Question 1319

Vectors II: Lines and Planes

2019 Paper 1 Question 12 Variant

Question

A ray of light passes from air into a material made into a rectangular prism. The ray of light is sent in direction

{\begin{pmatrix} - 4 \\ - 4 \\ 3 \end{pmatrix}}

from a light source at point

{P}

with coordinates

{\left( - 3, 5, 0 \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( - 9, - 1, - 1 \right).}

The acute angle between

{PQ}

and the normal to the top of the prism at

{Q}

{\theta}

and the acute angle between

{QR}

and the same normal is

{\beta.}

It is given that the top of the prism is a part of the plane

{x + y + 2 z = - 1,}

and that the base of the prism is a part of the plane

{x + y + 2 z = - 10.}

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.

(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 the prism.

[1]

(v)

What can be said about the value of

{k}

for a material for which

{\beta > \theta}

[1]