For now, without proof you can get the direction cosine matrix (DCM) from the axis angle of rotation by
[ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] = [ c o s ( α ) + e 1 2 ( 1 − c o s ( α ) ) e 1 e 2 ( 1 − c o s ( α ) ) + e 3 s i n ( α ) e 1 e 3 ( 1 − c o s ( α ) ) − e 2 s i n ( α ) e 1 e 2 ( 1 − c o s ( α ) ) − e 3 s i n ( α ) c o s ( α ) + e 2 2 ( 1 − c o s ( α ) ) e 2 e 3 ( 1 − c o s ( α ) ) + e 1 s i n ( α ) e 1 e 3 ( 1 − c o s ( α ) ) + e 2 s i n ( α ) e 2 e 3 ( 1 − c o s ( α ) ) − e 1 s i n ( α ) c o s ( α ) + e 3 2 ( 1 − c o s ( α ) ) ] {\displaystyle \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]}
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6ca944143959998d54bf310582f2cfc149109c)
