A good answer might be:

Direction to sun = ( -3, 4, 0)/5
Perpendicular to Bob's back = ( -1, 2, 2)/3
Perpendicular to Bill's back = ( -2, 1, 2)/3

To catch the most rays, your back should be pointed straight at the sun. You want the smallest angle between the perpendicular to your back and the direction to the sun.

For Bob: ( -1, 2, 2)/3 · ( -3, 4, 0)/5 = (3+8+0)/15 = 11/15; arc cos( 11/15 ) = 42.8 degrees

For Bill: ( -2, 1, 2)/3 · ( -3, 4, 0)/5 = (6+4+0)/15 = 10/15; arc cos( 14/15 ) = 48.2 degrees

Bob will tan faster.

More Practice

The diagram shows two more example vectors:

q = (-2, 4, 3)^T

p = (3, 1, -4)^T

For ease in visualization the tails of the vectors have been placed at the origin. The vectors point into two different octants of 3D space.

Rotate the viewpoint to get a better idea of the two vectors. See if you can find a viewpoint that best shows the angle between the vectors.

QUESTION 9:

What is the angle between the two vectors? (Work it out numerically.)