Find the equation of the plane orthogonal to L which contains the point (3, 2, 1).Compute the minimum distance from the point (0, 2, 6) to the line L.

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.
Page 2
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.
(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.
Page 5
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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