N2021 P1 Q8

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

\({{\mathbf{r}}_{1}}=\left( \begin{matrix} 3 \\ 2 \\ 0 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 2 \\ -3 \\ 1 \\ \end{matrix} \right)\) and \({{\mathbf{r}}_{2}}=\left( \begin{matrix} -4 \\ 1 \\ -3 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 1 \\ 2 \\ 4 \\ \end{matrix} \right)\)

respectively, where \(\lambda \) and \(\mu \) are parameters.

1. Find a cartesian equation of the plane containing \({{l}_{1}}\) and the point \(\left( 1,-3,-1 \right)\).[4]
2. Show that \({{l}_{1}}\) is perpendicular to \({{l}_{2}}\).[2]
3. 1. Find values of \(\lambda \) and \(\mu \) such that \({{\mathbf{r}}_{1}}-{{\mathbf{r}}_{2}}\) is perpendicular to both \({{l}_{1}}\) and \({{l}_{2}}\). State the position vectors of the points where the common perpendicular meets \({{l}_{1}}\) and \({{l}_{2}}\).[6]
   2. Find the length of this common perpendicular.[2]