Rotation Matrix

Definition

${\mathbf{R}}_{\theta }=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

Facts

All column vectors, or all row vectors of rotation matrix are orthonormal

$\text{Det}({\mathbf{R}}_{\theta })=1$

${\mathbf{R}}_{{\theta }_2}({\mathbf{R}}_{{\theta }_1}\mathbf{x})={\mathbf{R}}_{{\theta }_1}({\mathbf{R}}_{{\theta }_1}\mathbf{x})={\mathbf{R}}_{{\theta }_1+{\theta }_2}\mathbf{x}$

${\mathbf{R}}_{\theta }^{-1}={\mathbf{R}}_{-\theta }$