$$% Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}}$$

### Rotation around an arbitrary axis - Explicit matrix representation

The rotation formula implicit matrix representation (as derived here here) $\vec{p}'=\rejection{\vec{p}}{\hat{r}}\cos(\alpha) + (\cross{\hat{r}}{\vec{p}}) \sin(\alpha) + \projection{\vec{p}}{\vec{\hat{r}}}$ can be restated as $\vec{p}' = R \vec{p}$ using a matrix $R = \left[\begin{matrix} t \hat{r}_x^2 + c & t \hat{r}_y \hat{r}_x - s \hat{r}_z & t \hat{r}_x \hat{r}_z + s \hat{r}_y \\ t \hat{r}_x \hat{r}_y + s \hat{r}_z & t \hat{r}_y^2 + c & t \hat{r}_y \hat{r}_z - s \hat{r}_x \\ t \hat{r}_x \hat{r}_z - s \hat{r}_y & t \hat{r}_y \hat{r}_z + s \hat{r}_x & t \hat{r}_z^2 + c \end{matrix}\right]$ with $$c = \cos\alpha$$, $$s = \sin\alpha$$, and $$t = 1 - \cos\alpha$$.

#### Proof

Given $R = \cos(\alpha) I + \sin(\alpha) \widetilde{R} + \group{1 - \cos\group{\alpha}}\group{\widetilde{R}^2 + I}$ let $$c = \cos\alpha$$, $$s = \sin\alpha$$, and $$t = 1 - \cos\alpha$$ and obtain $R = c I + s \widetilde{R} + t \group{\widetilde{R}^2 + I}$ Expand $R = c \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix}\right] + s \left[\begin{matrix} 0 & -\hat{r}_z & \hat{r}_y \\ \hat{r}_z & 0 & -\hat{r}_x \\ -\hat{r}_y & \hat{r}_x & 0 \\ \end{matrix}\right] + t \left[\begin{matrix} \hat{r}_x^2 & \hat{r}_x \hat{r}_y & \hat{r}_x \hat{r}_z\\ \hat{r}_x \hat{r}_y & \hat{r}_y^2 & \hat{r}_y \hat{r}_z\\ \hat{r}_x \hat{r}_z & \hat{r}_y \hat{r}_z & \hat{r}_z^2 \end{matrix}\right]$ and rewrite to $R = \left[\begin{matrix} t \hat{r}_x^2 + c & t \hat{r}_x \hat{r}_y - s \hat{r}_z & t \hat{r}_x \hat{r}_z + s \hat{r}_y \\ t \hat{r}_x \hat{r}_y + s \hat{r}_z & t \hat{r}_y^2 + c & t \hat{r}_y \hat{r}_z - s \hat{r}_x \\ t \hat{r}_x \hat{r}_z - s \hat{r}_y & t \hat{r}_y \hat{r}_z + s \hat{r}_x & t \hat{r}_z^2 + c \end{matrix}\right]$