2021 DHS J2 MYE P1 Q10 [Modified]

A light ray passes from air to a vertical reflective panel, \({{p}_{1}}\) (see diagram). The reflective panel, \({{p}_{1}}\) is part of the surface which has the equation \(ax+y-z=3\), where \(a\) is a constant. Referenced to an origin \(O\), the light ray passes through point \(A\left( 2,1,4 \right)\) and hits the panel at point \(B\left( 4,1,2 \right)\). It then gets reflected to form another light ray along \(BC\). The path of the light ray and its reflected path make the same angle \(\theta \) with \({{p}_{1}}\).

1. Show that \(a=1\).[1]
2. Find the projection vector of the light ray along \(AB\) onto \({{p}_{1}}\).[3]
3. Find a cartesian equation of the reflected light ray along \(BC\).[4]
4. Two other distinct panels parallel to \({{p}_{1}}\) are each placed at a distance of \(\frac{10}{\sqrt{3}}\) units away from \({{p}_{1}}\). Find the vector equations of these panels.[3]
5. The light ray \(BC\) hits one of the panels. Determine which is the panel and provide a brief reason.[1]