2017 CJC Promo Q8

The line \(l\) has equation \(\mathbf{r}=4\mathbf{i}-9\mathbf{i}+9\mathbf{k}+\lambda \left( -2\mathbf{i}+\mathbf{j}-2\mathbf{k} \right)\), \(\lambda \in \mathbb{R}\).

1. The line \(l\) lies in the plane with cartesian equation \(ax+by-3z+1=0\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).[4]

The point \(P\) has position vector \(3\mathbf{i}+8\mathbf{j}+5\mathbf{k}\).

2. Find the foot of perpendicular from \(P\) to \(l\).[3]
3. Hence or otherwise, find the shortest distance from \(P\) to \(l\).[2]