6．The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1
10
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 5
4 \end{array} \right)
& L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 1
2
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
2
2 \end{array} \right) \end{aligned}$$ （a）Show that \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular．
（b）Show that \(L _ { 1 }\) and \(L _ { 2 }\) are skew lines． The point \(A\) with position vector \(- \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on \(L _ { 2 }\) and the point \(X\) lies on \(L _ { 1 }\) such that \(\overrightarrow { A X }\) is perpendicular to \(L _ { 1 }\)
（c）Find the position vector of \(X\) ．
（d）Find \(| \overrightarrow { A X } |\) The point \(B\)（distinct from \(A\) ）also lies on \(L _ { 2 }\) and \(| \overrightarrow { B X } | = | \overrightarrow { A X } |\)
（e）Find the position vector of \(B\) ．
（f）Find the cosine of angle \(A X B\) ．