Composition, Non-Commutativity, and Vector Decompositions of Finite Rotations

The need to use rotations occurs very often in different domains. We present a basic extensive treatment of rotations in 3D. The results are presented and derived in a coordinate-free setting, where no frames are required and no components of any matrix are manipulated. We start with the direct problem of establishing the finite rotation formula. Then we consider the composition and the vector decomposition of finite rotations. We conclude the paper by considering the inverse problem namely finding the axis of rotation and the angle of rotation from its effect on vectors.