So I have a rotating object a. It is rotating at non-constant speed around it's local x axis. It is parented to object b. When the object a rotates, it's local x axis stays facing the same direction relatively to object b, but not the local y and local z axis.
In image the problem is explained a bit better. So I need to get a vector that is relative to the local x axis, but rotated by 90 degrees. And it should always match the b object's local y arrow's rotation around it's local x arrow. And I need to obtain a vector in global axis. I hope that you understand my problem. So all I need is the math behind. Of course, if there is need for a dot product or cross product or something similar down there, please, don't start typing down all the math of that function:D

So I have a rotating object a. It is rotating at non-constant speed around it's local x axis. It is parented to object b. When the object a rotates, it's local x axis stays facing the same direction relatively to object b, but not the local y and local z axis.

In image the problem is explained a bit better. So I need to get a vector that is relative to the local x axis, but rotated by 90 degrees. And it should always match the b object's local y arrow's rotation around it's local x arrow. And I need to obtain a vector in global axis. I hope that you understand my problem.

So all I need is the math behind. Of course, if there is need for a dot product or cross product or something similar down there, please, don't start typing down all the math of that function:D