Is this book correct?

I’m currently reading this book on flight dynamics but I’m having some trouble getting my head around the way the author derives some equations of motion. He starts by showing the body in the first image as follows:

"Consider the motion referred to an orthogonal axis set (oxyz) with the origin o coincident with the cg of the arbitrary and, in the first instance, not necessarily rigid body shown in Fig. 4.1. The body, and hence the axes, is assumed to be in motion with respect to an external reference frame such as earth (or inertial) axes. The components of velocity and force along the axes ox, oy, and oz are denoted (U,V,W)
and (X,Y,Z), respectively. The components of angular velocity and moment about the same axes
are denoted (p,q,r) and (L,M,N), respectively. The point p is an arbitrarily chosen point within the
body with coordinates (x,y,z). The local components of velocity and acceleration at p relative to
the body axes are denoted (u,v,w) and (ax,ay, az), respectively."

Then he goes on defining the velocity components relative to "o" as follows:

He mentions that the additional elements on each of the velocity components correspond to the tangential velocities of the rotational motion and adds the following diagrams to "clear things up".

I do know that angular velocity times radius gives the tangential velocity. Now, concentrating on the "x" component of the local velocities, "u", the tangential velocities "qz" and "-rz" would only have those values if the point "p" was exactly on coordinates (0, y, z). Am I missunderstanding something?

Thanks in advanced for the help.