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Droite s-2" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire, c laude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 273 51 " Principales commandes utilis\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "vector, evalm, norm, dotprod, crossprod, assign, solve et la biblioth\350que geom3d avec point -line-Equation- distance" }}{PARA 0 "" 0 "" {TEXT -1 69 "Note : S\351lectionner un mot et utiliser l'aide pour plus d'information" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 266 39 "La distance d'un point M \340 une droite D" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT 269 39 "La distance d'un point M \340 une droite D" }{TEXT -1 7 " not\351e " }{TEXT 270 5 "d(M,D" } {TEXT 271 1 ")" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Soit B un poi nt mobile (quelconque) de la droite D " }}{PARA 257 "" 0 "" {TEXT -1 147 "La distance entre M et un point B de la droite est minimale si le vecteur MB est perpendiculaire au vecteur v // D et alors la distance d = ||MB|| " }}{PARA 0 "" 0 "" {TEXT -1 52 "On r\351soud dotprod(MB,v ) = 0 et alors d(M,D) = ||MB||" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 10 "Exemple 1)" }{TEXT -1 104 " Trouver la distance du point M(-1,2,5) la droite D passant par A(0,-2,-1) et // au vecteur v = (1,4,3)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Un point mobile B de la droite D v \351rifie l'\351quation de la droite D: OB = OA + t*v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "OA:=vector([0,-2,-1]);v:=vector([1, 4,3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "OB:=evalm(OA+t*v) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "MB = OB-OM" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "OM:=vector([-1,2,5]);MB:=evalm(OB-OM);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " La distance entre M et le point B est minimale lorsque MB est perpendiculaire au vecteur v" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dotprod(MB,v)=0; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sol:=solve(%,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(t=sol,op(MB));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d('M',D):=norm(%,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Deuxi\350me solution pour la distance d' un point M \340 une droite D: d = ||AM x v|| /||v|| o\371 A est un po int de D et v // D." }}{PARA 0 "" 0 "" {TEXT -1 238 "Remarque : ||AM \+ x v|| repr\351sente l'aire du parall\351logramme de c\364t\351s AM et \+ v et ||v|| repr\351sente la longueur de la base du parall\351logramme \+ donc ||AM x v|| /||v|| est la hauteur du parall\351logramme et la dist ance d'un point M \340 une droite D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AM:=evalm(OM-OA);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "prodvect:=crossprod(AM,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "d:=norm(prodvect,2)/norm(v,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " Tr oisi\350me solution avec " }{TEXT 267 6 "geom3d" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "restart:with(geom3d):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Trouver la distance du point M(-1,2,5) la droite D pa ssant par A(0,-2,-1) et // v=(1,4,3)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "point(M,-1,2,5):point(A, 0,-2,-1):v:=vector([1,4,3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "line(D,[A,v]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Equation(D, m);# donne l'\351quation vectorielle de D (avec le param\350tre m)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "distance(D,M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Si la droite est donn\351e \340 l'aide d e deux points. Trouver la distance du point M la droite passant par A et B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "point(A,0,-2,-1):p oint(B,1,2,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "line(D,[A ,B]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Equation(D,t);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "distance(D,M);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 53 "Les 2 points les plus rapproch\351s de 2 droites gauches" } {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 10 "Exemple 2)" } {TEXT -1 75 " Soit D1 et D2 les 2 droites dont on veut les 2 points le s plus rapproch\351s." }}{PARA 257 "" 0 "" {TEXT -1 36 "D1 : (x-2)/3 = (y-3)/2 = (z-1)/3 = k" }}{PARA 257 "" 0 "" {TEXT -1 33 "D2 : x-2 = (y -4)/-2 = (z-3)/3 = r" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 257 0 "" } {TEXT -1 92 "Soit A un point quelconque (mobile) de D1, B un point que lconque de D2, v1 // D1 et v2 // D2" }}{PARA 257 "" 0 "" {TEXT 258 104 "A et B sont les plus rapproch\351s <=> AB est perpendiculaire \+ \340 v1 et v2 <=> (AB)o(v1) = 0 et (AB)o(v2) = 0" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 90 "Un point quelconque A(x,y,z) de D1 v\351rifie : x \+ = 2+3*k, y = 3+2*k , z = 1+3*k tandis que :" }}{PARA 0 "" 0 "" {TEXT -1 75 "Un point quelconque B(x,y,z) de D1 v\351rifie : x = 2+r, y = 4- 2*r , z = 3+3*r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:=[2+3*k, 3+2*k,1+3*k];B:=[2+r,4-2*r,3+3*r];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "OA:=evalm(A);OB:=evalm(B);v1:=vector([3,2,3]);v2:=vec tor([1,-2,3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AB:=evalm (OB-OA);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "eq1:=dotprod(AB ,v1)=0; \neq2:=dotprod(AB,v2)=0; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{eq1,eq2\},\{r,k\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=subs(sol,op(OA));B:=subs(sol,op( OB));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Remarque : la distance \+ entre les 2 droites gauches est || AB||. Si ||AB|| = 0 alors on a 2 dr oites concourantes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "AB:=e valm(B-A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "norm(AB,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 261 " " 0 "" {TEXT 259 35 "La distance entre 2 droites gauches" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 263 10 "Exemple 3)" }{TEXT -1 1 " " }{TEXT 264 12 "V\351rification" }{TEXT -1 45 " : Calcu lons la distance entre les 2 droites " }{TEXT 265 7 "gauches" }{TEXT -1 64 " D1 Et D2, not\351e d(D1,D2), donn\351es plus haut, avec la for mule : " }{TEXT 261 39 "d(D1, D2) = |A1A2o(V1xV2)| / ||V1xV2|| " } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "o\371 A1 est un point co nnu de D1 et A2, un point connu de D2, v1 // D1 et v2 // D2" }}{PARA 0 "" 0 "" {TEXT -1 88 "Remarque : La formule n'est pas valable pour 2 \+ droites D1 et D2 parall\350les car v1xv2 = 0" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 84 "D1 : (x-2)/3 = (y-3)/2 =(z-1)/3 = k. Pour k = 0, on obt ient le point A1(2,3,1) de D1" }}{PARA 0 "" 0 "" {TEXT -1 82 "D2 : x-2 = (y-4)/-2 = (z-3)/3 = r. Pour r = 0, on obtient le point A2(2,4,3) d e D2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A1:=[2,3,1];A2:=[2,4, 3];A1A2:=evalm(A2-A1);v1:=vector([3,2,3]);v2:=vector([1,-2,3]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "v3:=crossprod(v1,v2);#crossp rod : produit vectoriel" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " d(D1,D2):=abs(dotprod(A1A2,v3))/norm(v3,2);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 22 "La distance entre les " }{TEXT 274 57 "2 points les plu s rapproch\351s A et B des 2 droites gauches" }{TEXT -1 61 " plus haut , est bien la distance entre les 2 droites gauches." }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 48 "V\351rification : La distance entre 2 droites avec " }{TEXT 260 6 "geom3d" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(geom3d):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "D1 passe p ar a1(2,3,1) et est // v1 = [3,2,3], D2 passe par a2(2,4,3) et est // \+ V2:=[1,-2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "point(a1,2 ,3,1):point(a2,2,4,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "l ine(D1,[a1,v1]):line(D2,[a2,v2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "distance(D1,D2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 275 10 "Exercices \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 278 5 "No 1)" }{TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "D1 : (x.y.z) = (1,5,-9) + k(2,7,11)" }}{PARA 0 "" 0 "" {TEXT -1 32 "D2 : (x-1)/2 = (y+3)/4 = (z-5)/9" }}{PARA 0 "" 0 "" {TEXT 282 2 "a)" }{TEXT -1 1 " " }{TEXT 280 52 "Trouver les 2 point s les plus rapproch\351s de D1 et D2" }}{PARA 0 "" 0 "" {TEXT 283 2 "b )" }{TEXT -1 35 " Trouver la distance entre D1 et D2" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 286 32 " 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 281 5 "No 2)" }{TEXT -1 25 " Voici 2 droi tes D1 et D2" }}{PARA 0 "" 0 "" {TEXT -1 47 "D1 passe par les points ( 4,-7,1 ) et (12,9,-43)" }}{PARA 0 "" 0 "" {TEXT -1 59 "D2 passe par l e point ( 5,5,-5) et est // \340 v2 = (2,4,-11)" }}{PARA 0 "" 0 "" {TEXT 284 2 "a)" }{TEXT -1 34 " Montrer que D1 est parall\350le \340 D 2" }}{PARA 0 "" 0 "" {TEXT 285 2 "b)" }{TEXT -1 35 " Trouver la distan ce entre D1 et D2" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 287 32 " 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 276 5 "No 3) " }{TEXT 277 1 " " }{TEXT -1 25 " Voici 2 droites D1 et D2" }}{PARA 0 "" 0 "" {TEXT -1 37 "D1 : (x.y.z) = (2,3,-5) + k(-2,-5,10)" }}{PARA 0 "" 0 "" {TEXT -1 37 "D2 : x = 2r, y = 8 + 6r, z = -12 + 7r" }}{PARA 0 "" 0 "" {TEXT -1 55 "Trouver un point A tel que d(A,D1) + d(A,D2) = d( D1,D2)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 288 32 " 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 }