Something about rotation matrices
Question: A matrix is called orthogonal iff . It is further called rotation matrix if . These are well known. But how to bridge the algebraic definite of rotation matrices and their geometric role?
I think the most essential geometric role of a rotation matrix is to rotate every vector in the same rotation angle around the same rotation axis. Therefore, geometrically, a rotation matrix is determined by a rotation angle and and rotation axis. Now the question is actually how to understand the relationship of the following algebraic and geometric definition:
Geometric: rotation angle and rotation axis
In other words, prove
1. Given , find the rotation angle and axis.
2. Given rotation angle and rotation axis , prove the corresponding matrix satisfy: