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

Rotation around an arbitrary axis - Implicit matrix representation

The rotation formula (as derived here) \[ \vec{v}'=\rejection{\vec{v}}{u}\cos(\alpha) + (\cross{\vec{u}}{\vec{v}}) \sin(\alpha) + \projection{\vec{v}}{\vec{u}} \] can be restated as \[ \vec{v}' = R \vec{v} \] using a matrix \[ R = \cos(\alpha) I + \sin(\alpha) \widetilde{U} + \left(1 - \cos(\alpha)\right)\left(\widetilde{U}^2 + I\right) \]

Proof

\[\begin{aligned} \vec{v}' \end{aligned}\]
Definition of \(\vec{v}'\).
\[\begin{aligned} &=\rejection{\vec{v}}{u}\cos(\alpha) + (\cross{\vec{u}}{\vec{v}}) \sin(\alpha) + \projection{\vec{v}}{\vec{u}} \end{aligned}\]
Commutativity.
\[\begin{aligned} &=\cos(\alpha) \rejection{\vec{v}}{u}+ \sin(\alpha) \left(\cross{\vec{u}}{\vec{v}}\right) + \projection{\vec{v}}{\vec{u}} \end{aligned}\]
Definition of the vector projection
Definition of \(\rejection{\vec{v}}{u}=\vec{v}-\projection{\vec{v}}{\vec{u}}\).
\[\begin{aligned} &=\cos(\alpha) (\vec{v} - \projection{\vec{v}}{\vec{u}}) + \sin(\alpha) (\cross{\vec{u}}{\vec{v}}) + \projection{\vec{v}}{\vec{u}} \end{aligned}\]
Distributivity of multiplication over addition.
\[\begin{aligned} &=\cos(\alpha)\vec{v} - \cos(\alpha)\projection{\vec{v}}{\vec{u}} + \sin(\alpha)(\cross{\vec{u}}{\vec{v}}) + \projection{\vec{v}}{\vec{u}} \end{aligned}\]
Commutativity of addition.
\[\begin{aligned} &=\cos(\alpha)\vec{v} + \sin(\alpha) (\cross{\vec{u}}{\vec{v}}) + \projection{\vec{v}}{\vec{u}} - \projection{\vec{v}}{\vec{u}}\cos(\alpha) \end{aligned}\]
Distributivity of multiplication over addition.
\[\begin{aligned} &=\cos(\alpha)\vec{v} + \sin(\alpha)(\cross{\vec{u}}{\vec{v}}) + (1 - \cos(\alpha))\projection{\vec{v}}{\vec{u}} \end{aligned}\]
Definition of the cross product
\(\cross{\vec{u}}{\vec{v}} = \widetilde{U}\vec{v}\).
Definition of the vector projection
\(\projection{\vec{v}}{\vec{u}} = \left(\widetilde{U}^2 + I\right)\vec{v}\).
\(\vec{v}\cos(\alpha) = I \vec{v} \cos(\alpha)\)
\[\begin{aligned} &=\cos(\alpha) I \vec{v} + \sin(\alpha)\widetilde{U}\vec{v} + (1 - \cos(\alpha))\left(\widetilde{U}^2 + I\right)\vec{v} \end{aligned}\]
\[\begin{aligned} &= \cos(\alpha) I \vec{v} + \sin(\alpha) \widetilde{U}\vec{v} + (1 - \cos(\alpha))\left(\widetilde{U}^2 + I\right)\vec{v} \end{aligned}\]
Distributivity of vector - matrix multiplication.
\[\begin{aligned} &= \left[\cos(\alpha) I + \sin(\alpha) \widetilde{U} + (1 - \cos(\alpha))\left(\widetilde{U}^2 + I\right)\right]\vec{v} \end{aligned}\]