Quaternion manipulations

[Michael Stevens]

Hello. I started to learn quaternions and my question is: How to represent a rotation of coordinate system through rotation of the vector that is starting from its origin? To be clear, I have two joints: root and child. If the root has some rotation (around “world” fixed zero center), child will be affected by that. I want to understand which rotation should the child has to achieve the same state as before while root is staying fixedly (so we can say that vector will rotate relatively “world” coordinate system). I know that to “summ” a few different rotations in single point we can just multiply correspondent quaternions, but how is it possible to add rotation to a distant point?

Initial data:

Root rotation (in quaternion): q = real + q1*i+ q2*j+ q3*k

Vector: u = u1*i + u2*j + u3*k