I am new to the topic of quaternions, but I have an excellent understanding of complex numbers and linear algebra.
I read that unitary quaternions (or versors) can be written in the form: $q = \exp(a\textbf{r}) = \cos(a) + \textbf{r}\sin(a)~,$ where $r^2 = -1$ and has no real part. This is similar to the complex formalism that is: $c = A \exp(i\theta) = A[cos(\theta) + i\sin(\theta)]$.
Can any quaternion be written in the same manner, by extending the formulation to: $q = A\exp(a\textbf{r}) = A[\cos(a) + \textbf{r}\sin(a)]$ ? If so, does the multiplication still work the same?
I have been trying to perform a quaternion rotation described here by hand, but I'm puzzeled by how to handle commutativity...