Wrong proof in textbook of optics?
In textbook on optics by Pedrotti (and I think Fowles does the same) on page 493 it is said that:
from which follows
where k, kr and kt are wave vectors of incident, reflected and transmitted wave.
Than Pedrotti concludes that kr and kt must lie in a plane determined by k and r. Must they?
Take, for example this:
k=(-1,0,1), r=(1,0,0), kr=(-1,5,3), kt=(-1,2,1).
If we put this vectors in the equation above, we will get 0, but kr and kt don't lie in the plane determined by k and r, which is y=0 plane.
So I conclude that Pedrotti is wrong. Am I right or wrong to conclude that? If wrong, why?