2022 RI J1 CT Q3

The line \({{L}_{1}}\) has equation \(x=3\), \(\frac{6-y}{3}=z\) and the line \({{L}_{2}}\) has equation \(x-2=y-6\), \(z=7\).

1. Find the acute angle between \({{L}_{1}}\) and \({{L}_{2}}\).[2]
2. Show that \({{L}_{1}}\) and \({{L}_{2}}\) are skew lines.[2]
3. Let \({{P}_{1}}\) and \({{Q}_{1}}\) be two points on \({{L}_{1}}\) such that distance between them is \(\sqrt{10}\). Let \({{P}_{2}}\) and \({{Q}_{2}}\) be the two respective points on \({{L}_{2}}\) such that \({{P}_{1}}{{P}_{2}}\) and \({{Q}_{1}}{{Q}_{2}}\) are minimum. Find the exact distance between \({{P}_{2}}\) and \({{Q}_{2}}\).[2]