Matrix Equation of Plane Rotation

Theorem
Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.

Let $r_\alpha$ rotate an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$

Then:


 * $\begin{bmatrix}

x' \\ y' \end{bmatrix}$ = $\begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$ $\begin{bmatrix} x \\ y \end{bmatrix}$

Proof
Let the coordinates of $P'$ be encoded as the elements of a $2 \times 1$ matrix.

We have:

Hence the result.