{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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q28:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=<2,1,0>: B:=<4,-1,0>: C:=<6,3,0>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "CrossProduct(B-A,C-A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")OuOL-%'MATRIXG6#7%7#\"\"!F+7#\"# 7&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Answer: 12 / 2 = 6" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q30:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=<1,3,0>: B:=<2,0,8> : C:=<0,2,2>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nrml:=Cr ossProduct(B-A,C-A)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nrmlG-%'R TABLEG6%\")))GYL-%'MATRIXG6#7%7#\"\"\"7#!\"&7#!\"#&%'VectorG6#%'column G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(nrml.)=(nrml.A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG\"\"\"*&\"\"&F&%\"yGF&! \"\"*&\"\"#F&%\"zGF&F*!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q32:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=<1,1,0>: B:=<-2,0,2>: \+ C:=<-2,0,-3>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A.CrossPro duct(B,C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#5" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "Answer: 10" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q34: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:=<1,3,6>: B:=<3,7,12 >: C:=<8,8,9>: d:=<2,2,8>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(A-d).CrossProduct(B-d,C-d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#!*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Answer: 90 / 6 = 15" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q36:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=<4,2,9>: B:=<3,2,1>: C:=<-4,6 ,9>: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A.CrossProduct(B,C );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$?#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 48 "Answer: Linearly independent (not in one plane)." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Q38a:" }}{PARA 0 "" 0 "" {TEXT -1 34 "Brute force (computational ) proof:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A:=: \+ B:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C:=Cros sProduct(A,B):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "expand(ad d(C[i]^2,i=1..3)-add(A[i]^2,i=1..3)*add(B[i]^2,i=1..3)+add(A[i]*B[i],i =1..3)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "b: (same brute-force approach)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " B:=: C:= : d:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "res:=CrossProduct(B,CrossProduct(C,d))-add(B[i]*d[i],i=1..3)*C+add (B[i]*C[i],i=1..3)*d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG-%'RTA BLEG6%\")gzF1F1F1F8F1F17#,**&F;F1,&*&F7F1F@F1F1*&F>F1F5F1F9F1F1*&FDF1F 2F1F9*&FBF1F7F1F9*&FHF1F5F1F17#,**&FDF1FF1F9* &FHF1F@F1F1&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(expand(res[i]),i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "c: Replacing Vector B in P art b by A x B yields: (A x B) x (C x D) = ((A x B) . D) C - (( A x B) . C) D = (A B D) C - (A B C) D (using the books' notation fo r triple product)." }}}}{MARK "34 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 33367436 33462888 34177960 }{RTABLE M7R0 I5RTABLE_SAVE/33367436X*%)anythingG6"6"[gl!#%!!!"$"$""!F'"#7F& } {RTABLE M7R0 I5RTABLE_SAVE/33462888X*%)anythingG6"6"[gl!#%!!!"$"$"""!"&!"#F& } {RTABLE M7R0 I5RTABLE_SAVE/34177960X*%)anythingG6"6"[gl!#%!!!"$"$,**&%#b2G""",&*&%#c1GF*%#d2 GF*F**&%#c2GF*%#d1GF*!""F*F**&%#b3GF*,&*&%#c3GF*F1F*F**&F-F*%#d3GF*F2F*F2*&,(*& %#b1GF*F1F*F**&F)F*F.F*F**&F4F*F9F*F*F*F-F*F2*&,(*&F=F*F-F*F**&F)F*F0F*F**&F4F* F7F*F*F*F1F*F*,**&F4F*,&*&F0F*F9F*F**&F7F*F.F*F2F*F**&F=F*F+F*F2*&F;F*F0F*F2*&F AF*F.F*F*,**&F=F*F5F*F**&F)F*FGF*F2*&F;F*F7F*F2*&FAF*F9F*F*F& }