PlanetPhysics/Axis Angle of Rotation to Direction Cosine Matrix

For now, without proof you can get the [[../DirectionCosineMatrix/|direction cosine matrix]] (DCM) from the [[../AxisAngleOfRotation/|axis angle of rotation]] by

$\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]=\left[\begin{array}{ccc}cos(\alpha )+e_1^2(1-cos(\alpha ))& e_1{e}_2(1-cos(\alpha ))+e_{3}sin(\alpha )& e_1{e}_3(1-cos(\alpha ))-e_{2}sin(\alpha )\\ e_1{e}_2(1-cos(\alpha ))-e_{3}sin(\alpha )& cos(\alpha )+e_2^2(1-cos(\alpha ))& e_2{e}_3(1-cos(\alpha ))+e_{1}sin(\alpha )\\ e_1{e}_3(1-cos(\alpha ))+e_{2}sin(\alpha )& e_2{e}_3(1-cos(\alpha ))-e_{1}sin(\alpha )& cos(\alpha )+e_3^2(1-cos(\alpha ))\end{array}\right]$