Dot Product Euclidian Space After Rotation

Question

Consider a three dimensional space with an orthogonal basis $\hat{i}$, $\hat{j}$, $\hat{k}$. Say I have a rotation matrix $\{\{0,0,1\},\{0,1,0\},\{-1,0,0\}\}$ that rotates a vector $\{1,1,0\}$ from the ${\hat{i},\hat{j}}$ plane to the ${\hat{j},\hat{k}}$ plane. The resulting vector is $\{0,1,-1\}$. Given this $\frac{\pi}{2}$ rotation, why does the dot product between the original vector, $\{1,1,0\}$ and the result, $\{0,1,-1\}$ give an angle of $\frac{\pi}{3}$?

Answer 1

Your matrix gives a rotation by $\pi/2$ about the $y$-axis. The vector that you act on does not lie in the perpendicular plane to the rotation, so the angle that it picks up is less than $\pi/2$.

Imagine you are on earth, not on the equator, and you wait $1/4$ of a day so that the earth rotates by $\pi/2$. You will not be at an angle $\pi/2$ to your earlier orientation.

The angle between the starting position and after the rotation needn't by $\pi/2$ (except in the $xz$-plane) - take the $y$-axis which does not move at all, for example!