A rotation is usually defined as a matrix $M$ meeting certain constraints, such that rotating vector $v$ by $M$ equals $Mv$.
Rotations, however, can be defined:
- The axis of rotation
- The angular distance of rotation
Thus, rotations have a direction and magnitude, and should be representable as vectors.
Is there a way to define a rotation as a vector $r$, so that rotating arbitrary vector $v$ by $r$ is readily defined in terms of typical vector operations (such as dot product)?
If not, what is the closest we can come?
The rotated vector $\mathbf{v}'$ of $\mathbf{v}$ with angle $\theta$ about the direction $\mathbf{u}$ (unit vector) can be computed by $$ \mathbf{v}'=\mathbf{v}\cos\theta + (\mathbf{u}\times \mathbf{v})\sin\theta+\mathbf{u}(\mathbf{u}\cdot\mathbf{v})(1-\cos\theta) $$ This is known as Rodrigues' formula.