# Quaternion Multiplication - Cross product of a $4\times1$ vector?

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.

• Yes, that page says "The imaginary part ${\displaystyle b\mathbf {i} +c\mathbf {j} +d\mathbf {k} }$ of a quaternion behaves like a vector ${\displaystyle {\vec {v}}=(b,c,d)}$ in three dimension vector space...When multiplying the vector/imaginary parts, ..." Commented Sep 10, 2020 at 2:58
• @J.W.Tanner So where does the real component from q go? Does the real component of a unit quaternion contain no useful information when multiplying quaternions? Commented Sep 10, 2020 at 3:00
• It says further on there that $(s+\vec v)(t + \vec w)=(st-\vec v\cdot \vec w)+(s\vec w + t\vec v+\vec v\times \vec w)$; the formula in your post works when $s=t=0$ Commented Sep 10, 2020 at 3:06
• @J.W.Tanner Ok thanks, I was actually having trouble understanding both of those equations, but when synthesized I understand what is being said. I wish I could mark your comment as an answer! Commented Sep 10, 2020 at 3:12

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.