2015 JJC P1 Q7

The plane \(\pi \) has equation \(13x-5y-7z=0\), and the line \({{l}_{1}}\) has equation \(\frac{x-11}{3}=\frac{y-7}{5}=\frac{z-4}{2}.\)

1. Calculate \(\left( \begin{matrix}3 \\5 \\2 \end{matrix} \right)\centerdot \left(\begin{matrix}13 \\-5 \\-7 \end{matrix} \right)\) and state the geometrical relationship between the line \({{l}_{1}}\) and the plane \(\pi \).
2. Given that \(P\) is the point \(\left( 11,\ 7,\ 4 \right)\), find the acute angle between the line \(OP\) and the plane \(\pi \).
3. Find the shortest distance from \(O\) to \({{l}_{1}}\).
4. The line \({{l}_{2}}\) has equation \(\mathbf{r}=\lambda \mathbf{d},\) where \(\mathbf{d}\ne \mathbf{0}\). Given that \(\left( \begin{matrix}3 \\5 \\2 \end{matrix} \right)\times\mathbf{d}=\mathbf{0}\), what can be deduced about the lines \({{l}_{1}}\) and \({{l}_{2}}\)?