2023 CJC Promo Q6

It is given that two lines, \(l\) and \(m\), with equations

\(\mathbf{r}=\left( \mathbf{i}+3\mathbf{k} \right)+\lambda \left( 2\mathbf{i}-4\mathbf{j}+7\mathbf{k} \right)\)

and \(\mathbf{r}=\left( 2\mathbf{i}+2\mathbf{j}-\mathbf{k} \right)+\mu \left( 3\mathbf{i}-2\mathbf{j}+3\mathbf{k} \right)\), where \(\lambda \), \(\mu \in \mathbb{R}\)

respectively, intersect at the point \(P\).

1. Verify that the coordinates of \(P\) are \(\left( -1,\ 4,\ -4 \right)\).[1]

A point \(A\) with position vector \(\mathbf{i}+3\mathbf{k}\) lies on line \(l\). The point \(Q\) on the line \(m\) is the closest point from \(A\) to the line \(m\).

2. Find the length of \(AQ\).[3]

The line \(n\) is a reflection of the line \(l\) in the line \(m\).

3. Find the position vector of the point of reflection of \(A\) in the line \(m\).
   Hence, find a cartesian equation of the line \(n\).[5]