1In this paper (equation 5), the author is presenting an equation of quaternion multiplication of three arrays. The second array is 3x1 while the other two are 4x1. Also, the author uses 'principle component'.
How is it possible to multiply quaternions with different dimensions?
Also, what is principle component? Is it the same as PCA? How is it calculated?
I don't know what is PCA, but the formula has sense if:
$\otimes$ represents the qaternions multiplication,
$\omega$ is a pure imaginary quaternion that represents a vector,
$q$ and $q^{-1}$ are the couple of quaternions that represent a rotation,
and the words principle components, indicates the correspondence from the pure imaginary quaternion that results from the given product and the corresponding vector in 3D space.
To see how this works and it is calculated you can see here.