N2019 P1 Q12

A ray of light passes from air into a material made into a rectangular prism. The ray of light is sent in direction \(\left( \begin{matrix}-2 \\-3 \\ -6 \end{matrix} \right)\) from a light source at the point \(P\) with coordinates \(\left( 2,\,2,\,4 \right).\) The prism is placed so that the ray of light passes through the prism, entering at the point \(Q\) and emerging at the point \(R\) and is picked up by a sensor at point \(S\) with coordinates \(\left( -5,\,-6,\,-7 \right).\) The acute angle between \(PQ\) and the normal to the top of the prism at \(Q\) is \(\theta \)and the acute angle between \(QR\) and the same normal is \(\beta\) (see diagram).

It is given that the top of the prism is a part of the plane \(x+y+z=1,\) and that the base of the prism is a part of the plane \(x+y+z=-9.\) It is also given that the ray of light along \(PQ\) is parallel to the ray of light along \(RS\) so that \(P,\,Q,\,R\) and \(S\) lie in the same plane.

1. Find the exact coordinates of \(Q\) and \(R.\)[5]
2. Find the values of \(\cos \,\theta \) and \(\cos \beta .\)[3]
3. Find the thickness of the prism measured in the direction of the normal at \(Q.\)[3]

Snell’s law states that \(\sin \theta =k\sin \beta\), where \(k\) is a constant called the refractive index.

4. Find \(k\) for the material of this prism.[1]
5. What can be said about the value of \(k\) for a material for which \(\beta >\theta\) ?[1]