11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$\begin{aligned}
& l _ { 1 } : \mathbf { r } = \left( \begin{array} { l }
7
4
9
\end{array} \right) + \lambda \left( \begin{array} { l }
1
1
4
\end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
- 6
- 7
3
\end{array} \right) + \mu \left( \begin{array} { l }
5
4
b
\end{array} \right)
\end{aligned}$$
where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),
- show that \(b = - 3\) and find the coordinates of \(X\).
The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\) - Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
- Using the result obtained in part (b), find the exact area of triangle \(A X B\).
Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.