I've come to see it more as the inverse of a division operator. Like a quaternion can be defined as the ratio of two vectors q = a/b. then qb = a and qc will

I've come to see it more as the inverse of a division operator. Like a quaternion can be defined as the ratio of two vectors q = a/b. then qb = a and qc will rotate c the same amount as needed to take b into a. dual quaternions have a similar motivating derivation.