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{SECT 0 {SECT 1 {PARA 205 "" 0 "" {TEXT 256 11 "Exercice 11" }}{PARA 
203 "" 0 "" {TEXT -1 0 "" }}{PARA 203 "" 0 "" {TEXT 210 3 "1) " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 
213 8 "restart;" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 26 "with
(geom3d):with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the na
me polar has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning,
 the name inverse has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 
"Warning, the protected names norm and trace have been redefined and u
nprotected\n" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 29 "plane(P
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{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 29 "plane(Q,x-y+4*z-2=0,[x,y
,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"QG" }}}{EXCHG {PARA 203 ">
 " 0 "" {MPLTEXT 1 213 20 "intersection(D,P,Q);" }{TEXT -1 26 " la dro
ite d'intersection " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 11 " des \+
plans " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 4 " et " }{XPPEDIT 18 
0 "Q;" "6#%\"QG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"DG" }}}{EXCHG 
{PARA 203 "> " 0 "" {MPLTEXT 1 218 10 "detail(D);" }{TEXT -1 34 " troi
s \351quations param\351triques de " }{XPPEDIT 18 0 "D;" "6#%\"DG" }}
{PARA 7 "" 1 "" {TEXT -1 69 "Warning, assume that the parameter in the
 parametric equations is _t\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%^rna
me~of~the~object:~~D|+form~of~the~object:~~line3d|+equation~of~the~lin
e:~~[x~=~-1/3+3*_t,~y~=~-7/3-9*_t,~z~=~-3*_t]G" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "L'axe de la rotation " }
{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 25 " passe donc par le point
 " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 62 " de coordonn\351es (-1/
3;-7/3;0) et il est dirig\351 par le vecteur " }{XPPEDIT 18 0 "u;" "6#
%\"uG" }{TEXT -1 29 " de coordonn\351es (3;-9;-3).   " }}{PARA 203 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 15 "poi
nt(A,6,1,2);" }{TEXT -1 21 " un point de l'espace" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 19 
"reflection(B,A,P); " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 40 " est
 son sym\351trique par rapport au plan " }{XPPEDIT 18 0 "P;" "6#%\"PG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"BG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "reflection(C,B,Q);" }{TEXT -1 24 " C est l'image de
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20 "6#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "projection(J
,A,D);" }{TEXT -1 23 " le projet\351 orthogonal " }{XPPEDIT 18 0 "J;" 
"6#%\"JG" }{TEXT -1 4 " de " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 
5 " sur " }{XPPEDIT 18 0 "D;" "6#%\"DG" }}{PARA 11 "" 1 "" {XPPMATH 
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18 0 "A;" "6#%\"AG" }{TEXT -1 22 " en un objet vectoriel" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"AG-%'vectorG6#7%\"\"'\"\"\"\"\"#" }}}{EXCHG 
{PARA 203 "> " 0 "" {MPLTEXT 1 213 34 "C:=convert(coordinates(C),vecto
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\")F+#!#SF+" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 213 34 "J:=conve
rt(coordinates(J),vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%
'vectorG6#7%#!#G\"#L#!#EF+#\"#<F+" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 213 16 "JA:=evalm(A-J); " }{TEXT -1 21 "le vecteur d'origin
e " }{XPPEDIT 18 0 "J;" "6#%\"JG" }{TEXT -1 16 " et d'extr\351mit\351 \+
" }{XPPEDIT 18 0 "A;" "6#%\"AG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J
AG-%'vectorG6#7%#\"$E#\"#L#\"#fF+#\"#\\F+" }}}{EXCHG {PARA 203 "> " 0 
"" {MPLTEXT 1 213 15 "JC:=evalm(C-J);" }}{PARA 11 "" 1 "" {XPPMATH 20 
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"> " 0 "" {MPLTEXT 1 213 61 "theta:=evalf(arccos(dotprod(JA,JC)/(norm(
JA,2)*norm(JC,2))));" }{TEXT -1 32 " l'angle form\351 par les vecteurs
 " }{XPPEDIT 18 0 "JA;" "6#%#JAG" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "J
C;" "6#%#JCG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG$\"+5r!fb#!\"
*" }}}{PARA 203 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "C
et angle (exprim\351 en radians) ne d\351pend pas du point " }
{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 8 " choisi." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v:=crossprod
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\"#**#\"$j)\"#L#F-F+" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 13 "Les vecteurs " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT 
-1 4 " et " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 23 " sont de sens \+
oppos\351s. " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 30 " est donc
 la rotation d'angle " }{XPPEDIT 18 0 "-theta;" "6#,$%&thetaG!\"\"" }
{TEXT -1 30 " autour de l'axe passsant par " }{XPPEDIT 18 0 "A;" "6#%
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}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "restart:with(geom3d):with(linalg):" }}{PARA 7 "
" 1 "" {TEXT -1 43 "Warning, the name polar has been redefined\n" }}
{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name inverse has been redefi
ned\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names nor
m and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "plane(P,2*x+y-z+3=0,[x,y,z]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%\"PG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "reflection(B,A,P);" }}{PARA 
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1 0 16 "coordinates(B); " }{TEXT -1 31 "les coordonn\351es de l'image \+
par " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 28 " d'un point de co
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" {MPLTEXT 1 0 34 "restart:with(geom3d):with(linalg):" }}{PARA 7 "" 1 
"" {TEXT -1 43 "Warning, the name polar has been redefined\n" }}{PARA 
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0 "" {MPLTEXT 1 0 29 "plane(P,2*x+y-z+3=0,[x,y,z]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%\"PG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "point(A,x,y,z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "reflection(C,reflection(B,A,P),Q);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%\"CG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s
implify(coordinates(C));" }{TEXT -1 32 " les coordonn\351es de l'image
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oordonn\351es (" }{XPPEDIT 18 0 "x,y,z;" "6%%\"xG%\"yG%\"zG" }{TEXT 
-1 1 ")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,*%\"xG#!\"#\"\"$*&#\"\"#
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