9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by
$$\begin{array} { l l }
l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) ,
l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) ,
\end{array}$$
where \(\lambda\) and \(\mu\) are parameters.
- Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
- Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\).
The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
- Find, in its simplest form, the exact area of the triangle \(P Q R\).
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