PlanetPhysics/Axis Angle of Rotation to Direction Cosine Matrix

For now, without proof you can get the direction cosine matrix (DCM) from the axis angle of rotation by

$$ \left[ \begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix} \right] = \left[ \begin{matrix} cos(\alpha) + e_1^2(1 - cos(\alpha)) & e_1 e_2 (1 - cos(\alpha)) + e_3 sin(\alpha) & e_1 e_3(1 - cos(\alpha)) - e_2 sin(\alpha) \\ e_1 e_2(1 - cos(\alpha)) - e_3 sin(\alpha) & cos(\alpha) + e_2^2(1 - cos(\alpha)) & e_2 e_3(1-cos(\alpha)) + e_1 sin(\alpha) \\ e_1 e_3(1 - cos(\alpha)) + e_2 sin(\alpha) & e_2 e_3(1-cos(\alpha)) - e_1 sin(\alpha) & cos(\alpha) + e_3^2(1 - cos(\alpha)) \end{matrix} \right] $$