2025 HCI J2 BT Q6

When a light ray passes from a transparent medium to air, it is deflected through an angle. A light ray is shone on a transparent object \(\mathrm{P}\) in the shape of a rectangular prism. The light ray hits the base of the object at point \(A\left( 2,2,-3 \right)\) relative to origin \(O\) and travels in the direction parallel to \(4\mathbf{i}+\mathbf{k}\).

It leaves the top of the object at point \(B\) and the deflected ray passes through point \(C\left( 2,-5,1 \right)\).

The top of the prism is part of a plane with normal vector \(\mathbf{n}\) modelled by the equation \(\mathbf{r}\cdot \left( \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right)=6\).

The diagram above shows a cross-section of the object \(\mathrm{P}\) in the plane of the light ray and \(\mathbf{n}\). The angle, \(\theta \), between the light ray and a line perpendicular to the surface of the object \(\mathrm{P}\) is known as the angle of incidence.

1. Find the coordinates of point \(B\).[3]
2. Determine the thickness of the glass object.[4]

A phenomenon known as total internal reflection occurs when the light ray does not leave the transparent medium and is instead reflected within the transparent medium. When object \(\mathrm{P}\) is replaced by another transparent object \(\mathrm{Q}\) with the same dimensions as object \(\mathrm{P}\), total internal reflection occurs. The reflected ray makes the same angle \(\theta \) with the line perpendicular to the surface of object \(\mathrm{Q}\) at the point \(B\) and exits object \(\mathrm{Q}\) at point \(A'\).

3. Find the coordinates of \(A'\).[4]
4. Hence, or otherwise, find the vector equation of the reflected ray.[2]