Math 142

Summer 2000

Exam 4

1. (8 pt) Let be a complex number of modulus 1 (that is, ). Show that .

2. (12 pt) Rewrite the following complex numbers in polar form.

a) b)

3. (15 pt) We wish to find the 4^th roots of –16i.

a) Use the half angle identities to find and

b) Use part a) to find the four 4^th roots of –16i in the form

4. (35 pt) Consider the points in the plane

Let

a) Find

b) Find

c) Find

d) Find

e) Find the angle between

f) For what value(s) of are the vectors perpendicular?

g) Find a unit vector in the direction of

5. (20 pt) A river flows from west to east. A swimmer on the north bank swims a straight course that makes an angle of 60° with the north bank. The swimmer swims at a constant speed of 4 miles per hour and ends up on the southern bank directly south of his starting position. How fast is the current of the river?

6. (20 pt) A woman pulls a cart by exerting 150 lbs of force on the handle, which makes an angle of 30° with the horizontal. How much work does she do when she pulls the cart 600 feet?