NJC Vectors II Tutorial Q5

The lines \({{l}_{1}}\) and \({{l}_{2}}\) have equations \(\frac{x+8}{2}=\frac{5-y}{2}=z+1\) and \(\mathbf{r}=\alpha \left( \begin{matrix} 6 \\ -4 \\ 1 \\ \end{matrix} \right)\), \(\alpha \in \mathbb{R}\) respectively.

Find

1. the acute angle between \({{l}_{1}}\) and the \(z\)-axis.
2. the position vector of the foot of perpendicular, \(N\), from the origin to \({{l}_{1}}\).
3. the coordinates of the point of intersection of \({{l}_{1}}\) and \({{l}_{2}}\).

Hence, find a vector equation of the reflection of line \({{l}_{2}}\) in the line \({{l}_{1}}\).