Math 241 - Quiz #2

1. Let a = 3i + j - k and b = i - j + 2k.

A. Determine if a and b are orthogonal, parallel or neither.

B. Compute the scalar a vector projection of b onto a, i.e. comp_a (b) and proj_a(b).

2. Find the area of the parallelogram ABCD with the points A(0,1), B(3,0), C(5,-2), D(2,-1).

Solutions:

1. A. a.b = (3,1,-1).(1,-1,2) = 3 -1 -2 = 0, so a and b are orthogonal.

B. Since a and b are orthogonal, there is no component of b in the direction of a, thus
comp_a(b) = 0 and proj_a(b) = 0(vector).

2. Let a = AB = (3,-1,0) and b = AD = (2,-2,0), then the area is |a x b|.
a x b = (0, 0, 3(-2)-(-1)2 = -4), thus the area is |-4| = 4.

Note that even if you used the wrong vectors, like a = AB and b = AC (for the parallelogram ABDC) you would get the same area!

ccollins@math.utk.edu
Last Modified: Jan. 25, 1999