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Math 225: Calculus III	Quiz 1    1/24&26/95
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*)
Needs["Calculus`Advanced`"];
i = {1,0,0}; j = {0,1,0}; k = {0,0,1};
(*
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1. Find the interior angles, theta, phi, of the parallelogram with vertices
{0,0}, {2,1}, {3,4}, and {5,5}.
:[font = subsubsection; inactive; preserveAspect]
Solution
:[font = text; inactive; preserveAspect]
The angle theta is the angle between the vectors
:[font = input; preserveAspect]
a = {2,1} - {0,0}; b = {3,4} - {0,0};
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theta = ArcCos[a.b/(Norm[a]Norm[b])] //N
:[font = output; output; inactive; preserveAspect; endGroup]
0.4636476090008061156
;[o]
0.463648
:[font = text; inactive; preserveAspect]
The angle phi is just the complementary angle,
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phi = Pi - theta //N
:[font = output; output; inactive; preserveAspect; endGroup]
2.677945044588987122
;[o]
2.67795
:[font = text; inactive; preserveAspect]
phi can also be calculated like theta as the angle between the vectors
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c = {0,0} - {2,1}; d = {5,5} - {2,1};
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phi = ArcCos[c.d/(Norm[c]Norm[d])] //N
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2.677945044588987121
;[o]
2.67795
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2.  Find a vector that is perpendicular to both of the vectors i + j + 3k and
2i + j + k.
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Solution
:[font = text; inactive; preserveAspect]
The cross product Cross[a,b] always gives a vector perpendicular to both a and b.
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a = i + j + 3k; b = 2i + j + k;
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Cross[a,b]
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{-2, 5, -1}
;[o]
{-2, 5, -1}
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Note that Cross[b,a] = -Cross[a,b] = {2,-5,1} is also a possiblility.
;[s]
3:0,0;10,1;45,0;70,-1;
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^*)