#### [[Determinant]] $\huge \det \pa{M_{R\theta}} = 1$ >[!info] Symmetric > $\huge \pa{M_{R\theta}}^{\intercal} \ne M_{R\theta}$ > > $\huge \pa{M_{R\theta}}^{\intercal} \ne M_{R\theta} \iff \theta \in n\pi$ ## 2D Rotating around the origin $\huge \begin{align*} R_{\theta}&= \mat{ \cos\theta & -\sin\theta \\ \sin\theta & \cos \theta} \end{align*}$ #### [[Inverse Matrices|Inverse]] $\huge \begin{align*} R_{\theta}^{-1}&= \mat{ \cos \theta & \sin\theta \\ -\sin\theta & \cos \theta} \end{align*}$ ### 3D Rotating around a basis vector $\huge R_\ang{\theta,x} \mat{ 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin \theta & \cos \theta }$ $\huge R_\ang{\theta,y} \mat{ \cos \theta &0& \sin \theta\\ 0&1&0\\ -\sin \theta &0&\cos\theta\\ }$ $\huge R_\ang{\theta,z} \mat{ \cos \theta & -\sin \theta&0 \\ \sin \theta & \cos \theta &0 \\ 0&0&1 \\ }$