N2014 P1 Q9

The planes \(p\) and \(q\) are perpendicular. The plane \(p\) has equation \(x+2y-3z=12\). The plane \(q\) contains the line \(l\) with equation \[\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{4}.\] The point \(A\) has coordinates \((1,-1,3)\).

1. Find a cartesian equation of \(q\).[4]
2. Find a vector equation of the line of intersection of \(p\) and \(q\).[4]
3. The point \(B\) lies on the line of intersection of \(p\) and \(q\). Find an expression for \(AB^2\) in terms of a parameter. Hence find the coordinates of the point on this line which is nearest to \(A\).[5]