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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "MTHSC 206H" }}{PARA 19 "
" 0 "" {TEXT -1 11 "Homework #5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 25 "with(linalg):with(plots):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT
-1 52 "Section 10.4, page 735 - 736 #28, 30, 34, 38, and 42" }
{MPLTEXT 1 0 1 "." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 95 "28 Verify th
at the following points are the vertices of a parallelogram and calcul
ate its area." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P := vector
(3,[2,-1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Q := vecto
r(3,[5,1,4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R := vecto
r(3,[0,1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S := vecto
r(3,[3,3,4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "A sketch shows \+
us that P, Q, S, and R would walk us around the figure with the inside
of the figure to the left. If the figure is a parallelogram, then the
vectors PQ and PR must add up to PS." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 19 "PQ := evalm(Q - P);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "PR := evalm(R-P);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "PS := evalm(S-P);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 15 "evalm(PQ + PR);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
119 "This does indeed verify that we have a parallelogram. The area i
s simply the length of the cross product of PQ and PR." }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "N := crossprod(PQ,PR);" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 282 "The length of the vector N is the square
root of the sum of the squares of the components. On the calculator w
e can compute this using MTH : VECTR : ABS. In Maple we must use the
norm function and specify in addition that the norm we're interested \+
in is the Euclidean or \"2-norm\"." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 10 "norm(N,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
19 "evalf(norm(N,2),6);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 63 "30 F
ind the area of the triangle with the vertices given below." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P := [2,-3,4];" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "Q := [0,1,2];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "R := [-1,2,0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 17 "PQ := evalm(Q-P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
17 "PR := evalm(R-P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "th
eArea := (1/2)*norm(crossprod(PQ,PR),2);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "evalf(theArea,6);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT
-1 35 " 34 Find the triple scalar product." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "u := [1,1,1];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "v := [2,1,0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 13 "w := [0,0,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "do
tprod(u,crossprod(v,w));" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 108 " 3
8 Use the triple scalar product to find the volume of the parallelpipe
d having adjacent edges u, v, and w." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "u := [1,3,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 13 "v := [0,5,5];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w \+
:= [4,0,4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dotprod(u,cr
ossprod(v,w));" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 85 " 42 Find the \+
torque on the crankshaft using the position and data shown in Figure 4
2." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Choose the coordinate system
so that the crankshaft is horizontal, in the positive " }{XPPEDIT 18
0 "y" "I\"yG6\"" }{TEXT -1 66 " direction. Then the force vector is be
ing applied at an angle of " }{XPPEDIT 18 0 "-120" ",$\"$?\"!\"\"" }
{TEXT -1 12 " degrees or " }{XPPEDIT 18 0 "-2*Pi/3" ",$*(\"\"#\"\"\"%#
PiGF%\"\"$!\"\"F(" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 19 "PQ := [0,16/100,0];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 40 "F := 2000*[0,cos(-2*Pi/3),sin(-2*Pi/3)];" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "norm(crossprod(PQ,F),2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(\",7);" }}}}}{SECT 0
{PARA 4 "" 0 "" {TEXT -1 51 "Section 10.5, page 744 - 745 #2, 4, 10, 1
2, and 14." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 354 "2 Given the parame
tric equations, plot the line and draw an arrow on the line indicating
the its orientation. Then find the coordinates of two points. P and Q
, on the line. Determine the vector PQ. What is its relation to the co
efficients of t in the parametric equation? Determine the coordinates \+
of any points of intersection wth the coordinate planes." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x := t -> 2 - 3*t;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y := t -> 2;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "z := t -> 1 - t;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 55 "p1 := spacecurve([x(t),y(t),z(t)],t=-3..3,axes=NORMAL
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P := [x(0),y(0),z(0)]
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Q := [x(1),y(1),z(1)];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PQ := evalm(Q-P);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "p2 := arrow(P, [-3,0,-1], [0,1,0], \+
.2, .4, .1, color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
19 "display3d(\{p1,p2\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Sinc
e we chose " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 18 " to correspon
d to " }{XPPEDIT 18 0 "t=0" "/%\"tG\"\"!" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Q" "I\"QG6\"" }{TEXT -1 18 " to correspond to " }
{XPPEDIT 18 0 "t=1" "/%\"tG\"\"\"" }{TEXT -1 76 " the components of th
e direction vector are the same as the cooeficients of " }{XPPEDIT 18
0 "t" "I\"tG6\"" }{TEXT -1 76 ". In general the components will always
be a multiple of the coeficients of " }{XPPEDIT 18 0 "t" "I\"tG6\"" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "The coordinates
of the intersections of the line with the coordinate planes can be de
termined by setting " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 2 "
, " }{XPPEDIT 18 0 "y=0" "/%\"yG\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT
18 0 "z=0" "/%\"zG\"\"!" }{TEXT -1 17 ", and solving to " }{XPPEDIT
18 0 "t" "I\"tG6\"" }{TEXT -1 14 " in each case." }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 25 "t[yz] := solve(x(t)=0,t);" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 29 "[x(t[yz]),y(t[yz]),z(t[yz])];" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t[xz] := solve(y(t)=0,t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The line is parallel to the " }
{XPPEDIT 18 0 "xz" "I#xzG6\"" }{TEXT -1 33 " plane, and there is no va
lue of " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 11 " for which " }
{XPPEDIT 18 0 "y=0" "/%\"yG\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 25 "t[xy] := solve(z(t)=0,t);" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 29 "[x(t[xy]),y(t[xy]),z(t[xy])];" }}}}{SECT 1
{PARA 5 "" 0 "" {TEXT -1 174 "4 Find the parametric equations and the \+
symmetric equations for the line through the given point and parallel \+
to the given vector - express the direction numbers as integers." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "P := [0,0,0];" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v := [-2,5/2,1];" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 18 "expand(2*v*t + P);" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 47 "In this case the symmetric equations are simply" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x:='x':y:='y':z:='z':" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x/(-4) = y/5;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y/5 = z/2;" }{XPPMATH 20 "6#/,$%\"y
G#\"\"\"\"\"&,$%\"zG#F'\"\"#" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1
103 "10 Find the parametric equations and the symmetric equations for \+
the line through the two given points." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 15 "P := [5,-3,-2];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 18 "Q := [-2/3,2/3,1];" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 16 "v := evalm(Q-P);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 30 "peq := evalm(expand(3*v*t)+P);" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 50 "The symmetric equatios can be obtained by setting "
}{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y" "I\"
yG6\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1
42 " to the preceding components, solving for " }{XPPEDIT 18 0 "t" "I
\"tG6\"" }{TEXT -1 26 ", and equating the results" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "solve(x=peq[1],t)=solve(y=peq[2],t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "17*11*9*\" - (495=495);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solve(x=peq[1],t)=solve(z=pe
q[3],t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "17*11*9*\" - (4
95=495);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 122 "12 Find the parame
tric equations for the line that passes through the given point and is
perpendicular to the given plane." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "P := [2,3,4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 36 "eq1 := (x,y,z) -> 3*x + 2*y - z - 6;" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 89 "If the line is perpendicular to this plane then it's di
rection is parallel to the normal." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "v := [3,2,-1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 23 "evalm(expand(v*t) + P);" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1
63 "14 Determine which of the given points lie on the line through " }
{XPPEDIT 18 0 "[2,0,-3]" "7%\"\"#\"\"!,$\"\"$!\"\"" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "[4,2,-2]" "7%\"\"%\"\"#,$F$!\"\"" }{TEXT -1 1 "." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P := [2,0,-3];" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q := [4,2,-2];" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "v := evalm(Q-P);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "ueq := evalm(expand(v*t) + P);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "U := t -> evalm(expand(v*t) + P);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "P1 := [4,1,-2];" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "t1 := solve(ueq[1]=P1[1],t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "U(t1)=P1;" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 38 "The line does not pass thru the point " }
{XPPEDIT 18 0 "P1" "I#P1G6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 22 "P2 := [5/2,1/2,-11/4];" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 28 "t2 := solve(ueq[1]=P2[1],t);" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 9 "U(t2)=P2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 34 "The line does pass thru the point " }{XPPEDIT 18 0 "P2" "I#P2G6
\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "P3 :=
[-1,-3,-4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "t3 := solve
(ueq[1]=P3[1],t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "U(t3)=P
3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The line does not pass thru
the point " }{XPPEDIT 18 0 "P3" "I#P3G6\"" }{TEXT -1 1 "." }}}}}}
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