EXAMINATION 1

MATH 2331

The point value of each problem is indicated in parentheses. There are 30 points all together. Show your work or otherwise explain what you are doing. Write in complete sentences, especially when stating definitions and theorems. Begin each problem on a new page of your blue book. Put the problems in your blue book in the order given.

1. (a) (2) How do you find the cross-product a × b of two vectors if you know their

lengths and the angle between them?

(b) (2) What if you know their components?

2. (2) True or False: The cross-product of two unit vectors is a unit vector. (If true, explain why. If false, give a counterexample).

3. Consider the planes x + y + z = 1 and x − 2y + 3z = 1.

(a) (2) Find the angle between them.

(b) (2) Find symmetric equations for L, their line of intersection.

4. Consider the quadric surface −x^{2} − y^{2} + z^{2} = 1.

(a) (2) Identify its vertical traces (parallel to a coordinate plane) as points, lines, parabolas, circles, ellipses or hyperbolas.

(b) (1) Identify the horizontal traces.

(c) (2) Sketch the graph of the surface.

5. (3) Find the volume of the parallelepiped with adjacent edges P Q, P R and P S, for P (1, 1, 1), Q(2, 0, 3), R(4, 1, 7) and S(3, −1, −2).

6. (4) Find the area enclosed by one loop of the four-leaved rose r = cos(2θ).

7. (4) Find the length of the polar curve r = e^{θ}, 0 ≤ θ ≤ 2π.

8. (a) (3) Find a vector function r(t) that represents the curve of intersection of the

cone z = \sqrt{x^{2} + y^{2}} and the plane z = 1 + y.

(b) (1) Find its second derivative, r′′(t).

Date: Friday, February 5, 1999.