7. Relative to a fixed origin, two lines have the equations
and
$$\begin{aligned}
& \mathbf { r } = \left( \begin{array} { c }
7
0
- 3
\end{array} \right) + s \left( \begin{array} { c }
5
4
- 2
\end{array} \right)
& \mathbf { r } = \left( \begin{array} { l }
a
6
3
\end{array} \right) + t \left( \begin{array} { c }
- 5
14
2
\end{array} \right) ,
\end{aligned}$$
where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,
- find the position vector of their point of intersection,
- find the value of \(a\).
Given also that \(\theta\) is the acute angle between the lines,
- find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.