Angle between Two Directions

It might sound odd to measure the angle between things that have no position. However, many phenomena have an orientation, but no position. The wind might come from the West one day and from the North the next (a difference of 90°). At noon in Summer the light from the Sun arrives at a different angle than at noon in Winter. Shading in 3D graphics uses vectors for the orientation of a surface and for the direction of each light source.

Bob and Bill have skipped class and are at the beach working on their sun tans. A convenient coordinate frame points the x-axis North, the y-axis straight up, and the z-axis East.

In this frame, the direction to the Sun is (-3, 4, 0)^T / 5.

A unit vector perpendicular to Bob's back is ( -1, 2, 2)^T / 3.

A unit vector perpendicular to Bill's back is ( -2, 1, 2)^T / 3.