C4 June 2005 Q7

7. The line \(l_1\) has vector equation

\[\mathbf{r} = \begin{pmatrix}3\\1\\2\end{pmatrix} + \lambda\begin{pmatrix}1\\-1\\4\end{pmatrix}\]

and the line \(l_2\) has vector equation

\[\mathbf{r} = \begin{pmatrix}0\\4\\-2\end{pmatrix} + \mu\begin{pmatrix}1\\-1\\0\end{pmatrix},\]

where \(\lambda\) and \(\mu\) are parameters.

The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).

The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\).
The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\).
The point \(D\) is such that \(ABCD\) is a parallelogram.