Multiplication of Rotation Matrices in quaternion

Question

Given Data and specifications

NB : * means multiplication

1. Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is just take the product $Q*P$. If we want to perform the same in the quaternion domain, what I do is, take the quaternion $q$ of $Q$, then convert the point $P$ to a quaternion $p$ as $(w,x,y,z)=(0,x,y,z)$ then do the following operation $q*p*q^{-1}$ in quaternion. The resultant will be a quaternion with $w = 0$. That will be a position vector after rotation.

2. Suppose we need to multiply a rotation matrix ${S}_{3\times3}$ with rotation matrix ${Q}_{3\times3}$ then what we do is just take the product $S*Q$. That will be a new rotation matrix.

Question

If we want to multiply a rotation matrix $S_{3\times3}$ with rotation matrix ${Q}_{3\times3}$ in the quaternion domain (after converting $Q$, $S$ to quaternion q,s respectively) as I did for position vector/point in example (1), how will I do that? Is it just $s*q$?

Answer 1

Yes, because the conversion map from matrices to quaternions has to be a homomorphism.

In words, the product of the quaternions equals the quaternion for the product of the matrices.