2011 MI P2 Q4

The plane \({{\pi }_{1}}\) contains the point \(A\left( 2,2,-2 \right)\), \(B\left( 1,3,-3 \right)\) and is parallel to the line \(\mathbf{r}=\left( \begin{matrix}5 \\6 \\7 \end{matrix} \right)+\lambda \left( \begin{matrix}-2 \\0 \\1 \end{matrix} \right)\).

Find

1. the equation of \({{\pi }_{1}}\) in scalar product form,[2]
2. the position vector of the foot of the perpendicular form the point \(P\left( -1,7,6 \right)\) to the plane \({{\pi }_{1}}\),[3]
3. the position vector of the point \(P'\), which is the reflection of \(P\) about \({{\pi }_{1}}\).[2]

The plane \({{\pi }_{2}}\)has the equation \(x+y-2z=2\).

4. Find the equation of the line of intersection of \({{\pi }_{1}}\) and \({{\pi }_{2}}\).[2]