Rotation Transformation, Rotation Matrix
Definition
In the two-dimensional plane $\mathbb{R}^{2}$, the transformation that rotates an arbitrary vector counterclockwise by $\theta$ is given by
$$ \begin{bmatrix} x^{\prime} \\ y^{\prime} \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$
Explanation
The matrix $\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ is called the rotation matrix or the rotation transformation.
Derivation
Let $x = r \cos \phi$ and $y = r \sin \phi$. Let $(x^{\prime}, y^{\prime})$ denote the point obtained by rotating the point $(x, y)$ by $\theta$. By the trigonometric addition formulas (link), $x^{\prime}, y^{\prime}$ are respectively given by
$$ \begin{align*} x^{\prime} &= r \cos(\phi + \theta) \\ &= r\cos\phi \cos\theta - r\sin\phi \sin\theta \\ &= x \cos\theta - y \sin\theta \\ \end{align*} $$
$$ \begin{align*} y^{\prime} &= r \sin(\phi + \theta) \\ &= r\sin\phi \cos\theta + r\cos\phi \sin\theta \\ &= y \cos\theta + x \sin\theta \\ &= x \sin\theta + y \cos\theta \end{align*} $$
$$ \implies \begin{cases} x^{\prime} = x \cos\theta - y \sin\theta \\ y^{\prime} = y \cos\theta + x \sin\theta \end{cases} $$
Writing this system of equations as a matrix (link) yields
$$ \begin{bmatrix} x^{\prime} \\ y^{\prime} \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$
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