\newcommand{\pstLineABD}[4][]{\pstLineAB[#1]{#2}{#3}\bput(1){$\cal #4$}}
\newcommand{\pstLineABd}[4][]{\pstLineAB[#1]{#2}{#3}\bput(1){$#4$}}
\newcommand{\pstick}[1]{\rput(#1){\psline(0,-.05)(0,0.05)}}
% \pstplan{lx}{ly}{nom}
\newcommand{\pstplan}[3]{\psline(0,0)(#2;70)\psline(0,0)(#1;0)
\rput(#2;70){\psline(0,0)(#1;0)\uput[dl](#1;0){$\cal #3$}}
\rput(#1;0){\psline(0,0)(#2;70)}}
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\begin{pspicture}(4,3.5)
\psset{dotscale=0.1}
\pstGeonode[PosAngle=190](1,3){M}\pstGeonode(3,4){M'}\psline{->}(M)(M')
\pstGeonode[PosAngle=190](2,2){A}\pstGeonode(4,3){B} \psline{->}(A)(B)
\psset{PointSymbol=none}
\pstGeonode[PosAngle=190](1.8,1){u}\pstGeonode(3.8,2){v}
\pcline{->}(u)(v)\Bput{$\vec u$}
\end{pspicture}
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\begin{pspicture}(-1,-1)(5.5,2)%\psgrid
\psset{dotscale=0.1, labelsep=3pt, arrows=->}
\pstGeonode[PosAngle=-130]{A}
\pstGeonode[PosAngle=90](3;20){B}
\pstGeonode(3.2;-10){C}
\ncline{A}{B}\Aput{$\vec u$}
\ncline{B}{C}\Aput{$\vec v$}
\ncline{A}{C}\aput{:U}(.6){$\vec u+\vec v$}
\psset{PointSymbol=none}
\pstGeonode(.2,1.2){uA}
\pstTranslation{A}{B}{uA}{uB}
\ncline{uA}{uB}\Aput{$\vec u$}
\pstGeonode(4.5,2){vB}
\pstTranslation{B}{C}{vB}{vC}
\ncline{vB}{vC}\Aput{$\vec v$}
\end{pspicture}
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\begin{pspicture}(8,2)%\psgrid
\small
\rput{10}(3,1){
% \psaxes[labels=none, ticks=x, ticksize=1pt]{<->}(-2,0)(3,0)
\pcline{<->}(-2,0)(3,0)
\multips(-1,0)(1,0){4}{\psline(0,-.05)(0,.05)}
\bput(0){N}\bput(1){M}\bput(0.4){A}\aput(0.6){B}
\psline{->}(0,0)(1,0)
\pcline{->}(-.7,.5)(.3,.5)\Aput{$\vec u$}
\pcline{->}(1.2,-1)(4.2,-1)
\bput(0){A'}\bput(1){M'}
\pcline{->}(1.2,-1)(2.2,-1)\bput(1){B'}
}
\end{pspicture}
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\begin{pspicture}(-2,-.5)(5,5.5)%\psgrid
\psset{dotscale=0.1}
% triangle
\pstGeonode[PosAngle=-110](-1,3){A}
\pstGeonode(2,3){C}
\pstGeonode[PosAngle=90](-0.5,5){B}
\pcline{->}(A)(B)\Aput{$\vec u$}
\pcline{->}(A)(C)\Aput{$\vec v$}
\pcline{->}(B)(C)\Aput{$\vec X = \vec v - \vec u$}
% parallélogramme
\pstGeonode[PosAngle=-90](0,0){A}
\pstGeonode[PosAngle=180](2;110){B}
\pstGeonode(3;0){C}
\pstMiddleAB{B}{C}{I}
\pstTranslation[PointName=none]{A}{C}{B}{D}
\psline{->}(A)(B)\psline{->}(A)(C)
\psset{linestyle=dashed}
\psline{->}(A)(D)\psline(B)(D)(C)
\pstSegmentMark{B}{I}\pstSegmentMark{I}{C}
\end{pspicture}
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\begin{pspicture}(5,3)
\psset{PointSymbol=+, PosAngle=-90}
\pstGeonode(1,1){A}
\pstGeonode(4,1.5){M}
\pstLineAB[nodesep=-.5,nodesepB=-.8]{A}{M}
\bput(1){$\cal D$}
\pstHomO[HomCoef=.2]{A}{M}{B}
\pcline[linewidth=1.5\pslinewidth]{->}(A)(B)
\psset{PointSymbol=none}
\pstGeonode(2,2){u}
\pstTranslation{A}{B}{u}{uu}
\pcline{->}(u)(uu)\Aput{$\vec u$}
\end{pspicture}
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\begin{pspicture}(-1,-1.5)(5,2)
\psset{dotscale=0.1}
\pspolygon(-1,-1.5)(0,2)(6,2)(5,-1.5)
% A, B, C
\pstGeonode[PosAngle=-170](0,0){A}
\pstGeonode[PosAngle=110](1.5;30){C}
\pstGeonode[PosAngle=-90](1;-15){B}
\pcline{->}(A)(C)\Aput{$\vec v_2$}
\pcline{->}(A)(B)\Bput{$\vec v_1$}
% B', C'
\pstHomO[PosAngle=90, HomCoef=1.8]{A}{C}{C'}
\pstHomO[PosAngle=-60, HomCoef=2.2]{A}{B}{B'}
\psset{linestyle=dashed}
\pcline{->}(A)(C')\aput(.8){$\lambda \vec v_2$}
\pcline{->}(A)(B')\bput(.8){$\lambda \vec v_1$}
% M
\pstTranslation{A}{B'}{C'}{M}
\psline(C')(M)(B')
\end{pspicture}
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\begin{pspicture}(5,1.5)
\psset{PosAngle=-90, PointSymbol=+}
\pstGeonode(2,1.2){A}
\pstGeonode(5,.5){A'}
% droites
\psset{PointSymbol=none}
\pstGeonode(7,0.7){BB}
\pstTranslation{A'}{BB}{A}{AA}
\pstLineABD[nodesepA=-.8, nodesepB=-4]{A}{AA}{D}
\pstLineABD[nodesepA=-4, nodesepB=-.8]{A'}{BB}{D'}
% vecteurs
\psset{linewidth=1.5\pslinewidth}
\pstHomO[HomCoef=.4]{A}{AA}{Au}
\pcline{->}(A)(Au)\Aput{$\vec u$}
\pstHomO[HomCoef=-1]{A'}{BB}{A'u}
\pcline{->}(A')(A'u)\Aput{$\vec u'$}
\end{pspicture}
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\begin{pspicture}(6,3)
\psset{dotscale=0.1}
\rput{10}(2,0){
\pstGeonode[PosAngle=90](1,2){A}
\pstick{A}
\pstGeonode[PointSymbol=none](3,2){aa}
\ncline{->}{A}{aa}\Aput{$\vec u$}
\pstLineABD[nodesep=-1]{A}{aa}{D}
}
\rput(3,1){
\pspolygon(-1,-.8)(0,1.1)(4,1.1)(3,-.8)
\uput[dl](4,1.1){$\cal P$}
\pstGeonode[PosAngle=-160]{B}
\psset{PointSymbol=none}
\pstGeonode(1;-10){v1}
\pstGeonode(2;-10){v11}
\ncline{->}{B}{v1}\Bput{$\vec v_1$}
\pstLineAB[nodesepB=-.2]{B}{v11}
\pstGeonode(.9;35){v2}
\pstGeonode(1.2;35){v22}
\ncline{->}{B}{v2}\Aput{$\vec v_1$}
\pstLineAB[nodesepB=-.1]{B}{v22}
\pstTranslation{B}{v11}{v22}{u}
\ncline{->}{B}{u}\Aput{$\vec u$}
\psline[linestyle=dashed](v22)(u)(v11)
}
\end{pspicture}
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\begin{pspicture}(-2,0)(9,4)
\psset{dotscale=0.1, labelsep=3pt}
\rput(0,2){
\pstplan{4}{1.5}{P}
\rput(1,.5){
\pstGeonode[PosAngle=-170]{A}
\pcline{->}(A)(2;0)\lput(1.2){$\vec u_1$}
\pcline{->}(A)(1;25)\lput(1.3){$\vec u_2$}
}}
\pstplan{4.2}{1.6}{P'}
\rput(1,.5){
\pstGeonode[PosAngle=-170]{B}
\pcline{->}(B)(2.2;-6)\lput(1.2){$\vec v_1$}
\pcline{->}(B)(2;3)\lput(1.2){$\vec u_1$}
\pcline{->}(B)(1;30)\lput(1.2){$\vec u_2$}
\pcline{->}(B)(1.1;60)\Aput{$\vec v_2$}
}
\end{pspicture}
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\begin{pspicture}(6,3)
\psset{dotscale=0.1}
\pstplan{5}{2}{P}
\rput(1,.6){
\pstGeonode[PosAngle=-170]{O}
\pstGeonode(2;-10){A} \ncline{->}{O}{A}
\pstGeonode(3.5;3){B} \ncline{->}{O}{B}
\pstGeonode(1.5;40){C} \ncline{->}{O}{C}
}
\end{pspicture}
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\begin{pspicture}(-4,-4)(4,4)
\psset{dotscale=0.1}
\pstGeonode[PosAngle=-80]{O}
\pcline{->}(O)(1;0)\bput(.8){$\vec j$}
\pcline{->}(O)(.8;90)\aput(.8){$\vec k$}
\pcline{->}(O)(2.5;-130)\bput(.8){$\vec i$}
\psset{PointSymbol=none}
\pstGeonode(3;0){yj} \pcline{->}(O)(yj)\aput(.8){$\vec y_j$}
\pstGeonode(2.5;90){zk} \pcline{->}(O)(zk)\bput(.8){$\vec z_k$}
\pstGeonode(1.5;-130){xi}\pcline{->}(O)(xi)\bput(.8){$\vec x_i$}
\pcline(O)(4;0)\bput(1){$y$}
\pcline(O)(3.5;90)\aput(1){$z$}
\pcline(O)(3.5;-130)\aput(1){$x$}
\pstTranslation{O}{yj}{xi}{xy}
\pstTranslation{O}{xi}{zk}{zx}
\pstTranslation{O}{yj}{zk}{zy}
\pstTranslation[PointSymbol=*]{O}{xi}{zy}{M}
\psline{->}(O)(M)
\psline[linestyle=dashed](xy)(xi)(zx)(zk)(zy)(M)(zx)
\psline[linestyle=dashed](zy)(yj)(xy)(M)
\end{pspicture}
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\begin{pspicture}(-4,-1)(4,4)
\psset{PosAngle=-90}
\pstGeonode{A}
\pstGeonode(1.5;-35){B}
\pstGeonode[PosAngle=10](2.5;5){C}
\pstTranslation{A}{B}{C}{M}
\pstGeonode[PosAngle=80](2.5;62){D}
\psset{CodeFig=true, CodeFigColor=black}
\pstMiddleAB[PosAngle=170]{A}{D}{I}
\pstMiddleAB[PosAngle=-80]{B}{C}{J}
\pstMiddleAB[SegmentSymbol=pstslash,PosAngle=10]{D}{J}{K}
\pstTranslation[PointSymbol=none]{C}{A}{A}{N'}
\pstTranslation[PosAngle=90]{A}{I}{N'}{N}
\pspolygon(A)(B)(M)(C)(B)(D)
\psline(D)(J)\psline(D)(C)\psline(J)(M)
\psline(A)(N')(N)
\psset{linestyle=dashed}
\psline(A)(J)\psline(A)(C)\psline(I)(K)
\psline(I)(M)
\pcline[nodesepA=1.3, linestyle=solid](I)(M)
\end{pspicture}
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\begin{pspicture}(-1,-1)(4,3)
\psset{PosAngle=-70}
\pstGeonode[PosAngle=-100]{A}
\pstGeonode(3;0){B}
\pstGeonode(1.5;40){D}
\pstTranslation[PosAngle=-30]{A}{B}{D}{C}
\pstGeonode[PosAngle=180](3;90){E}
\pstTranslation[PosAngle=-10]{A}{B}{E}{F}
\pstTranslation[PosAngle=100]{A}{D}{E}{H}
\pstTranslation[PosAngle=10]{A}{B}{H}{G}
{\psset{CodeFig=true, CodeFigColor=black}
\pstMiddleAB[PosAngle=170]{E}{H}{J}
\pstMiddleAB[PosAngle=-10]{F}{G}{I}
\pstMiddleAB[PosAngle=-90]{A}{B}{M}
\pstMiddleAB[PosAngle=-10]{B}{C}{P}
\pstMiddleAB[PosAngle=-10]{B}{F}{N}
}
\pspolygon(A)(B)(F)(E)
\psline(E)(F)(G)(H)\psline(B)(C)(G)
\pstInterLL[PosAngle=-90]{E}{G}{H}{F}{O}
\pstInterLL[PosAngle=90]{H}{I}{E}{G}{K}
\psline(E)(G)\psline(H)(F)\psline(H)(I)
\pstTranslation[PosAngle=-90]{E}{A}{K}{L}
\psset{linestyle=dashed}
\pspolygon(A)(D)(C)
\psline(D)(H)\psline(K)(L)
\end{pspicture}
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