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.