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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 44 "Area, Volume, and Torque
 in Three Dimensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 151 "The cell below illustrates how Maple can compute the d
ot product and cross product of two vectors and how it can compute the
 determinant of a matrix.  " }{TEXT 256 16 "Evaluate it now." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r
estart;with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := [[1, 2
, 3], [4, 5, 6], [1, 2, 1]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(
A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "x := [1, 2, 3]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"y := [4, 5, 6]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "dotprod(x, y);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "crossprod(x, y);" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!\"$\"\"'F'" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
150 "You can use Maple to perform some of the algebraic computations n
eeded to verify the theorems in this module.  The following cell shows
 one example.  " }{TEXT 257 16 "Evaluate it now." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 90 "**Exercise: Prove Lagrun
ge Identity: (F x G) . (H x K) = (F . H)(G . K) - (F . K)(G . H). " }
{TEXT -1 46 "[Note. The following example might be useful.]" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "assume(x1,real):assume(x2,re
al):assume(x3,real):assume(y1,real):assume(y2,real):assume(y3,real):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x := [x1, x2, x3];y := [y
1, y2, y3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7%%$x1|irG%$x2|ir
G%$x3|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7%%$y1|irG%$y2|irG%
$y3|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "LeftSide  := do
tprod(crossprod(x, y), crossprod(x, y));\nRightSide := dotprod(x, x) *
 dotprod(y, y) - dotprod(x, y)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%)LeftSideG,(*$),&*&%$x2|irG\"\"\"%$y3|irGF+F+*&%$x3|irGF+%$y2|irGF+!
\"\"\"\"#\"\"\"F+*$),&*&F.F2%$y1|irGF+F+*&%$x1|irGF+F,F2F0F1F2F+*$),&*
&F9F2F/F2F+*&F*F2F7F2F0F1F2F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Ri
ghtSideG,&*&,(*$)%$x1|irG\"\"#\"\"\"\"\"\"*$)%$x2|irGF+F,F-*$)%$x3|irG
F+F,F-F-,(*$)%$y1|irGF+F,F-*$)%$y2|irGF+F,F-*$)%$y3|irGF+F,F-F-F-*$),(
*&F*F-F7F-F-*&F0F-F:F-F-*&F3F-F=F-F-F+F,!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "simplify(LeftSide - RightSide);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}}{MARK "2 2 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }