2016 CJC CT Q5

In computer graphics rendering, ray tracing is the technique of tracing the paths of incident light rays onto surfaces and calculating the paths of their reflected lights rays to achieve a realistic image to a given point of view.

The point \(A\) has coordinates \(\left( 3,4,2 \right)\) and the line \(l\) has equation \(x-1=\frac{y+2}{3}=\frac{3-z}{2}\).

1. Show that the cartesian equation of the plane \(\Pi \) which contains \(A\) and the line \(l\) is \(3x-y=5\).

A light ray travels in a straight line, \({{l}_{SA}}\), from a point source at \(S\) with coordinates \(\left( 4,2,3 \right)\) towards point \(A\) and is reflected off the surface of the plane \(\Pi \). The reflected light ray travels along a line \(m\) and the angles that lines \({{l}_{SA}}\) and \(m\) make with the normal vector of the plane \(\Pi \) are the same. Then line \({{l}_{SA}}\) and \(m\) lie in a plane that is perpendicular to \(\Pi \).

2. Find the foot of the perpendicular, \(F\), from \(S\) to the plane \(\Pi \).
3. Hence by finding \(\overrightarrow{FA}\), or otherwise, show that a cartesian equation for the line \(m\) is \(\frac{x-3}{2}=y-4=2-z\).
4. The viewing window in this digital simulation is the rectangle in the \(xy\)-plane starting at the origin with \(4\) units length in the positive \(x\)-direction and \(3\) units height in the positive \(y\)-direction. Determine whether the reflected light ray along line \(m\) will reach the viewing window.