2020 ACJC Promo Q8

The lines \(l\) and \(m\) are defined by the equations

\(l:\mathbf{r}=\mathbf{i}-\mathbf{k}+\lambda (2\mathbf{i}-6\mathbf{j}+3\mathbf{k})\),

\(m:\frac{x-1}{4}=\frac{a-y}{a}=\frac{z+3}{4}\)

1. Given that the lines intersect, show that \(a=6\).[2]
2. Find the position vector of \(N\), the foot of perpendicular from the point \(A\left( 5,0,1 \right)\) to the line \(l\).[3]
3. Hence or otherwise, find the position vector of the two points on \(l\) that are \(5\) units from \(A\).[3]