2017 AJC P2 Q3

Planes \({{\Pi }_{1}}\) and \({{\Pi }_{2}}\) are defined by

\({{\Pi }_{1}}:x-2y+2z=7\), \({{\Pi }_{2}}:\mathbf{r}\cdot \left( \begin{matrix} a \\ 3 \\ -1 \\ \end{matrix} \right)=8\), where \(a\) is a constant.

1. The point \(P\) has position vector \(-2\mathbf{i}+\mathbf{j}+\mathbf{k}\). Find the position vector of \(F\), the foot of the perpendicular from \(P\) to plane \({{\Pi }_{1}}\).
   Hence, or otherwise, find the shortest distance from \(P\) to plane \({{\Pi }_{1}}\).[5]
2. Line \(m\) passes through the point \(F\) and is parallel to both planes \({{\Pi }_{1}}\) and \({{\Pi }_{2}}\). Find the vector equation of line \(m\).[2]
3. It is given that the point \(Q\left( 1,-4,-1 \right)\) lies on line \(m\). Find the value of \(a\).[3]
4. Find the length of projection of \(\overrightarrow{PQ}\) on the \(x-y\) plane.[3]