2012 ACJC P2 Q1

The lines \({{l}_{1}}\) and \({{l}_{2}}\) have equations

\(\mathbf{r}=\left( \begin{matrix} 2.5k \\ k \\ 2k \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 3 \\ 1 \\ \end{matrix} \right)\) and \(\mathbf{r}=\left( \begin{matrix} 3 \\ 0.5k \\ 0.5k \\ \end{matrix} \right)+\mu \left( \begin{matrix} 4 \\ 7 \\ 5 \\ \end{matrix} \right)\), \(\lambda \), \(\mu \in \mathbb{R}\)

respectively, where \(k\) is a constant.

1. Given that \({{l}_{1}}\) and \({{l}_{2}}\) intersect at a point, find the value of \(k\).[3]
2. Hence find the coordinates of the point of intersection.[1]