Quote by Simon Bridge
... so what is ##\vec{r}## ?

Well,
r is vector from the origin the point on the boundary where wave is reflected and transmitted. It is the point where the wave "touches" the boundary.
Well, I think Pedrotti takes only the equations I have written in his argument (Introduction to Optics, third edition, page 493).
And what does that mean three vectors to have the same extent against fourth vector? How rotation helps in proving coplanarity?