How to proof the polarity of the reflected wave of Oblique incident.

As shown in the attachment, the book assumes ##\hat E_{||}^r=\hat y_r=(\hat x
\cos \theta_r +\hat z \sin\theta_r)##. Why? How do you proof this. I have another post here about the Normal Incidence and no luck.

I am not even convinced that the reflected E is even in the Plane of Incidence, how do you even proof this. I have 5 EM books only Griffiths even attempt to proof in a way I don't even agree for the Normal Incidence. Every book pretty much just give the polarity. If anyone have article or notes, please share with me.

Thanks

Alan

Attached Images

Oblique L.jpg (159.8 KB)