2006 AJC P1 Q8 [Modified]

The plane \(\pi \) contains the origin and is parallel to vectors \(2\mathbf{i}+6\mathbf{j}+\mathbf{k}\) and \(\mathbf{i}+2\mathbf{j}\). A point \(P\) with position vector \(3\mathbf{i}+\mathbf{j}+2\mathbf{k}\) is reflected about the plane \(\pi \). The line \(l\) passes through the point \(P\) and is parallel to \(\mathbf{i}-\mathbf{j}+3\mathbf{k}\).

1. Find the position vector of the intersection point between \(l\) and \(\pi \).[3]
2. Find the position vector of the foot \(N\) of the perpendicular from \(P\) to the plane \(\pi \). Hence, find the position vector of \(P'\), the reflection of \(P\) about \(\pi \).[4]
3. Hence or otherwise, find the equation of the line \(l'\), the reflection of line \(l\) about \(\pi \).[2]