I am self-studying quaternions for an engineering project I want to conduct, but I am having difficulty interpreting the information regarding quaternion multiplication.
The wikipedia article I am referencing is: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternions
So if I have two quaternions which I want to apply sequentially to a point p, I can instead combine the quaternions in the sequence I want them to be applied and then apply them to point p all at once. However, I am having difficulty interpreting the instructions on the page… specifically:
$$ \vec{v}\vec{w} = \vec{v} \times \vec{w} – \vec{v} \bullet \vec{w} $$
Where (to my knowledge) $ \vec{v} $ and $\vec{w}$ are quaternions. In that case how can I take the cross product? Does the page intend to suggest taking the cross product of just the imaginary components and then scalar multiply just the real components?
Thanks in advance for any insight.
Best Answer
As the page you referenced says, that formula you gave is only for multiplication of the ($3$-dimensional) vector/imaginary part of the quaternion. There is a somewhat more complicated formula for multiplication of quaternions that have non-zero real parts given further on in that page.