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.
(i)
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]
(ii)
Find the angle between the plane $PQR$ and the horizontal, giving your answer correct to the nearest ${{0.1}^{\circ }}$.
[2]
(iii)
Find the position vector of the point $A’$, which is obtained when point $A$ is reflected about the plane $PQR$.
[5]
(iv)
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]
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