2021 SAJC Promo Q9

When referred to the origin \(O\), the points \(A\) and \(B\) have position vectors \(5\mathbf{j}+15\mathbf{k}\) and \(\mathbf{i}-\mathbf{j}+3\mathbf{k}\) respectively.

The line \({{l}_{1}}\) has equation \(\mathbf{r}=\left( \begin{matrix}-2 \\1 \\3 \end{matrix} \right)+\lambda \left( \begin{matrix}-1 \\2 \\3 \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).

1. Find a vector equation of line \({{l}_{2}}\) passing through the points \(A\) and \(B\) .[2]
2. Find the coordinates of the point \(C\), where \({{l}_{1}}\) and \({{l}_{2}}\) intersect.[3]
3. Find the position vector of the point \(F\), the foot of the perpendicular from \(A\) to the line \({{l}_{1}}\).[4]
4. Find the vector equation of the line of reflection of \({{l}_{2}}\) in the line \({{l}_{1}}\).[3]
5. Find a cartesian equation of the plane that contains the lines \({{l}_{1}}\) and \({{l}_{2}}\).[3]