Rotational Transformation
2D
The transformation that rotates a vector $\theta$ counter-clockwise on a 2D plane $\mathbb{R}^{2}$ is as follows.
$$ \begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$
Derivation
By the addition formulas for trigonometric functions, $x^{\prime}, y^{\prime}$ can be written as follows.
$$ \begin{align*} x^{\prime} =&\ \dfrac{1}{r} \cos(\phi + \theta) \\ =&\ \dfrac{\cos \phi}{r} \cos \theta - \dfrac{\sin \phi}{r} \sin \theta \\ =&\ x \cos \theta - y \sin \theta \end{align*} $$
$$ \begin{align*} y^{\prime} =&\ \dfrac{1}{r} \sin(\phi + \theta) \\ =&\ \dfrac{\cos \phi}{r} \cos \theta - \dfrac{\sin \phi}{r} \sin \theta \\ =&\ x \cos \theta - y \sin \theta \end{align*} $$
When represented as a matrix, it looks like this.
$$ \begin{pmatrix} x^{\prime} \\ y^{\prime} \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$
■