NJC Vectors II Tutorial Q3

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

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

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

1. Verify that \({{l}_{1}}\) passes through point \(P\) with position vector \(\left( \begin{matrix} 1 \\ -4 \\ 2 \\ \end{matrix} \right)\).
2. Hence, show that the distance between point \(P\) and any point on the line \({{l}_{2}}\) is given by \(\sqrt{6{{\mu }^{2}}-12\mu +20}\).
   Deduce the shortest distance between point \(P\) and the line \({{l}_{2}}\).