N2023 P1 Q9

The line \(l_1\) contains the point \(A\) with coordinates \((-3,1,2)\) and is parallel to the vector \(\begin{pmatrix}2\\1\\a\end{pmatrix}\), where \(a\) is a constant. The line \(l_2\) has equation \(\frac{x+2}{3}=\frac{y-1}{2}=z-5\). It is given that \(l_1\) and \(l_2\) cross at the point \(B\).

1. Find the value of \(a\) and the coordinates of \(B\).[5]
2. The plane \(\pi_1\) contains the point \(A\) and is perpendicular to \(l_2\).
   1. Find the shortest distance from \(B\) to \(\pi_1\).[3]
   2. Hence find the acute angle between \(l_1\) and \(\pi_1\).[2]
3. The plane \(\pi_2\) is perpendicular to \(\pi_1\) and contains \(l_1\). Find a cartesian equation of \(\pi_2\).[3]