how to combine 2 rotaion quaternions

Question

if I have 2 quaternions that represent rotation in 2 different axis say one that rotate 30deg around the x and another that rotate 15deg around the y how can I combine them in one quaternion

Answer 1

You can just multiply them using the rules for quaternion multiplication.

These rules are exactly such that multiplication is (under a group homomorphism between the quaternions and SO(3)) combination of rotations.

Important note: There is not one way to combine two rotations. When you first rotate 30 degrees around the x axis and then 15 around the y axis, you end up in a different configuration than when doing it the other way around.

Indeed, quaternion multiplication has the same so-called non-commutative behaviour.