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```Mathematics 320
First Test
This pleasant 50 minute test covers sections 12:1—6 of Calculus by Stewart (5ed). Relax
and do well. Unless otherwise stated, problems are five points. Indicate your answers.
1.
Let A, B and C be vectors and let k be a scalar. Circle either T for always true or F
for at least once false.
(1 point each)
T F A × A = 1 if A is a unit vector
T F
If A and B are perpendicular, then projAB = 0.
T F
If A and B are parallel, then A • B = 0.
T F
If A is a unit vector, then so is −A.
T F
A × B + B × A = 0.
T F
kA • B = kB • A
T F
A • (B × C) = A × (B • C)
T F
The cross product B × A of the two vectors
A
B
on the right is pointing out of the page.
T F
The cross product of two unit vectors is a unit vector.
2.
Find the equation of the sphere that passes through the point (3,8,1) and has center
(−2,3,1).
3.
Give a geometric description of the set of points which satisfy the given inequality: x2
+ 4y2 + z2 ≤ 9.
4.
Find two unit vectors parallel to 5i − 3j + 4k.
5.
Let A = 5i + 12j − 7k, B = 3i − 5j. Find the following.
|A| =
|B| =
A•B =
cos θ =
projAB =
compAB =
(2 points each)
6.
Find the length and direction of A×B given that A = 3i + j + 4k and B = 2i + j − 2k.
7.
Find a parameterization for the line segment from (3, 2, 0) to (−2, 5, 2).
8.
Find the equation of the plane through the points (1,2,3), (2,1,3) and (3,2,1).
9.
Find the symmetric equation for the following lines.
a) The line through (1, 2, 3) and (3, 2, 1).
b) The line through (1, 2, 3) and perpendicular to the plane x + 2y − z = 4.
c) The line of intersection of the planes x + 2y – z = 2 and 2x – y + 3z = 1.
10. Find the distance from the point (0, 0, 0) to the plane x – 2y – 2z = 1.
11. Find the equation of the plane through the point (1,2,3) and containing the line:
x = 2 + 3t, y = 5 – 3t,
z = t.
12. Determine if the following lines are the same line, parallel, skew, or intersecting lines.
x = 4 + t, y = 1 – 3t, z = 2 + 5t
x = 3 + 2t, y = 1 – 6t, z = 10t
13. Identify both of the following surfaces
(2 points each)
a) x2 – z2 = 1.
b) x2 − 2y + z = 1.
14. Sketch one of the surfaces in the previous problem. Clearly indicate which you are
doing! Label at least four points.
(7 points)
```