You have 95 minutes to complete this exam. Follow the directions carefully and SHOW
ALL RELEVANT WORK to receive full credit.
Question Points Score
1 15
2 12
3 20
4 16
5 6
6 6
Total: 75
1. (15 points) Let x = ⟨−2, 1⟩, y = ⟨1, 0⟩, and z = ⟨3, 4⟩ be vectors in R
2
. Compute
(a) 2z − 5x
(b) A unit vector in the direction of x.
(c) The scalar projection scalz(y).
(d) The vector projection projy
(x).
(e) The cosine of the angle between y and z.
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2. (12 points) Let a = ⟨−2, 0, 0⟩, b = ⟨0, 1, 3⟩, and c = ⟨0, p, 0⟩ be vectors in R
3
. Compute
each of the following, if possible. If the given expression is not defined, briefly explain
why (Note: The bold dot · means dot product.)
(a) a × c. For which p ∈ R will a and c be parallel?
(b) 5c × a
(c) 3a + c · (b × c)
(d) ∥b∥
5
· 5b
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3. (20 points) Let S denote the plane which contains the points P = (−1, 2, 2), Q =
(2, 2, 0), and R = (4, 1, −1).
(a) Find an equation for the plane S in the form ax + by + cz = d.
(b) Compute the minimum distance from the plane S to the point (1, 0, 2).
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(c) Find the parametric equations for the line which is orthogonal to the plane S and
passes through the point R.
(d) Find the area of the triangle ∆P QR in the plane S.
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4. (16 points) Consider the line L with parametric equations x = −2t, y = 1+t, z = 4−3t.
(a) Find the equation of the plane orthogonal to L which contains the point (3, 2, 1)
(b) Compute the minimum distance from the point (0, 2, 6) to the line L.
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(c) Write down the symmetric equations for the line L.
(d) Find the point where the line L intersects the xz-plane.
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5. (6 points) Find and identify the trace of the surface given by 4x
2+ (y+2)2−(z+1)2 = 0
in the xy-plane.
6. (6 points) Write down the equation of a sphere of radius 2 centered at (3, 4, 2). Sketch
this sphere in R
3 with the center labeled.
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