2021 RI CT Q7

To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to the origin \(O\), every point in the virtual world is represented as coordinates \(\left( x,y,z \right)\). Perspective projection is a technqiue used to project points in the virtual world onto a virtual screen while respecting the scaling properties of objects at various distances. Getting this right in VR helps in the perception of depth and scale. In addition, only points that lie within a zone in front of the eye positioned at \(O\), known as the viewing frustum (see Fig. 1), will be projected onto the virtual screen.

In Fig. 1, a triangular object \(ABC\) in the viewing frustum with \(A\left( 36,36,12 \right)\) and \(B\left( 24,48,72 \right)\) is projected onto the virtual screen to give an image of \(A'B'C'\). It is given that the coordinates of \(A'\) are \(\left( 3,3,1 \right)\) and the line that passes through \(A'\) and \(B'\) is parallel to the vector \(2\mathbf{i}+\mathbf{j}-2\mathbf{k}\).

1. Show that the coordinates of \(B'\) are \(\left( 1,2,3 \right)\).[4]
2. Given that the coordinates of \(C'\) are \(\left( 1,4,2 \right)\), find the equation of the virtual screen in scalar product form.[2]
3. Find the exact area of triangle \(A'B'C'\).[2]
4. If the length of projection of \(\overrightarrow{BC}\) onto the normal of the virtual screen is \(\frac{57}{\sqrt{29}}\) units, find the possible position vector(s) of \(C\), leaving your answer(s) in exact form.[4]