A Simple Algebraic Way to understand Quaternions and Rotations in 3D

Before doing this study, using quaternions to represent rotation ( v′=qvq−1 ) is a magical formula to me.
It’s easy to verify its correctness by expanding the identity to the vector rotation formula simply using the multiplication rule of quaternions in the language of vector cross and dot products : Proof of the quaternion rotation identity.
I can easily understand the vector rotation formula. Because it can be derived from the geometry by very elementary vector operations.
In fact, the proof shows these two formulas are mathematically equivalent. Its reversed process is just Factorization. We can get the “quaternion rotation formula” from the “vector rotation formula” by the factorization. If the factorization is understandable:
For the first time, I can figure out the quaternion rotation formula from the first place.

Created with Raphaël 2.1.0 Axis-angle choose an axis vector choose the positive angle of rotation vector algebra vector rotation formula Factorization(vector cross and dot products can be represented in terms of quaternion) quaternion rotation formula

P. S.
There is another method. It uses 2D rotation analogy instead of the pure factorization. See:

• 3D Rotations and Quaternion Exponentials: Special Case
• 3D Rotations in General: Rodrigues Rotation Formula and Quaternion Exponentials

The Next: Remake Quaternions