2023 YIJC P2 Q10

A security system in a museum uses laser to protect a valuable artefact. The path of the laser beam can be represented by the line passing through the points \(\left( 0,\,-3,\,1 \right)\) and \(\left( 2,\,-1,\,2 \right)\). The museum also installed a reflective shield, represented by the plane passing through the points \(\left( 6,\,9,\,3 \right)\), \(\left( -2,\,13,\,1 \right)\) and \(\left( 4,\,10,\,0 \right)\).

1. Show that a cartesian equation of the plane representing the reflective shield is \(x+2y=24\).[2]
2. Find \(\theta \), the acute angle between the laser beam and the reflective shield.[2]
3. Find the position vector of the point where the laser beam hits the reflective shield.[2]

The reflective shield deflects the laser beam such that the angle between the laser beam and the reflective shield equals to the angle between the deflected laser beam and the reflective shield. The laser beam, the deflected laser beam and the normal of the reflective shield lie in the same plane (see diagram).

4. Find a vector equation of the line representing the deflected laser beam.[6]