Swap order of quaternion multiplication

Question

I have three quaternions, A, B and C such that A = B * C.

I would like to be able to swap the order of B and C so that A = C * B but of course quaternions aren't commutitive so just swapping the order results in a different value for A.

Is there a way to transform B so that after the swap A = C * B' is true? I would prefer a transform that just uses B, but if need be I can settle for a transform that would also uses A or C.

Answer 1

You can derive a formula by setting up the equality $$B\cdot C=A=C\cdot B'$$ From here, it is clear that if we take the inverse of C on the left side of each equation, we obtain a formula for B' $$B'=C^{-1}\cdot B\cdot C$$

Answer 2

$B'=C^{-1}\cdot B\cdot C$

In addition to @Tyberius answer, the geographic meaning of $B'$ is actually representing the rotation operation of $B$ back in coordinates before $C$ rotation has occurred.

In another word, $B\cdot C$ means apply $B$ then in the resultant coordinates apply $C$ rotation, while $C\cdot B$ means apply $B$ then in the original coordinates apply $C$ rotation.