2017 NJC P2 Q4

The diagram (not drawn to scale) shows the structure of a partially constructed building that is built on a horizontal ground. The building has a square base foundation of \(7\) m in length. Points \(O\), \(A\), \(B\) and \(C\) are the corners of the foundation of the building. The building currently consists of three vertical pillars \(OD\), \(AE\) and \(CF\) of heights \(8\) m, \(5\) m and \(5\) m respectively. A canvas is currently attached at \(D\), \(E\) and \(F\), forming a temporary shelter for the building. \(O\) is taken as the origin and vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), each of length \(1\) m, are taken along \(\overrightarrow{OA}\), \(\overrightarrow{OC}\) and \(\overrightarrow{OD}\) respectively.

1. Find a Cartesian equation of the plane that represents the canvas \(DEF\).[3]
2. Find the acute angle which the canvas \(DEF\) makes with the horizontal ground.[2]
3. Given that the canvas is to be extended along the plane \(DEF\) till it touches the horizontal ground, explain why point \(B\) will lie beneath the canvas.[2]

A cement roof is to be built to replace the extended canvas. A vertical partition wall is also to be built such that it is \(d\) m away from and parallel to the plane \(ODFC\), where \(0 < d < 7\).

4. Find the exact vector equation of the line where the roof meets the partition wall. Show your working clearly, leaving your answer in terms of \(d\).[4]
5. A lighting point, \(P\), is to be placed on the roof such that it is closest to \(B\). Find the position vector of \(P\).[3]