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](http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation) by hand, but I'm puzzeled by how to handle commutativity...



Can any quaternion be written in exponential form?
Multiplication ... Note
$$
\exp(a\_1\textbf{r}\_1) \exp(a\_2\textbf{r}\_2)= \exp(a\_1\textbf{r}\_1+a\_2\textbf{r}\_2) 
$$
provided $\textbf{r}\_1$ and $\textbf{r}\_2$ commute, which is not always true.


