8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
2
- 3
4
\end{array} \right) + \lambda \left( \begin{array} { r }
- 1
2
1
\end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
2
- 3
4
\end{array} \right) + \mu \left( \begin{array} { r }
5
- 2
5
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters.
- Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)
The point \(A\) has position vector \(\left( \begin{array} { l } 0
1
6 \end{array} \right)\). - Show that \(A\) lies on \(l _ { 1 }\)
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
- Write down the coordinates of \(X\).
- Find the exact value of the distance \(A X\).
The distinct points \(B _ { 1 }\) and \(B _ { 2 }\) both lie on the line \(l _ { 2 }\)
Given that \(A X = X B _ { 1 } = X B _ { 2 }\) - find the area of the triangle \(A B _ { 1 } B _ { 2 }\) giving your answer to 3 significant figures.
Given that the \(x\) coordinate of \(B _ { 1 }\) is positive,
- find the exact coordinates of \(B _ { 1 }\) and the exact coordinates of \(B _ { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}