Are quaternions generally multiplied in an order opposite to matrices?

Question

I was reading Jason Gregory's "Game Engine Architecture". Since he uses row vectors, there's an example that goes

v' = v*R1*R2*R3

Rn being matrices. Instead, since I'm using column vectors, I would write v' = R3*R2*R1*v. He then proceeds

v' = q3*q2*q1*v*~q1*~q2*~q3 Notice how the quaternion product must be performed in an order opposite to that in which the rotations are applied

Does that mean that I should compute q1*q2*q3*v*~q3*~q2*~q1 instead? Also, is the quaternion product associative?

Answer 1

is the quaternion product associative?

The * operator is also called Hamilton product and it is associative.

Does that mean that I should compute q1*q2*q3*v*~q3*~q2*~q1 instead?

No. You are interested in applying first rotation 1, then 2, and finally 3, right? So, for the sake of clarity, you can think of your operation as q3*(q2*(q1*v*~q1)*~q2)*~q3. Considering that Hamilton product is not commutative you should keep that order.

Note that I used parentheses thinking that it would help interpret the equation easier. Since Hamilton product is associative, rearranging the parentheses will not change the result.