For some given angle $\theta$ and [[Axis]] $\vec u$, the rotation matrix is the [[Counter-Clockwise]] Rotation upon the $\vec u$-[[Axis]]. #### Vector Formula $\huge \begin{align} R_\ang{\theta, \vec u}(\vec v) &= \cos(\theta)\vec v + \pa{\frac{1-\cos\theta}{\lvert \vec u \rvert ^2}}\pa{\vec u \cdot \vec v}\vec u + \frac{\sin \theta}{\lvert \vec u \rvert }\pa{\vec u \times \vec v} \end{align}$ $ \huge R_{\ang{\theta,\vec u}}(\vec v_{\parallel}) = \vec v_{\parallel} $ #### Matrix Formula $\huge R_{\ang{\theta,\vec u}} = I_{3}\cos(\theta)+ \frac{1-\cos\theta}{\lvert \vec u \rvert ^2} \pa{\vec u \otimes \vec u} + \frac{\sin\theta}{\lvert \vec u \rvert } \Lambda \vec u $ >[!note] $\Lambda \vec u$ is the [[Cross Product Matrix]] of $\vec u$ >[!note] $\vec u \otimes \vec u$ is the [[Outer Product]] of $\vec u$ and itself Properties of $R=R_{\braket{\theta, \vec u}}$ 1. $\huge \det R = 1$ 2. $R$ is [[Orthogonal]] / the rows and columns are mutually [[Orthonormal]] 3. $\huge R^{-1} = R^{\intercal}$