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-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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 293 24 "Laboratoire 2 - Vecteur s" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 52 "Par Claude St-Hilaire, cla ude.sthilaire@videotron.ca" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 295 51 "Principales comm andes utilis\351es dans ce laboratoire" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "vector de la biblioth\350que linalg, randvector, evalm, \+ subs avec op, " }{TEXT 296 6 "equate" }{TEXT -1 20 " de la biblioth \350que " }{TEXT 297 7 "student" }{TEXT 299 2 ", " }{TEXT 298 11 "seq, norm, " }{TEXT -1 12 "normalize, " }}{PARA 0 "" 0 "" {TEXT 300 37 "R epr\351sentation graphique des vecteurs" }{TEXT -1 40 " : arrow de la \+ biblioth\350que plottools. " }}{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 256 37 "Vecteurs et manipulations de vec teurs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "with(linalg):#bibli oth\350que contenant des programmes d'alg\350bre lin\351aire" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Un " }{TEXT 263 7 "vecteur" }{TEXT -1 20 " est repr\351sent\351 par " }{TEXT 294 23 "vector([a1,a2,...,an ]) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A:=vector([3,7,5]);t ype(A,vector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Un vecteur peu t \352tre d\351fini par une fonction : vector(n, x->f(x)); donne un ve cteur de dimension n dont les composantes sont f(1), f(2), f(3),..f(n) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u1:=vector(3,x->x^3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 14 "randvector(n) " }{TEXT -1 28 " d\351finie un vecteur avec des " }{TEXT 275 7 "nombres" }{TEXT -1 1 " \+ " }{TEXT 267 10 "al\351atoires" }{TEXT -1 13 " de -99 \340 99." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "u2:=randvector(5);# donne un vecteur de 5 composantes avec des entr\351es al\351atoires entre -99 \+ \340 99" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Si on veut des entr \351es entre 1 et 10 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r andvector(5,entries = rand(1..10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 46 "u[i] d\351signe la i i\350me composante du vecteur u" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "u2[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 14 "Op\351rations sur" }{TEXT 277 5 " les " }{TEXT 269 8 "vecteurs" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 27 "Additi on et soustraction de" }{TEXT -1 1 " " }{TEXT 271 8 "vecteurs" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "u:=vector([-3,5,1]);v:=vecto r([4,4,4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "u+v;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Pour avoir la somme de u+v : evalm (u+v); ou matadd(u,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e valm(u+v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 60 "Multiplication d'u n vecteur v par un scalaire k : evalm(k*v)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "evalm(10*v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 19 "Combinaison linaire" }{TEXT 288 21 " des vecteurs u et v," }{TEXT -1 12 " au + bv : " }{TEXT 257 16 "evalm(a*u+b*v); " }{TEXT -1 26 "ou bien : matadd(u,v,a,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(5*u+7*v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 28 "Substitution dans un vecteur" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:=vector([x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Su bstituons x=5, y=10 et z=15 dans le vecteur u;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "subs(\{x=5,y=10,z=15\},u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Nous devons indiquer \340 Maple de substituer sur les composantes (" }{TEXT 262 2 "op" }{TEXT -1 21 "\351rands) du vect eur u " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(\{x=5,y=10,z =15\},op(u));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 20 "\351galit\351 d e vecteurs:" }{TEXT -1 28 " Trouver x,y,z si v1 = v2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "v1:=vector([2*x,5,3+y]);v2:=vector( [12,3*z-5,4*y-3]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "On peut \+ \351galiser, composante par composante, 2 vecteurs (ou matrices) avec \+ " }{TEXT 264 6 "equate" }{TEXT -1 20 " de la biblioth\350que " }{TEXT 265 7 "student" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(student,equate):equate(v1,v2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(%,\{x,y,z\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Une autre mani\350re avec sequence : " }{TEXT 266 3 "seq " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(v1[i]=v2[i], i=1..3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(\{%\},\{x,y,z\} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 21 "Vecteurs parall\350les :" }{TEXT -1 43 " u //v s'il existe un r\351el k tel que u = kv" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 301 10 "Exemple 1)" }{TEXT -1 70 " Trouve r les vecteurs parall\350les parmi u1 = [12.459, -3.782, 32,746], " }} {PARA 0 "" 0 "" {TEXT -1 5 "u2 = " }{XPPEDIT 18 0 "vector([42.98355, - 13.04790, 110.40, 2573.70]);" "6#-%'vectorG6#7&$\"(b$)H%!\"&,$$\"(!z/8 F)!\"\"$\"&S5\"!\"#$\"'qtDF0" }{TEXT -1 6 "\nu3 = " }{XPPEDIT 18 0 "ve ctor([-32.01963, 9.71974, -82.24, -1917.32]);" "6#-%'vectorG6#7&,$$\"( j>?$!\"&!\"\"$\"'u>(*F*,$$\"%C#)!\"#F+,$$\"'K<>F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "u1:=vector([12.459, -3.782, 32,746]);u2: =vector([42.98355, -13.04790, 110.40, 2573.70]);u3:=vector([-32.01963, 9.71974, -82.24, -1917.32]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " u1 est-il parall\350le \340 u2 ? a-t-on u1 = k*u2?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(k*u2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "equate(%,u1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,k);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 " u1 = " } {XPPEDIT 18 0 ".2898550725;" "6#$\"+D2b)*G!#5" }{TEXT -1 67 " u2 donc \+ u1 // u2, et de m\352me sens car k >0 . Est-ce que u3 // u1 ?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "equate(u1,r*u3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Maple ne r\351pond rien, donc il n'y a pas de soluti on alors u3 n'est pas parall\350le \340 u1." }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 32 "Norme d'un vecteur v: norm(v,2);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 302 83 "La norme d'un vecteur u = (u1,u2,..un) est ||u|| = racine carr\351e(u1^2+u2^2+..un^2) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Remarque : Dans Maple, norm(u,k) donne la racine k i\350 me de (u1^k+u2^k+...un^k) donc || u || = norm(u,2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v:=vector([3,2,-4]);norm(v,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sqrt(3^2+2^2+(-4)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 46 "Vecteur unitaire \340 un vecteur v : nor malize(v)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Les 2 vecteurs unita ires // \340 un vecteur v sont u = v/||v|| et -u " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "u:=evalm(v/norm(v,2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "normalize(u);norm(%,2);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 292 16 "Vecteur et point" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Un " }{TEXT 312 7 "vecteur" }{TEXT -1 20 " est repr\351sent\351 par " }{TEXT 313 22 "vector([a1,a2,...,an])" } {TEXT -1 54 ". Cependant un vecteur peut aussi \352tre repr\351sent \351 par " }{TEXT 314 24 "une liste [a1,a2,..,an]." }{TEXT -1 274 " A ttention: Certaines commandes Maple (norm, basis, GramSchmidt, etc) ne fonctionneront pas si les vecteurs sont d\351finis par des listes con trairement \340 d'autres commandes (dotprod, crossprod dans R^3, etc). On ne rencontre pas ce probl\350me si les vecteurs sont d\351finis av ec " }{TEXT 315 8 "vector. " }}{PARA 0 "" 0 "" {TEXT -1 17 "Nous utili serons " }{TEXT 309 23 "vector([a1,a2,...,an]) " }{TEXT -1 27 "pour re pr\351senter un vecteur" }{TEXT 310 1 " " }{TEXT -1 2 "et" }{TEXT 311 15 " [a1,a2,...,an]" }{TEXT -1 26 " pour repr\351senter un point" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Remarque: le vecteur OA et le poi nt A = [a1,a2,..,an] sont le m\352me n-uple dans R^n, o\371 O est le p oint [0,0,...,0]. " }}{PARA 0 "" 0 "" {TEXT -1 166 "Dans Maple, evalm( A) transforme un point A en vecteur : evalm(A) repr\351sente le vecteu r OA. De m\352me, si A et B sont des points alors evalm(B-A) repr\351s ente le vecteur " }{TEXT 291 12 "AB = OB - OA" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 289 10 "Exemple 2)" }{TEXT -1 61 " Montre r que le triangle ABC est isoc\350le o\371 A,B,C sont les 3 " }{TEXT 261 6 "points" }{TEXT -1 46 " suivants : A(4,-7,8), B(7,-3,9) et C(12, -3,8)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Premi\350re solution :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "OA:=vector([4,-7,8]);OB:= vector([7,-3,9]);OC:=vector([12,-3,8]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "AB:=evalm(OB-OA);;AC:=evalm(OC-OA);BC:=evalm(OC-OB); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "norm(AB,2);norm(AC,2);n orm(BC,2);;" }}}{PARA 0 "" 0 "" {TEXT -1 38 "||AB|| = ||BC||, donc ABC est isoc\350le." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Deuxi\350me so lution " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=[4,-7,8];B:=[ 7,-3,9];C:=[12,-3,8];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "AB :=evalm(B-A);;AC:=evalm(C-A);BC:=evalm(OC-OB);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "norm(AB,2);norm(AC,2);norm(BC,2);;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "||AB|| = ||BC||, donc ABC est isoc\350le. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 10 "Exemple 3)" }{TEXT -1 92 " T rouvons le point P situ\351 au 4/5 de MN \340 partir de M o\371 M = [1 2,-36,89], N = [35,-11,-78]. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 " On a OP = OM + (4/5)MN " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " OM:=vector([12,-36,89]);ON:=vector([35,-11,-78]);MN:=evalm(ON-OM);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "OP=evalm(OM+(4/5)*MN);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "donc P = " }{XPPEDIT 18 0 "vector([ 152/5, -16, -223/5]);" "6#-%'vectorG6#7%*&\"$_\"\"\"\"\"\"&!\"\",$\"#; F+,$*&\"$B#F)F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT 278 10 "Exercice A" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 100 "Voici 5 points : A(4,6,-9), B(-5,6,0), C(1,-5,6), E(3,-5,4); F(x + y - z, 3x + 5y + 4z,-2x + y - 9z)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 5 "No 1)" }{TEXT 317 1 " " }{TEXT -1 84 "a) Trouve r la norme de AB b) Trouver 3AB - 4CE c) Trouver x,y et z tels que AB \+ = EF " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 279 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 -1 0 "" }{TEXT 318 5 "No 2)" }{TEXT -1 48 " Trouver deux vecteurs unitaires parall\350les \340 BC" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 303 31 "Espace de travail d e 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 319 5 "No 3)" } {TEXT -1 66 " Montrer que ABCE est un trap\350ze (un trap\350ze a 2 c \364t\351s parall\350les)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 304 31 "Es pace 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 328 5 "No 4)" }{TEXT 326 1 " " }{TEXT 323 4 "Soit" }{TEXT 327 1 " " } {TEXT 322 3 "A =" }{TEXT 324 1 " " }{TEXT 325 48 "[-5,7,3], B = [2,11, -9] et C = [2,0,5]. 3 points" }}{PARA 0 "" 0 "" {TEXT 320 2 "a)" } {TEXT -1 76 " Trouver P le point milieu de BC. Note : AP est la m\351d iane issue du point A " }}{PARA 0 "" 0 "" {TEXT 321 2 "b)" }{TEXT -1 70 " Trouver le point M situ\351 au 2/3 de la m\351diane AP, \340 part ir du point A" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 305 31 "Espace de trav ail 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 330 3 "No " } {TEXT 329 2 "5)" }{TEXT -1 154 " V\351rifier que le point d'intersecti on des m\351dianes du triangle ABC du No 4) se rencontrent en un point situ\351 au 2/3 de chaque m\351diane \340 partir du sommet. " }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 306 31 "Espace de travail de l'\351tudia nt" }}{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 3 "" 0 "" {TEXT 280 59 "Repr\351sentation graphiq ue des vecteurs g\351om\351triques dans R^2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 351 "En math\351matiques, dans R^2, un couple (a,b) repr\351s ente tout aussi bien un point de coordonn\351es (a,b) que le vecteur ( a,b) ayant comme origine le point (0,0) et comme extr\351mit\351 le po int (a,b). Le vecteur (a,b) = a*i + b*j o\371 i et j sont les vecteurs unitaires port\351s respectivement par l'axe des x et l'axe des y. O n a une situation analogue dans R^3." }}{PARA 0 "" 0 "" {TEXT -1 216 " Avec la commande arrow de la biblioth\350que (package) plottools, on p eut d\351finir une fl\350che repr\351sentant un vecteur et le tracer a vec display de la biblioth\350que (package) plots. Dans R^2, un point \+ est repr\351sent\351 par : " }{TEXT 283 5 "[a,b]" }{TEXT -1 36 " tandi s que le vecteur (a,b) sera : " }{TEXT 284 13 "vector([a,b])" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plo ttools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A:=[4,7];B:=[-2 ,5];C:=[3,-2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 26 "arrow(point1,p oint2,a,b,c)" }{TEXT -1 231 " trace un vecteur allant du point1 au poi nt2. \"a\" repr\351sente la largeur de la fl\350che, \"b\" repr\351sen te la largeur de la pointe de la fl\350che et \"c\" repr\351sente la l ongueur de la pointe de la fl\350che divis\351e par la longueur de la \+ fl\350che." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 29 "arrow(point1,vector(v),a,b,c)" }{TEXT -1 50 " trace le vecteur v ayant comme origine le point1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "AB:=arrow(A,B,0.01,0.4,0.1,color=red):\nCflecheB:=ar row(C,vector(B),0.01,0.4,0.1,color=blue):\nnoms:=textplot(\{[4,7,'A'], [-2,5,'B'],[3,-2,'C']\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(AB,CflecheB,noms,view=[-5..5,-5..10]);" }}{PARA 13 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "AB repr\351sen te le vecteur joignant le point A au point B et CflecheB repr\351sente le vecteur B avec comme origine le point C." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 73 "Repr\351sentation de la somme de 2 vecteurs u et v avec la loi du triangle :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "or igine:=[0,0];A:=[2,5];B:=[3,4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "u:=arrow(origine,vector(A),0.01,.4,.1,color=blue):\nv:=arrow( A,vector(B),0.01,.4,.1,color=red):\nuplusv:=arrow(origine,vector(A+B), 0.01,0.4,0.1,color=yellow):\ntexte:=textplot(\{[1,3.5,` u `],[3,7,` v \+ `],[3,4.5,` u+v `]\}):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(\{u,v,uplusv,texte\},view=[-2..7,-2..10]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 285 22 "Exercice B (optionnel)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 331 5 "No 1 )" }{TEXT -1 117 " Illustrer la loi du parall\351logramme en ajoutant au graphique ci-haut, les 2 vecteurs compl\351tant le parall\351logra mme " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 307 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 332 5 "No 2)" }{TEXT -1 46 " Soit 3 points A:=[2,5]; B:=[-3,5]; C:=[4, -2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 334 2 "a)" } {TEXT 335 1 " " }{TEXT -1 83 "Tracer, sur un m\352me graphique, les fl \350ches AB, AC, BC, tra\347ant le triangle ABC . " }}{PARA 0 "" 0 " " {TEXT 333 4 " " }{TEXT 336 2 "b)" }{TEXT -1 49 " Ajouter les fl \350ches repr\351sentant les 3 m\351dianes " }}}{SECT 1 {PARA 3 "" 0 " " {TEXT 308 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 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 337 59 "Repr\351sentation graphique des vecteurs g\351om \351triques dans R^3" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plottools):with(plots):with(linalg):" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 286 36 "Repr\351sentation de vecteurs da ns R^3 " }{TEXT -1 49 ": point par [a,b,c] , vecteur par vector([a,b,c ])" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "OrigineR3:=[0,0,0];A: =[2,-3,5];B:=[1,4,-2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Tracer \+ les vecteurs AB et le vecteurA = OA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "flecheAB:=arrow(A,B,0.08,0.4,.1,color=red):\nflechev ecteurA:=arrow([0,0,0], vector(A), .08, .4, .1,color=blue):\nplots[dis play](flecheAB,flechevecteurA,orientation=[45,45],axes=boxed);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 639 "Dans plots[display], on doit ajouter orientation([theta, phi]). Theta et phi sont 2 angles en degr\351s sp\351cifiant la positi on de l'oeil regardant le graphique. \nRemarque: Theta et phi sont les angles dans les coordonn\351es sph\351riques. Theta est l'angle dans \+ le plan XY dont un c\364t\351 est la partie positive de l'axe des X et l'autre c\364t\351 est la demi-droite obtenue en faisant tourner la p artie positive de l'axe des X d'un angle theta antihoraire. Phi est l 'angle dont un c\364t\351 est la partie positive de l'axe des z et l'a utre c\364t\351 est une demi-droite partant de l'origine et faisant un angle de phi degr\351s avec la partie positive de l'axe des z." }}}}} {MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }