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l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 
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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 22 "Groupe orthogonal : TD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning,
 the protected names norm and trace have been redefined and unprotecte
d\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 4" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=vector([2,1]):x:=vector([x1,x2])
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(dotprod(x-a*f,f)
=0,\{a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"aG,&%#x1G#\"\"#\"
\"&*&#\"\"\"F*F-%#x2GF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(x-2
*a*f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$,&%#x1G#!\"$\"
\"&*&#\"\"%F+\"\"\"%#x2GF/!\"\",&F0#\"\"$F+*&#F.F+F/F(F/F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "genmatrix(convert(%,list),[x1,x2]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$#!\"$\"\"&#!\"%F*7
$F+#\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "C'est bien une mat
rice orthogonale, sym\351trique, de trace nulle." }}}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 10 "Exercice 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "a:=vector([3,4]):b:=vector([1,-2]):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "arccos(dotprod(a,b)/sqrt(dotprod(a,a)*dotprod(b,b)
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#PiG\"\"\"-%'arccosG6#,$*$-%
%sqrtG6#\"\"&F%#F%F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "det(matrix([a,b]))/sqrt(dotprod(a,a)*dotprod(b,b));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"&\"\"\"#!\"#F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 45 "Donc l'angle orient\351 est arccos(1/sqrt
(5))-Pi" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 8" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:=<<3 | -4> , <4 | 3>>/5:B:
=<<3 | 4> , <4 | -3>>/5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 
"evalm(A&*transpose(A)),evalm(B&*transpose(B));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'matrixG6#7$7$\"\"\"\"\"!7$F)F(F#" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "det(A),det(B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "A e
st donc une rotation et B une sym\351trie par rapport \340 :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "kernel(B-Matrix(2,shape=iden
tity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7$\"\"#\"\"\"
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 9" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "x:=vector([x1,x2,x3]):a:=vector([a1,a2,a3
]):b:=vector([b1,b2,b3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "evalm(crossprod(a,x)-b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec
torG6#7%,(*&%#a2G\"\"\"%#x3GF*F**&%#a3GF*%#x2GF*!\"\"%#b1GF/,(*&F-F*%#
x1GF*F**&%#a1GF*F+F*F/%#b2GF/,(*&F5F*F.F*F**&F)F*F3F*F/%#b3GF/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(%[1],\{x1,x2,x3\});" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%#x2G,$*&,&*&%#a2G\"\"\"%#x3GF+!\"
\"%#b1GF+F+%#a3GF-F-/F,F,/%#x1GF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
259 "Mouais... a voir... Si b est non nul et colin\351aire \340 a, l'
\351quation initiale n'a clairement pas de solution : Maple a du faire
 des simplification illicites... c'est-\340-dire diviser par quelque c
hose de potentiellement nul. Il faut donc reprendre cela \340 la main.
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 11" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "Faisons cela tout en finesse..." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "u:=vector([u1,u2,u3]):v:=vec
tor([v1,v2,v3]):w:=vector([w1,w2,w3]):s:=vector([s1,s2,s3]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dotprod(crossprod(u,v),cross
prod(w,s),orthogonal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&*&%#u2
G\"\"\"%#v3GF(F(*&%#u3GF(%#v2GF(!\"\"F(,&*&%#w2GF(%#s3GF(F(*&%#w3GF(%#
s2GF(F-F(F(*&,&*&F+F(%#v1GF(F(*&%#u1GF(F)F(F-F(,&*&F3F(%#s1GF(F(*&%#w1
GF(F1F(F-F(F(*&,&*&F:F(F,F(F(*&F'F(F8F(F-F(,&*&F?F(F4F(F(*&F0F(F=F(F-F
(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "dotprod(u,w,orthogon
al)*dotprod(v,s,orthogonal)-dotprod(u,s,orthogonal)*dotprod(v,w,orthog
onal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,(*&%#u1G\"\"\"%#w1GF(F(
*&%#u2GF(%#w2GF(F(*&%#u3GF(%#w3GF(F(F(,(*&%#v1GF(%#s1GF(F(*&%#v2GF(%#s
2GF(F(*&%#v3GF(%#s3GF(F(F(F(*&,(*&F'F(F3F(F(*&F+F(F6F(F(*&F.F(F9F(F(F(
,(*&F2F(F)F(F(*&F5F(F,F(F(*&F8F(F/F(F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "%-%%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&,(*&
%#u1G\"\"\"%#w1GF(F(*&%#u2GF(%#w2GF(F(*&%#u3GF(%#w3GF(F(F(,(*&%#v1GF(%
#s1GF(F(*&%#v2GF(%#s2GF(F(*&%#v3GF(%#s3GF(F(F(F(*&,(*&F'F(F3F(F(*&F+F(
F6F(F(*&F.F(F9F(F(F(,(*&F2F(F)F(F(*&F5F(F,F(F(*&F8F(F/F(F(F(!\"\"*&,&*
&F+F(F8F(F(*&F.F(F5F(FCF(,&*&F,F(F9F(F(*&F/F(F6F(FCF(FC*&,&*&F.F(F2F(F
(*&F'F(F8F(FCF(,&*&F/F(F3F(F(*&F)F(F9F(FCF(FC*&,&*&F'F(F5F(F(*&F+F(F2F
(FCF(,&*&F)F(F6F(F(*&F,F(F3F(FCF(FC" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "C'est bien la g\351om\351trie
, non ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "evalm(crossprod(
crossprod(u,v),crossprod(w,s))-det(matrix([u,v,s]))*w+det(matrix([u,v,
w]))*s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,**&,&*&%#u3
G\"\"\"%#v1GF,F,*&%#u1GF,%#v3GF,!\"\"F,,&*&%#w1GF,%#s2GF,F,*&%#w2GF,%#
s1GF,F1F,F,*&,&*&F/F,%#v2GF,F,*&%#u2GF,F-F,F1F,,&*&%#w3GF,F8F,F,*&F4F,
%#s3GF,F1F,F1*&,.*(F/F,F<F,FCF,F1*(F/F,F0F,F5F,F,*(F-F,F>F,FCF,F,*(F-F
,F+F,F5F,F1*(F8F,F>F,F0F,F1*(F8F,F+F,F<F,F,F,F4F,F,*&,.*(F/F,F<F,FAF,F
,*(F/F,F0F,F7F,F1*(F-F,F>F,FAF,F1*(F-F,F+F,F7F,F,*(F4F,F>F,F0F,F,*(F4F
,F+F,F<F,F1F,F8F,F,,**&F:F,,&*&F7F,FCF,F,*&FAF,F5F,F1F,F,*&,&*&F>F,F0F
,F,*&F+F,F<F,F1F,F2F,F1*&FEF,F7F,F,*&FMF,F5F,F,,**&FZF,F?F,F,*&F)F,FVF
,F1*&FEF,FAF,F,*&FMF,FCF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"
!F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "h\351h\351 ! Bon, cela d
it, on demande une preuve \"humaine\" de ces r\351sultats..." }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 12" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 92 "Essayons une m\351thode brutale en calculant dans un
e b.o.n.d obtenue par orthonormalisation..." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 78 "a:=vector([alpha,0,0]):b:=vector([beta,gamma,0]):c
:=vector([delta,lambda,mu]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 "x:=vector([x1,x2,x3]):y:=vector([y1,y2,y3]):z:=vector([z1,z2,z3]
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(crossprod(x,y)-
c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,(*&%#x2G\"\"\"%#
y3GF*F**&%#x3GF*%#y2GF*!\"\"%&deltaGF/,(*&F-F*%#y1GF*F**&%#x1GF*F+F*F/
%'lambdaGF/,(*&F5F*F.F*F**&F)F*F3F*F/%#muGF/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "op(convert(%,list));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%,(*&%#x2G\"\"\"%#y3GF&F&*&%#x3GF&%#y2GF&!\"\"%&deltaGF+
,(*&F)F&%#y1GF&F&*&%#x1GF&F'F&F+%'lambdaGF+,(*&F1F&F*F&F&*&F%F&F/F&F+%
#muGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "e1:=%:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e2:=op(convert(evalm(crossprod(y,z)
-a),list));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G6%,(*&%#y2G\"\"\"
%#z3GF)F)*&%#y3GF)%#z2GF)!\"\"%&alphaGF.,&*&F,F)%#z1GF)F)*&%#y1GF)F*F)
F.,&*&F4F)F-F)F)*&F(F)F2F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "e3:=op(convert(evalm(crossprod(z,x)-b),list));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#e3G6%,(*&%#z2G\"\"\"%#x3GF)F)*&%#z3GF)%#x2GF)!\"
\"%%betaGF.,(*&F,F)%#x1GF)F)*&%#z1GF)F*F)F.%&gammaGF.,&*&F4F)F-F)F)*&F
(F)F2F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(\{e1,e2,
e3\},\{x1,x2,x3,y1,y2,y3,z1,z2,z3\});" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#<+/%#z2G\"\"!/%#x3G*(,&*&%'lambdaG\"\"\"%%betaGF-F-*&%&deltaGF-%&ga
mmaGF-!\"\"F--%'RootOfG6#,&*&%&alphaGF-%#muGF-F-*&F1F-)%#_ZG\"\"#F-F2F
2F1F2/%#z3G*&F8F-F3F2/%#x1G*&F9F-F3F2/%#z1GF&/%#x2G,$*(F.F-F3F-F8F2F2/
%#y1GF&/%#y2GF3/%#y3G,$*(F,F-F3F-F9F2F2" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 525 "Tr\350s franchement je n'y croyais pas beaucoup...\nalph
a, gamma et mu \351tant non nuls, le r\351sultat a au moins une petite
 chance d'\352tre correct. Cela dit, on ne sait pas si dans son calcul
, Maple n'a pas divis\351 par quelque chose de potentiellement nul. Il
 reste donc du travail.\nOn sait au moins ce qu'on veut montrer : z es
t colin\351aire \340 \"a vectoriel b\", et par r\364le sym\351trique, \+
x et y sont respectivement colin\351aires \340 \"b vectoriel c\" et \"
a vectoriel c\". Quelques doubles produits vectoriels devraient permet
tre d'\351tablir cel\340 !" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "E
xercice 13" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "v1:=vector([1,
1,0]):v2:=vector([1,0,-1]):v3:=vector([3,4,0]):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "arccos(dotprod(v1,v2)/sqrt(dotprod(v1,v1)*dotp
rod(v2,v2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "arccos(dotprod(v1,v3)/sqr
t(dotprod(v1,v1)*dotprod(v3,v3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'arccosG6#,$*$-%%sqrtG6#\"\"#\"\"\"#\"\"(\"#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "arccos(dotprod(v2,v3)/sqrt(dotprod(v2,v2)*dotp
rod(v3,v3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arccosG6#,$*$-%%sq
rtG6#\"\"#\"\"\"#\"\"$\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "evalf(%+%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_Vau7!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi/3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+^v>Z5!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 11 "Exercice 14" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f:=vec
tor([2,1,0]):x:=vector([x1,x2,x3]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "solve(dotprod(x-a*f,f)=0,\{a\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#/%\"aG,&%#x1G#\"\"#\"\"&*&#\"\"\"F*F-%#x2GF-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(x-2*a*f);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,&%#x1G#!\"$\"\"&*&#\"\"%F+\"\"\"%#x2GF/!
\"\",&F0#\"\"$F+*&#F.F+F/F(F/F1%#x3G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "genmatrix(convert(%,list),[x1,x2,x3]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#!\"$\"\"&#!\"%F*\"\"!7%F+#\"\"$
F*F-7%F-F-\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "C'est bien u
ne matrice orthogonale, sym\351trique, de trace 1, qui stabilise e3 (q
ui est orthogonal \340 f)" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Ex
ercice 15" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "generateur_exo
:=proc(v0,theta)\nlocal v,nv0,x,y,z;\nv:=vector([x,y,z]):nv0:=sqrt(dot
prod(v0,v0,orthogonal)): RETURN(genmatrix(convert(evalm(dotprod(v,v0,o
rthogonal)/nv0^2*v0+cos(theta)*(v-dotprod(v,v0,orthogonal)/nv0^2*v0)+s
in(theta)*crossprod(v0,v)/nv0),list),[x,y,z]))\nend:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "generateur_exo(vector([1,0,1]),Pi/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"\"\"\"#,$*$-%%sqr
tG6#F*F)#!\"\"F*F(7%,$F,F(\"\"!F+7%F(F3F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "generateur_exo(vector([1,0,-1]),Pi/3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"$\"\"%,$*&-%%sqrtG6#F)\"\"
\"-F.6#\"\"#F0#F0F*#!\"\"F*7%,$F,F5#F0F3F87%F5F+F(" }}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 11 "Exercice 16" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 299 "analyse_rotation:=proc(A)\nlocal v0,theta,w:\nv0:=op
(kernel(A-Matrix(3,shape=identity))):\ntheta:=arccos((trace(A)-1)/2);
\nif v0[1]=0 then w:=vector([1,0,0]) else w:=vector([v0[2],-v0[1],0]) \+
fi:\nif evalf(dotprod(A&*w,crossprod(v0,w)))>0 then RETURN(evalm(v0),t
heta) else RETURN(evalm(-v0),theta) fi\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "A:=<<8 | 1 | -4> , <-4 | 4 | -7> , <1 | 8 | 4>>/
9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"(!)yu%-%'MAT
RIXG6#7%7%#\"\")\"\"*#\"\"\"F0#!\"%F07%F3#\"\"%F0#!\"(F07%F1F.F6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(A&*transpose(A));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7
%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "analyse_rotation(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$-%'vectorG6#7%\"\"$!\"\"F(-%'arccosG6##\"\"(\"#=" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 73 "B:=<<3 | 1 | sqrt(6)> , <1 | 3 | -sqrt(6)
> , <-sqrt(6) | sqrt(6) | 2>>/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"BG-%'RTABLEG6$\"(#4dW-%'MATRIXG6#7%7%#\"\"$\"\"%#\"\"\"F0,$*$-%%sqrt
G6#\"\"'F2F17%F1F.,$F4#!\"\"F07%F:F3#F2\"\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "evalm(B&*transpose(B));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "an
alyse_rotation(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'vectorG6#7%\"
\"\"F'\"\"!,$%#PiG#F'\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
74 "C:=<<3 | 1 | sqrt(6)> , <1 | 3 | -sqrt(6)> , <sqrt(6) | -sqrt(6) |
 -2>>/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6$\")gw\"y
\"-%'MATRIXG6#7%7%#\"\"$\"\"%#\"\"\"F0,$*$-%%sqrtG6#\"\"'F2F17%F1F.,$F
4#!\"\"F07%F3F:#F<\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
evalm(C&*transpose(C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6
#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "det(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Normal : on a chang\351 L3 en -L3 \+
dans B !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "kernel(C-Matrix
(3,shape=identity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#
7%\"\"\"F(\"\"!-F%6#7%F),$*$-%%sqrtG6#\"\"'F(!\"\"F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "kernel(C+Matrix(3,shape=identity));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7%!\"\"\"\"\"*$-%%sqrtG6
#\"\"'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "C est donc une r\351f
lexion.\nN.B. :" }}{PARA 0 "" 0 "" {TEXT -1 131 " - Le calcul du secon
d noyau \351tait inutile; pourquoi ?\n - En fait, C \351tant sym\351tr
ique, c'\351tait forc\351ment une r\351flexion (pourquoi ?)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "D:=<<8 | -1 | -4> , <-4 | -4
 | -7> , <1 | -8 | 4>>/9;" }}{PARA 8 "" 1 "" {TEXT -1 54 "Error, attem
pting to assign to `D` which is protected\n" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 46 "Classique : D est l'op\351rateur de d\351rivation..." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "DD:=<<8 | -1 | -4> , <-4 |
 -4 | -7> , <1 | -8 | 4>>/9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DDG
-%'RTABLEG6$\")%3zy\"-%'MATRIXG6#7%7%#\"\")\"\"*#!\"\"F0#!\"%F07%F3F3#
!\"(F07%#\"\"\"F0#!\")F0#\"\"%F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
100 "DD est obtenu en changent C2 par con oppos\351e dans A : DD va do
nc \352tre orthogonale de d\351terminant -1." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "evalm(DD&*transpose(DD));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(DD);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "DD n'est \+
pas sym\351trique donc ne correspond pas \340 une r\351flexion : le pr
emier calcul va le confirmer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "kernel(DD-Matrix(3,shape=identity));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "D\351termin
ons l'axe de la rotation (qui est aussi l'orthogonal du plan de r\351f
lexion)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "kernel(DD+Matri
x(3,shape=identity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6
#7%#\"\"\"\"\"$#\"\"&F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 
"f:=op(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%'vectorG6#7%#\"
\"\"\"\"$#\"\"&F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "thet
a:=arccos((trace(DD)+1)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thet
aG-%'arccosG6##\"#<\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "On or
iente l'axe par f. On va regarder l'image d'un vecteur v du plan : est
-elle \"du cot\351 de \"f vectoriel v\" ?" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "v:=vector([5,-1,0]):dotprod(crossprod(f,v),DD&*v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!$b%\"#F" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 47 "Le sinus est donc n\351gatif. Pour v\351rification...
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "%/dotprod(v,v)/sqrt(dot
prod(f,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"#N\"\"
\"#!\"\"\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "%^2+(17/18)
^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 44 "Ouf ! (sauriez-vous expliquer le calcul ???)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Ainsi, DD est la r\351flexion par
 rapport \340 l'orthogonal de f, compos\351e avec la rotation d'axe di
rig\351 et orient\351 par f et d'angle -arccos(17/18)." }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 11 "Exercice 17" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "evalm(crossprod(vector([a,b,c]),vector([x1,x2,x3])));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&*&%\"bG\"\"\"%#x3G
F*F**&%\"cGF*%#x2GF*!\"\",&*&F-F*%#x1GF*F**&%\"aGF*F+F*F/,&*&F4F*F.F*F
**&F)F*F2F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "genmatrix(
convert(%,list),[x1,x2,x3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat
rixG6#7%7%\"\"!,$%\"cG!\"\"%\"bG7%F*F(,$%\"aGF+7%,$F,F+F/F(" }}}}}
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