- Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l }
2
0
7
\end{array} \right) + \lambda \left( \begin{array} { r }
2
- 2
1
\end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
2
0
7
\end{array} \right) + \mu \left( \begin{array} { l }
8
4
1
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).
- Write down the coordinates of \(A\).
Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\),
- show that \(\sin \theta = k \sqrt { 2 }\), where \(k\) is a rational number to be found.
The point \(B\) lies on \(l _ { 1 }\) where \(\lambda = 4\)
The point \(C\) lies on \(l _ { 2 }\) such that \(A C = 2 A B\). - Find the exact area of triangle \(A B C\).
- Find the coordinates of the two possible positions of \(C\).