 ": Trouver le volume du parall\351l\351pip\350de dont 3 c\364t \351s issus d'un point sont les vecteurs a = (1,6,-6), b = (4,2,-7) et c = ( 9,0,12)" }}{PARA 0 "" 0 "" {TEXT -1 40 "a) En utilisant le prod uit mixte ao(bxc)" }}{PARA 0 "" 0 "" {TEXT -1 75 "b) En utilisant le d \351terminant de la matrice dont les lignes sont a, b et c" }}{PARA 0 "" 0 "" {TEXT -1 42 "c) En utilisant (aire de la base)* hauteur" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# a)\na:=vector([1,6,-6]);b: =vector([4,2,-7]);c:=vector([9,0,12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "ao(bxc) = +/- volume du parall\351l\351pip\350de ou volum e du parall\351lipip\350de = abs(ao(bxc))" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "dotprod(a,crossprod(b,c));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "volume=abs(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "b) Avec le d\351terminant de la matrice dont les lignes sont a, b et c" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=matrix([a,b,c ]);det(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "volume=abs(%) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "c) Trouvons le volume du par all\351l\351pip\350de en calculant (aire de la base)* hauteur" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Prenons a) comme base du parall \351l\351pip\350de, le parall\351lograme de c\364t\351s a et b et " }} {PARA 0 "" 0 "" {TEXT -1 116 "b) comme hauteur du parall\351l\351pip \350de, la norme de la projection du vecteur c sur un vecteur perpendi culaire \340 a et b " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "a) L'aire de la base du parall\351l\351pip\350de est || axb ||" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "airebase:=norm(crossprod(a,b),2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "b) La hauteur du parall\351l \351pip\350de est la norme de la projection du vecteur c sur un vecteu r perpendiculaire \340 a et b, Prenons v = axb comme vecteur perpendic ulaire \340 a et b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v:=cr ossprod(a,b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "La projection de c sur v : proj(c/v) = (cov/vov)*v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "proj(c/v):=evalm((dotprod(c,v)/dotprod(v,v))*v);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "La hauteur du parall\351l\351pip \350de = || proj(c/v) ||" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "hauteur:=norm(proj(c/v),2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "L e volume du parall\351l\351pip\350de = (aire de la base)* hauteur" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "volume:=airebase*hauteur;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 297 39 "Exemple 5 : Propri\351t\351s du produit mixte" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 2 "1)" }{TEXT -1 35 " ao(bxc) = - ao(cxb) car bxc = -cxb" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ao(bxc)=dotprod(a,crossprod(b,c));\nao(cxb)=dotprod(a ,crossprod(c,b));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 2 "2)" }{TEXT -1 95 " Une permutation circulaire des vecteurs dans ao(bxc) ne change pas la valeur du produit mixte " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "ao(bxc)=dotprod(a,crossprod(b,c));\nco(axb)=dotprod( c,crossprod(a,b));\nbo(cxa)=dotprod(b,crossprod(c,a));" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT 339 8 "Exercice" }{TEXT 340 3 "s B" }{TEXT -1 3 " : " }{TEXT 362 30 "Produit mixte uo(vxw) dans R^3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Pour les num\351ros 1 \340 3 : Soit v1 = (3,3,3), v2 = (-2,7,11), v3 = (8,9,10) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 341 5 "No 1)" }{TEXT 347 1 " " }{TEXT -1 69 "Trouver le volu me du parall\351l\351pip\350de construit sur les c\364t\351s v1,v2,v3 " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 352 0 "" }{TEXT 351 1 " " }{TEXT 350 31 "Espace de travail de l'\351tudiant" }}{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 -1 0 "" }{TEXT 344 5 "No 2)" }{TEXT 348 1 " " }{TEXT -1 106 "Tro uver le volume de la pyramide dont la base est le parall\351logramme d e c\364t\351s v1,v2 et dont un c\364t\351 est v3" }}{PARA 0 "" 0 "" {TEXT -1 69 " Remarque : le volume de la pyramide = (aire de la base)( hauteur) / 3" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 355 0 "" }{TEXT 354 1 " " }{TEXT 353 31 "Espace de travail de l'\351tudiant" }}{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 -1 1 " " }{TEXT 345 2 "No" }{TEXT 349 1 " " }{TEXT 342 2 "3)" }{TEXT 343 1 " " }{TEXT -1 109 "Trouver le volume du t\351t ra\350dre dont les sommets sont les points A(2,3,-2), B(3,4,-1), C(-4, 5,8) et E(3,3,3) " }}{PARA 0 "" 0 "" {TEXT -1 67 "Remarque : le volum e du t\351tra\350dre = (aire de la base)(hauteur) / 3 " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 358 0 "" }{TEXT 357 1 " " }{TEXT 356 31 "Espace \+ de travail de l'\351tudiant" }}{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 346 5 "No \+ 4)" }{TEXT -1 50 " Soit u = ( 5,-5,3) v = ( 9,-2 8) et w = ( 4,7,10)" }}{PARA 0 "" 0 "" {TEXT -1 108 "V\351rifier qu'une permutation circula ire des vecteurs dans le produit mixte ne change pas la valeur obtenue : " }}{PARA 0 "" 0 "" {TEXT -1 79 "On a les formules : uo(vxw) = wo( uxv) = vo(wxu) et uo(wxv) = vo(uxw) = wo(vxu)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 361 0 "" }{TEXT 360 1 " " }{TEXT 359 31 "Espace de travail \+ de l'\351tudiant" }}{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 "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }