The angle between two unit vectors is not what I expected Ok imagine a vector with only X and Z components that make a 45 degree angle with the positive X axis. It's a unit vector. Now also imagine a unit vector that has the same direction as the positive x axis. Now imagine rotating both of these around the Z axis. 
I expect the angle between these vectors to still be 45 degrees. But it's not. If you don't believe me look here. Angle between two 3D vectors is not what I expected.
Another way to think about it is to draw a 45 degree angle between two lines on a piece of paper. Now stand the paper up, and rotate the paper. The angle between the lines are still 45 degrees.
Why is my way of thinking wrong?

If you start with the vectors $(1,0,0)$ and $(1/\sqrt{2},0,1/\sqrt{2})$, and rotate both by $45^\circ$ about the $z\text{-axis}$, then you end up with $(1/\sqrt{2},1/\sqrt{2},0)$ and $(1/2,1/2,1/\sqrt{2})$.  The second point is not $(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$ as you imagined.  If you think about it, the $z\text{-coordinate}$ cannot be changed by this rotation.  If the $z\text{-axis}$ is vertical, and the $x\text{-}y$ plane is horizontal, then the height of the point above the plane is not changed by rotation about the $z\text{-axis}$.  The height remains $1/\sqrt{2}$, and the length of the horizontal coordinate remains $1/\sqrt{2}$ as well.  That would not be the case if the final vector were what you thought it was.
