# Angular velocities in moving reference frames

Please refer the image below for a clear idea of my question.

So, the instructor has written the angular velocity of the wheel, w.r.t. ground and the unit vectors he used are the ones for the frame B. I can convert the angular velocity (which are currently in terms of unit vectors of frame B ie, i,j,k) in terms of the unit vectors for the ground frame (I,J,K).

It seems pretty logical, write i,j,k in terms of I,J,K and then substitute. But my question is what if i go the reverse way around? Can I, according to the figure, say that the net angular velocity of the wheel in terms of unit vectors of ground frame is

ω = ω1(I) + ω2(J)

and then write I and J in terms of i and j and substitute? It is evident that i'll get a different result this time! So will I be incorrect if I choose to start writing the expression for ω in ground frame first and then proceeding to write it in the frame B or is there some silly mistake I am making in the process?

Thanks!