# 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:

Algebraic:

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: