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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 7 " Planes" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plo
ts):" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 326 13 "Unit Vectors:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "i:=vector([1,0,0]); j:=vector([0,1,0]); k:=vector([0,0,1]);" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 20 " Equation of a Plane" }}{PARA 0 
"" 0 "" {TEXT -1 89 "Find an equation for the plane that contains the \+
point P(1,-2,0) and has a normal vector " }{TEXT 293 1 "N" }{TEXT -1 
4 " = 3" }{TEXT 294 1 "i" }{TEXT -1 4 " - 2" }{TEXT 295 1 "j" }{TEXT 
-1 3 " + " }{TEXT 296 1 "k" }{TEXT -1 19 ".  Graph the plane." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " The equa
tion for the plane is the dot-product " }{TEXT 324 2 "N " }{TEXT 325 
0 "" }{TEXT -1 67 "and the vector [(x - 1), (y + 2),  z] set equal to \+
zero. It results" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" 
{TEXT -1 20 "3x - 2y + z - 7 = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "z1:=(x,y)->2*y-3*x+7;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot3d(z1(x,y),x=-10..10,y=-
10..10,axes=framed,shading=zhue,labels=([x,y,z]));" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "P
oints on the plane" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "t1:=v
ector([0,0,7]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t2:=vect
or([-3,-3,10]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Vector in the \+
normal direction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h1:=ve
ctor([3,-2,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "n1:=arro
w(t1,h1,width=.2,color=red,length=7):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Vector lying in the plane
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "n2:=arrow((t1,t2),width
=.2,color=red,difference=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 104 "plane1:= plot3d(z1(x,y),x=-5..5,y=-5..5,axes=framed,shading=z
hue,labels=([x,y,z]),orientation=[107,49]):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "display3d([n1,n2,plane1],axes=framed,labels=[x,y,z
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 27 " Intersection of Two Plan
es" }}{PARA 0 "" 0 "" {TEXT -1 10 "The planes" }}{PARA 263 "" 0 "" 
{TEXT -1 21 "p1: x + 2y + 3z = 0  " }}{PARA 265 "" 0 "" {TEXT -1 3 "an
d" }}{PARA 264 "" 0 "" {TEXT -1 20 "p2: -3x + 4y + z = 0" }}{PARA 0 "
" 0 "" {TEXT -1 75 "intersect at a line.  Graph the planes and find an
 equation for the line.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "z2:=(x,y)->(-1/3)*x-(2/3)*y;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
19 "Normal to plane z2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "
n2:=arrow([0,0,0],[6,12,18],width=.3,color=black):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "z3:=(x,y)->3*x-4*y;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 18 "Normal to plane z3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "n3:=arrow([0,0,0],[-9,12,3],width=.7,color=black):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "Z2:=plot3d(z2(x,y),x=-10.
.10,y=-10..10,color=blue): Z3:=plot3d(z3(x,y),x=-10..10,y=-10..10,colo
r=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "display3d([Z2,n2
],axes=framed,labels=[x,y,z],orientation=[-144,44]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 68 "display3d([Z3,n3],axes=framed,labels=[x,y
,z],orientation=[-144,44]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
74 "display3d([Z2,n2,Z3,n3],axes=framed,labels=[x,y,z],orientation=[-1
44,44]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 327 0 "" }
{TEXT 328 28 "Angle between the two planes" }}{PARA 0 "" 0 "" {TEXT 
-1 103 "The angle between the two planes can be defined as the acute a
ngle between their corresponding normals." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N1:=vector([1,2,3]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "N2:=vector([-3,4,1]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "DP:=dotprod(N1,N2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "NN1:=norm(N1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "NN2:=norm(N2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "theta:=evalf(arccos(abs(DP/(NN1*NN2))));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(convert(theta,degrees));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
306 54 "A Plane that passes through three non-collinear points" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Find an equation of a plane that c
ontains the points A(1,0,1), B(2,1,0), and C(1,1,1)." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "First, name the vectors
 in the plane that are defined by these points." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "v1:=vector([1,1,-1]); v2:=vector([0,1,0]);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The crossproduct of these two vec
tors will provide the normal vector to the plane." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "N:=crossprod(v1,v2);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 204 "Pick an arbitrary point in the plane, P(x,y,z).  Cr
eate a vector using P and a point we know is in the plane, say A (or B
 or C).  P is a point in the plane if the dot product of N and PA is e
qual to zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dotprod(N,
[x-1,y-0,z-1])=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The equation
 for the plane is:" }}{PARA 0 "" 0 "" {TEXT -1 9 "x + z = 2" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "z
4:=(x,y)->2-x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plot3d(z4
(x,y),x=-5..5,y=-5..5,axes=framed,shading=zhue,orientation=[65,78],lab
els=[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "n4:=arrow(
[1,0,1],[2,1,0],width=.2,color=black,length=10):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 74 "n41:=arrow([1,0,1],[2,1,0],width=.2,color=bla
ck,length=4,difference=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 74 "n42:=arrow([1,0,1],[1,1,1],width=.2,color=black,length=4,differe
nce=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Z4:=plot3d(z4
(x,y),x=-5..5,y=-5..5,axes=framed,shading=zhue,orientation=[62,94],lab
els=[x,y,z]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "display3d(
[Z4,n4,n41,n42],axes=framed,labels=[x,y,z],orientation=[69,88]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "1.  Graph the planes" }}
{PARA 0 "" 0 "" {TEXT -1 22 "p1: x -4y + 3z = 2 and" }}{PARA 0 "" 0 "
" {TEXT -1 18 "p2: 2x + y + 3z =5" }}{PARA 0 "" 0 "" {TEXT -1 45 "  an
d find the angle at which they intersect." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}}{MARK "56" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }