MAT 205 SPRING 2000 MIDTERM I

NAME (first and last; readable): SSN:

THERE ARE FOUR PROBLEMS PLUS ONE EXTRA CREDIT PROBLEM SHOW YOUR WORK!!!

1) Let

a =< 2, 1, −1 >, b =< 1, −2, 3 >, c =< 1, 0, −2 > .

A) Compute b · (a + c).

B) Let d =< 2, m, 1 > be a vector varying with the parameter m. Find the values of m, if any, that make d and a perpendicular.

NAME (first and last; readable): SSN:

2) Let

P = (−1, 1, −1), Q = (1, −1, 2), R = (4, 0, 3)

be three points in space. Find the plane passing through P, Q and R.

NAME (first and last; readable): SSN:

3) Sketch the surface

y = \sqrt{x^{2} + z^{2}}.

Give also a brief explanation in words as to why it looks like you have drawn it.

NAME (first and last; readable): SSN:

4) A particle is moving in space with position vector r(t). The acceleration is

a(t) = i − 2k.

The velocity at time t = 0 is

v(0) = i − j + k.

The position at time t = 0 is

r(0) = 2i − 2k.

Find the position vector r(2) at time t = 2.

NAME (first and last; readable): SSN: THIS IS FOR EXTRA CREDIT. ATTEMPT THE FIRST FOUR PROB-

LEMS FIRST.

5) Let a and b be non zero vectors. Show that the vector

b − proj_{a}b

is orthogonal to the vector a.