A ray of light passes from air into a material made into a rectangular prism. The ray of light is sent in direction $\left(\begin{array}{l}-2 \ -3 \ -6\end{array}\right)$ from a light source at the point $P$ with coordinates $(2,2,4)$. 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 $(-5,-6,-7)$. The acute angle between $P Q$ and the normal to the top of the prism at $Q$ is $\theta$ and the acute angle between $Q R$ 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 $P Q$ is parallel to the ray of light along $R S$ so that $P, Q, R$ and $S$ lie in the same plane.

(i) Find the exact coordinates of $Q$ and $R$.

[5]

(ii) Find the values of $\cos \theta$ and $\cos \beta$.

[3]

(iii) 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.

(iv) Find $k$ for the material of this prism.

[1]

(v) What can be said about the value of $k$ for a material for which $\beta>\theta$ ?

[1]