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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 25 "Volume of Parallelepipeds" }}
{PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1
21 "School of Mathematics" }}{PARA 259 "" 0 "" {TEXT -1 21 "herod@math
.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 259 " In order to understand the setting for this wo
rksheet, there is an elementary fact from high school geometry you wil
l need to remember: The area of the parallelogram pictured below, havi
ng sides of length X and Y, and with the indicated angle " }{XPPEDIT
18 0 "alpha" "6#%&alphaG" }{TEXT -1 5 ", is " }}{PARA 0 "" 0 "" {TEXT
-1 60 " X*Y*sin(" }
{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 ""
{TEXT -1 123 "We will compute the volume of a parallelepiped using thi
s formula and using the cross product, dot product, and projection." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {GLPLOT2D 256 199
199 {PLOTDATA 2 "6&-%'CURVESG6$7'7$\"\"!F(7$$\"+++++:!\"*$\"\"\"F(7$$
\"+++++MF,F*7$$\"+++++>F,$\"+++++]!#5F'-%'COLOURG6&%$RGBGF(F($\"*++++
\"!\")-%%TEXTG6$7$$\"#7!\"\"$\"#B!\"#%\"XG-F@6$7$$\"\"(FEFM%\"YG-F@6$7
$$\"#bFH$\"#CFH%\"aG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " We need the linea
r algebra and the plots packages for this worksheet." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots
): with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 16 "" 0 "
" {TEXT -1 117 " We choose two vectors, u and v, in the plane and draw
the parallelogram they determine having corners u, v, and u+v." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=[1,2]; v:=[2,1]; w:=u+v;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "polygonplot([[0,0],u,w,
v,[0,0]],color=BLUE);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 ""
0 "" {TEXT -1 83 "We compute the area of this parallelogram by the hig
h school method recalled above." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 50 "alpha:=arccos(dotprod(u,v)/(norm(u,2)*norm(v,2)));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "area:=norm(u,2)*norm(v,2)*sin(alpha
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT -1 69 "W
e compute the area of this parallelogram by using the cross product."
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "crossprod([u[1],u[2],0],[v
[1],v[2],0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "area:=norm
(%,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0
"" {TEXT -1 117 "If all went well, you got the same answer both ways. \+
This happens because the length of the cross product is given as" }}
{PARA 0 "" 0 "" {TEXT -1 40 " |u| |v| sin(
" }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1
80 " is the angle between u and v. Try to see if you can establish why
this is true." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 ""
0 "" {TEXT -1 42 "How to draw the parallelogram with labels." }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u:
=[3/2,1];v:=[19/10,1/2];w:=u+v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 36 "J:=polygonplot([[0,0],u,w,v,[0,0]]):" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 57 "L:=textplot(\{[1.2,.23,'X'],[.7,.7,'Y'],[0.55,0.
24,'c']\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(\{J,
L\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "The volume of \+
a parallelepiped" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 37 "We take the vectors u and v to be in " }{XPPEDIT 18 0 "R^
3" "6#*$%\"RG\"\"$" }{TEXT -1 98 ". For illustration, we change the ve
ctors u and v above by appending zero in the third component. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "u3:=[u[1],u[2],0]; \nv3:=[v[
1],v[2],0];" }}}{PARA 0 "" 0 "" {TEXT -1 26 "We now add a third vector
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w3:=[1,2,3];" }}}{PARA
0 "" 0 "" {TEXT -1 224 "These three vectors define a parallelepiped - \+
a three dimensional figure with base in the x-y plane. To find the vol
ume of this figure, we multiply the area of the base time the height. \+
The area of the base was found above. " }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 15 "BaseArea:=area;" }}}{PARA 0 "" 0 "" {TEXT -1 177 "To \+
get the height, we find the length of the projection of w3 onto a vect
or perpendicular to the base. Such a vector perpendicular to the base \+
is the cross product of u3 and v3." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "perpen:=crossprod(u3,v3);" }}}{PARA 0 "" 0 "" {TEXT
-1 91 "We understand how to get the length of the projection of w3 ont
o this perpendicular vector." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
43 "ht:=abs(dotprod(w3,perpen)/norm(perpen,2));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "ht*area;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 24 "Exercise for the student" }}{PARA 0 "" 0 "" {TEXT -1 97 "
Find the volume of the parallelepiped determined by the vectors [1, 1,
0], [3, -1, 0], [1, 0, 1]." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 28 "How to draw a parallelepiped" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Here is how you
could draw a parallelepiped" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "u:=[1,2
,0];v:=[2,1,0];zero:=[0,0,0];w:=[1,2,3];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 192 "J:=polygonplot3d([zero,u,u+w,w,zero,v,v+w,w],axes=NO
RMAL,\n color=green,orientation=[15,65]):\nK:=polygonplot3d([u,v+
u,u+v+w,u+w],color=blue):\nL:=polygonplot3d([v,u+v,u+v+w,v+w],color=re
d):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{J,K,L\});
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 102 "Touch the figure abo
ve with the mouse. Then, observe that you can rotate it. Look down ins
ide the box." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0" 0 }{VIEWOPTS 1 1
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }