Rotation on a plane

by lendav_rott
Tags: plane, rotation
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,345 Yes, the rotation matrix, $$\begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$$ is orthogonal- its columns or rows, considered as vectors, are "orthonormal", perpendicular and of length 1. Also, the transpose is, as you say, equal to the reverse rotation: $$\begin{bmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{bmatrix}= \begin{bmatrix}cos(-\theta) & -sin(-\theta) \\ sin(-\theta) & cos(-\theta)\end{bmatrix}$$ because cosine is an "even" function and sine is an "odd" function: $cos(-\theta)= cos(\theta)$ and $sin(-\theta)= -sin(\theta)$.