2017 AJC Promo Q8

In order to construct a temporary shelter, three vertical poles \(OP\), \(AQ\) and \(BR\) of heights \(6\)m, \(8\)m and \(10\)m respectively are planted firmly their bases on the horizontal ground.

The base \(O\) is taken as the origin and the other bases are such that \(OA=OB=8\)m.

The mutually perpendicular unit vectors \(\mathbf{i}\) , \(\mathbf{j}\) and \(\mathbf{k}\) are defined with \(\mathbf{i}\) along \(OA\), \(\mathbf{j}\) along \(OB\) and \(\mathbf{k}\) vertically upwards as shown in the diagram.

1. Verify that the plane with equation \(\mathbf{r}\cdot \left( \begin{matrix} -1 \\ -2 \\ 4 \\ \end{matrix} \right)=24\) passes through \(P\), \(Q\) and \(R\).[3]
2. Find the angle between the plane \(PQR\) and the horizontal, giving your answer correct to the nearest \({{0.1}^{\circ }}\).[2]
3. Find the position vector of the point \(A'\), which is obtained when point \(A\) is reflected about the plane \(PQR\).[5]
4. A point \(N\) lying on the horizontal ground has coordinates \(\left( \alpha ,9,0 \right)\) where \(\alpha \) is a positive constant. Given that the shortest distance from point \(N\) to the plane \(PQR\) is \(12\) m, find the value of \(\alpha \).[3]