Relative Velocities: Engineering Mechanics Dynamics - Hibbeler

[serverxeon]

Source: Engineering Mechanics Dynamics - Hibbeler
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My friends and I have discussed and have no idea how to solve.
So we looked at the textbook.

We were told to use this equation.

In this equation, A/B means 'A respect to B'
Also, the equation is derived based on an intermediate frame attached to point B, which is rotating at $\omega$.

In the textbook's solution below, however, the intermediate frame is attached to point B, but the $\omega$ is to the right of the frame (the centre of the prescribed circle). The $\omega$ is no longer about the frame through point B.

Hence, we have our suspicious of the textbook solution.
Anyway has got any idea if we're wrong, or is the textbook wrong?

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[haruspex]

ω is the same everywhere in a rotating frame.