{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Tim es" 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 129 "USING MAPLE TO DETERMIN E THE ROTATION MATRIX FOR A COUNTERCLOCKWISE ROTATION BY ANGLE t ABOUT THE AXIS [p,q,r] THROUGH THE ORIGIN" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Define the axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "axis1:=Matrix([[p],[q],[r]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&axis1G-%'RTABLEG6%\"+y(>%[X-%'MATRIXG6#7%7#%\"pG7#%\"qG7#%\"rG%'Ma trixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Write the matrix A1 that rotates about the z axis and zeroes out the y component of the axis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "n1:=sqrt(p^2+q^2); A1:= Matrix([[p/n1,q/n1,0],[-q/n1,p/n1,0],[0,0,1]]);Determinant(A1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G*$,&*$)%\"pG\"\"#\"\"\"F+*$)%\"q GF*F+F+#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'RTABLEG6%\"+1 *>%[X-%'MATRIXG6#7%7%*&%\"pG\"\"\",&*$)F/\"\"#F0F0*$)%\"qGF4F0F0#!\"\" F4*&F7F0F1F8\"\"!7%,$F:F9F.F;7%F;F;F0%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ax is2:=simplify(Multiply(A1,axis1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&axis2G-%'RTABLEG6%\"+M+U[X-%'MATRIXG6#7%7#*$,&*$)%\"pG\"\"#\"\"\"F4 *$)%\"qGF3F4F4#F4F37#\"\"!7#%\"rG%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Write the matrix A2 that rotates about the y axis and ro tates the axis in the direction of the x axis." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 118 "n2:=sqrt(p^2+q^2+r^2); A2:=Matrix([[axis2(1)/ n2,0,axis2(3)/n2],[0,1,0],[-axis2(3)/n2,0,axis2(1)/n2]]);Determinant(A 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n2G*$,(*$)%\"pG\"\"#\"\"\"F+ *$)%\"qGF*F+F+*$)%\"rGF*F+F+#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#A2G-%'RTABLEG6%\"+)4?%[X-%'MATRIXG6#7%7%*&,&*$)%\"pG\"\"#\"\"\"F4*$) %\"qGF3F4F4#F4F3,(*$F1F4F4*$F6F4F4*$)%\"rGF3F4F4#!\"\"F3\"\"!*&F>F4F9F ?7%FAF4FA7%,$FBF@FAF.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "axis3:=simplify(Multi ply(A2,axis2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&axis3G-%'RTABLEG 6%\"+E-U[X-%'MATRIXG6#7%7#*$,(*$)%\"pG\"\"#\"\"\"F4*$)%\"qGF3F4F4*$)% \"rGF3F4F4#F4F37#\"\"!F<%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Write the matrix A3 that rotates by the angle t about the x axis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A3:=Matrix([[1,0,0],[0, cos(t),-sin(t)],[0,sin(t),cos(t)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#A3G-%'RTABLEG6%\"+!H?%[X-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/-%$cosG6 #%\"tG,$-%$sinGF3!\"\"7%F/F6F1%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Calculate the matrix M = A1^(-1) A2^(-1) A3 A2 A1. This rotates the axis to the direction of the x axis, performs the rotatio n by the angle t, and then rotates the axis back to its original posit ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "M:=simplify(Multipl y(MatrixInverse(A1),Multiply(MatrixInverse(A2),Multiply(A3,Multiply(A2 ,A1)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"+mCU[ X-%'MATRIXG6#7%7%*&,(*&-%$cosG6#%\"tG\"\"\")%\"qG\"\"#F5F5*&F1F5)%\"rG F8F5F5*$)%\"pGF8F5F5F5,(*$F=F5F5*$F6F5F5*$F:F5F5!\"\",$*&,(*(F1F5F>F5F 7F5F5*(-%$sinGF3F5F?#F5F8F;F5F5*&F>F5F7F5FCF5F?FCFC,$*&,,**F>F5F;F5F?F KF1F5F5*(FIF5F=F5F7F5FC*&FIF5)F7\"\"$F5FC*(FIF5F7F5F:F5FC*(F>F5F;F5F?F KFCF5F?#!\"$F8FC7%,$*&,(FGF5FHFCFLFCF5F?FCFC*&,(*&F1F5F=F5F5F9F5*$F6F5 F5F5F?FC,$*&,,**F7F5F;F5F?FKF1F5F5*&FIF5)F>FTF5F5*(FIF5F>F5F6F5F5*(FIF 5F>F5F:F5F5*(F7F5F;F5F?FKFCF5F?FWFC7%,$*&,(*(F1F5F;F5F>F5F5*(FIF5F7F5F ?FKF5*&F;F5F>F5FCF5F?FCFC,$*&,(*(F1F5F;F5F7F5F5*(FIF5F>F5F?FKFC*&F;F5F 7F5FCF5F?FCFC*&,(FinF5F0F5*$F:F5F5F5F?FC%'MatrixG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 65 "Do some algebraic simplifications to get the equiv alent matrix N." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "M(1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,**%\"pG\"\"\"%\"rGF(,(*$)F'\" \"#F(F(*$)%\"qGF-F(F(*$)F)F-F(F(#F(F--%$cosG6#%\"tGF(F(*(-%$sinGF6F(F, F(F0F(!\"\"*&F9F()F0\"\"$F(F;*(F9F(F0F(F2F(F;*(F'F(F)F(F*F3F;F(F*#!\"$ F-F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "N13:=-(p*r*cos(t)-s in(t)*q*n2-p*r)/n2^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N13G,$*&,( *(-%$cosG6#%\"tG\"\"\"%\"rGF-%\"pGF-F-*(-%$sinGF+F-%\"qGF-,(*$)F/\"\"# F-F-*$)F3F7F-F-*$)F.F7F-F-#F-F7!\"\"*&F.F-F/F-F=F-F4F=F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "M(1,3)-N13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,,**%\"pG\"\"\"%\"rGF(,(*$)F'\"\"#F(F(*$)%\"qGF-F(F (*$)F)F-F(F(#F(F--%$cosG6#%\"tGF(F(*(-%$sinGF6F(F,F(F0F(!\"\"*&F9F()F0 \"\"$F(F;*(F9F(F0F(F2F(F;*(F'F(F)F(F*F3F;F(F*#!\"$F-F;*&,(*(F4F(F)F(F' F(F(*(F9F(F0F(F*F3F;*&F)F(F'F(F;F(F*F;F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "M(2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,**%\"qG\"\"\"%\"rGF(,(*$)%\"pG\"\"#F(F(*$)F' F.F(F(*$)F)F.F(F(#F(F.-%$cosG6#%\"tGF(F(*&-%$sinGF6F()F-\"\"$F(F(*(F9F (F-F(F0F(F(*(F9F(F-F(F2F(F(*(F'F(F)F(F*F3!\"\"F(F*#!\"$F.F@" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "N23:=-(q*r*cos(t)+sin(t)*p*n 2-q*r)/(n2^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$N23G,$*&,(*(-%$cosG6#%\"tG\"\"\"%\"rGF-%\"qGF-F-* (-%$sinGF+F-%\"pGF-,(*$)F3\"\"#F-F-*$)F/F7F-F-*$)F.F7F-F-#F-F7F-*&F.F- F/F-!\"\"F-F4F>F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "M(2,3) -N23;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,,**%\"qG\"\"\"%\"rGF(,(* $)%\"pG\"\"#F(F(*$)F'F.F(F(*$)F)F.F(F(#F(F.-%$cosG6#%\"tGF(F(*&-%$sinG F6F()F-\"\"$F(F(*(F9F(F-F(F0F(F(*(F9F(F-F(F2F(F(*(F'F(F)F(F*F3!\"\"F(F *#!\"$F.F@*&,(*(F4F(F)F(F'F(F(*(F9F(F-F(F*F3F(*&F)F(F'F(F@F(F*F@F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "N:=Matrix([[M[1,1],M[1,2],N13],[M[2,1],M[2,2],N23],[M[3,1],M[3,2], M[3,3]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'RTABLEG6%\"+IDU [X-%'MATRIXG6#7%7%*&,(*&-%$cosG6#%\"tG\"\"\")%\"qG\"\"#F5F5*&F1F5)%\"r GF8F5F5*$)%\"pGF8F5F5F5,(*$F=F5F5*$F6F5F5*$F:F5F5!\"\",$*&,(*(F1F5F>F5 F7F5F5*(-%$sinGF3F5F?#F5F8F;F5F5*&F>F5F7F5FCF5F?FCFC,$*&,(*(F1F5F;F5F> F5F5*(FIF5F7F5F?FKFC*&F;F5F>F5FCF5F?FCFC7%,$*&,(FGF5FHFCFLFCF5F?FCFC*& ,(*&F1F5F=F5F5F9F5*$F6F5F5F5F?FC,$*&,(*(F1F5F;F5F7F5F5*(FIF5F>F5F?FKF5 *&F;F5F7F5FCF5F?FCFC7%,$*&,(FPF5FQF5FRFCF5F?FCFC,$*&,(FhnF5FinFCFjnFCF 5F?FCFC*&,(FYF5F0F5*$F:F5F5F5F?FC%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Now assume that p^2+q^2+r^2=1 to make the formulas look n icer, getting matrix N2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N1:=N*(p^2+q^2+r^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1G-%'RTAB LEG6%\"+AFU[X-%'MATRIXG6#7%7%,(*&-%$cosG6#%\"tG\"\"\")%\"qG\"\"#F4F4*& F0F4)%\"rGF7F4F4*$)%\"pGF7F4F4,(*(F0F4F=F4F6F4!\"\"*(-%$sinGF2F4,(*$F< F4F4*$F5F4F4*$F9F4F4#F4F7F:F4F@*&F=F4F6F4F4,(*(F0F4F:F4F=F4F@*(FBF4F6F 4FDFHF4*&F:F4F=F4F47%,(F?F@FAF4FIF4,(*&F0F4F " 0 "" {MPLTEXT 1 0 158 "N2:=Matrix([[c*q^2+c*r^2+p^2,-c*p*q-s*r+p*q,-c*r*p+s*q+r*p],[-c*p *q+s*r+p*q,c*p^2+c*r^2+q^2,-c*r*q-s*p+r*q],[-c*r*p-s*q+r*p,-c*r*q+s*p+ r*q,c*p^2+c*q^2+r^2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N2G-%'RT ABLEG6%\"+1b%)GX-%'MATRIXG6#7%7%,(*&%\"cG\"\"\")%\"qG\"\"#F1F1*&F0F1)% \"rGF4F1F1*$)%\"pGF4F1F1,(*(F0F1F:F1F3F1!\"\"*&F:F1F3F1F1*&F7F1%\"sGF1 F=,(*(F0F1F:F1F7F1F=*&F:F1F7F1F1*&F3F1F@F1F17%,(*(F0F1F:F1F3F1F=*&F:F1 F3F1F1*&F7F1F@F1F1,(*&F0F1F9F1F1*&F0F1F6F1F1*$F2F1F1,(*(F0F1F3F1F7F1F= *&F:F1F@F1F=*&F3F1F7F1F17%,(*(F0F1F:F1F7F1F=*&F:F1F7F1F1*&F3F1F@F1F=,( *(F0F1F3F1F7F1F=*&F:F1F@F1F1*&F3F1F7F1F1,(*&F0F1F9F1F1*&F0F1F2F1F1*$F6 F1F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "It turns out we can represent thi s formula using quaternions--expressions of the form a+bi+cj+dk, where i^2=j^2=k^2=ijk=-1. " }}{PARA 0 "" 0 "" {TEXT -1 112 "First verify t hat if w=cos(t/2)+(pi+qj+rk)sin(t/2) where p^2+q^2+r^2=1, then w^(-1)= cos(t/2)=(pi+qj+rk)sin(t/2)." }}{PARA 0 "" 0 "" {TEXT -1 190 "Then ver ify (takes longer) that if u=xi+yj+zk and u'=wuw^(-1), then the i,j,k \+ coordinates of u' are the coordinates of u rotated by the angle t abou t the axis [p.q.r] given by the matrix N2." }}}{EXCHG }{PARA 11 "" 1 " " {TEXT -1 0 "" }}{EXCHG }}{MARK "0 0 0" 28 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 4548419778 4548419906 4548420034 4548420098 4548420226 4548420290 4548422466 4548422530 4548422722 4528845506 }{RTABLE M7R0 I7RTABLE_SAVE/4548419778X,%)anythingG6"F%[gl!"%!!!#$"$""%"pG%"qG%"rGF% } {RTABLE M7R0 I7RTABLE_SAVE/4548419906X,%)anythingG6"F%[gl!"%!!!#*"$"$*&%"pG""",&*$F'""#F(*$% "qGF+F(#!""F+,$*&F-F(F)F.F/""!F1F&F2F2F2F(F% } {RTABLE M7R0 I7RTABLE_SAVE/4548420034X,%)anythingG6"F%[gl!"%!!!#$"$""*$,&*$%"pG""#"""*$%"qGF *F+#F+F*""!%"rGF% } {RTABLE M7R0 I7RTABLE_SAVE/4548420098X,%)anythingG6"F%[gl!"%!!!#*"$"$*&,&*$%"pG""#"""*$%"qGF *F+#F+F*,(*$F)F*F+*$F-F*F+*$%"rGF*F+#!""F*""!,$*&F3F+F/F4F5F6F+F6F8F6F&F% } {RTABLE M7R0 I7RTABLE_SAVE/4548420226X,%)anythingG6"F%[gl!"%!!!#$"$""*$,(*$%"pG""#"""*$%"qGF *F+*$%"rGF*F+#F+F*""!F1F% } {RTABLE M7R0 I7RTABLE_SAVE/4548420290X,%)anythingG6"F%[gl!"%!!!#*"$"$"""""!F'F'-I$cosG6$%*pr otectedG%(_syslibG6#%"tG-I$sinG6$F+F,F-F',$F/!""F(F% } {RTABLE M7R0 I7RTABLE_SAVE/4548422466X,%)anythingG6"F%[gl!"%!!!#*"$"$*&,(*&-I$cosG6$%*protec tedG%(_syslibG6#%"tG"""%"qG""#F0*&F)F0%"rGF2F0*$%"pGF2F0F0,(*$F6F2F0*$F1F2F0*$F 4F2F0!"",$*&,(*(F)F0F6F0F1F0F0*(-I$sinG6$F,F-F.F0F7#F0F2F4F0F;*&F6F0F1F0F;F0F7F ;F;,$*&,(*(F)F0F4F0F6F0F0*(FAF0F1F0F7FDF0*&F4F0F6F0F;F0F7F;F;,$*&,(F?F0F@F0FEF; F0F7F;F;*&,(*&F)F0F6F2F0F3F0*$F1F2F0F0F7F;,$*&,(*(F)F0F4F0F1F0F0*(FAF0F6F0F7FDF ;*&F4F0F1F0F;F0F7F;F;,$*&,,**F6F0F4F0F7FDF)F0F0*(FAF0F6F2F1F0F;*&FAF0F1""$F;*(F AF0F1F0F4F2F;*(F6F0F4F0F7FDF;F0F7#!"$F2F;,$*&,,**F1F0F4F0F7FDF)F0F0*&FAF0F6FinF 0*(FAF0F6F0F1F2F0*(FAF0F6F0F4F2F0*(F1F0F4F0F7FDF;F0F7F\oF;*&,(FQF0F(F0*$F4F2F0F 0F7F;F% } {RTABLE M7R0 I7RTABLE_SAVE/4548422530X,%)anythingG6"F%[gl!"%!!!#*"$"$*&,(*&-I$cosG6$%*protec tedG%(_syslibG6#%"tG"""%"qG""#F0*&F)F0%"rGF2F0*$%"pGF2F0F0,(*$F6F2F0*$F1F2F0*$F 4F2F0!"",$*&,(*(F)F0F6F0F1F0F0*(-I$sinG6$F,F-F.F0F7#F0F2F4F0F;*&F6F0F1F0F;F0F7F ;F;,$*&,(*(F)F0F4F0F6F0F0*(FAF0F1F0F7FDF0*&F4F0F6F0F;F0F7F;F;,$*&,(F?F0F@F0FEF; F0F7F;F;*&,(*&F)F0F6F2F0F3F0*$F1F2F0F0F7F;,$*&,(*(F)F0F4F0F1F0F0*(FAF0F6F0F7FDF ;*&F4F0F1F0F;F0F7F;F;,$*&,(FIF0FJF;FKF;F0F7F;F;,$*&,(FVF0FWF0FXF;F0F7F;F;*&,(FQ F0F(F0*$F4F2F0F0F7F;F% } {RTABLE M7R0 I7RTABLE_SAVE/4548422722X,%)anythingG6"F%[gl!"%!!!#*"$"$,(*&-I$cosG6$%*protecte dG%(_syslibG6#%"tG"""%"qG""#F/*&F(F/%"rGF1F/*$%"pGF1F/,(*(F(F/F5F/F0F/!""*(-I$s inG6$F+F,F-F/,(*$F5F1F/*$F0F1F/*$F3F1F/#F/F1F3F/F/*&F5F/F0F/F/,(*(F(F/F3F/F5F/F 8*(F:F/F0F/F=FAF8*&F3F/F5F/F/,(F7F8F9F8FBF/,(*&F(F/F5F1F/F2F/*$F0F1F/,(*(F(F/F3 F/F0F/F8*(F:F/F5F/F=FAF/*&F3F/F0F/F/,(FDF8FEF/FFF/,(FLF8FMF8FNF/,(FIF/F'F/*$F3F 1F/F% } {RTABLE M7R0 I7RTABLE_SAVE/4528845506X,%)anythingG6"F%[gl!"%!!!#*"$"$,(*&%"cG"""%"qG""#F)*&F (F)%"rGF+F)*$%"pGF+F),(*(F(F)F/F)F*F)!""*&F/F)F*F)F)*&F-F)%"sGF)F),(*(F(F)F/F)F -F)F2*&F/F)F-F)F)*&F*F)F5F)F2,(*(F(F)F/F)F*F)F2*&F/F)F*F)F)*&F-F)F5F)F2,(*&F(F) F/F+F)*&F(F)F-F+F)*$F*F+F),(*(F(F)F*F)F-F)F2*&F/F)F5F)F)*&F*F)F-F)F),(*(F(F)F/F )F-F)F2*&F/F)F-F)F)*&F*F)F5F)F),(*(F(F)F*F)F-F)F2*&F/F)F5F)F2*&F*F)F-F)F),(*&F( F)F/F+F)*&F(F)F*F+F)*$F-F+F)F% }