2D Point Rotation

In 2D, the rotated point $(x',y')$ is computed from the original point $(x,y)$ with a rotation by angle $\theta$ by

$\begin{eqnarray} x' &=& x \, \cos \theta - y \, \sin \theta \\ y' &=& x \, \sin \theta + y \, \cos \theta \end{eqnarray}$

(Note that this describes a rotation in the counter-clockwise direction. For a clockwise rotation, $\theta$ can be negated.)

As a $2 \times 2$ matrix this is very simple. If you want to rotate a column vector $v$ by $\theta$ degrees using matrix $M$ , use

$M = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$

so that $v' = M v$ .

If you have a row vector, use the transpose of $M$ (turn rows into columns and vice versa).

If you want to combine rotations, in 2D you can just add their angles, but in higher dimensions you must multiply their matrices.