<p>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.</p>
<p>The point $A$ has coordinates $\left( 3,4,2 \right)$ and the line $l$ has equation $x-1=\frac{y+2}{3}=\frac{3-z}{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the cartesian equation of the plane $\Pi $ which contains $A$ and the line $l$ is $3x-y=5$.</div></li></ol>
<p>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 $.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1761054614716-2pkneluj13r.png" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the foot of the perpendicular, $F$, from $S$ to the plane $\Pi $.</div></li><li><div class="list-item-paragraph">Hence by finding $\overrightarrow{FA}$, or otherwise, show that a cartesian equation for the line $m$ is $\frac{x-3}{2}=y-4=2-z$.</div></li><li><div class="list-item-paragraph">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.</div></li></ol>

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 $x-1=\frac{y+2}{3}=\frac{3-z}{2}$.

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 $.

Find the foot of the perpendicular, $F$, from $S$ to the plane $\Pi $.

Hence by finding $\overrightarrow{FA}$, or otherwise, show that a cartesian equation for the line $m$ is $\frac{x-3}{2}=y-4=2-z$.

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.

2016 CJC CT Q5

2016 CJC CT Q5

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