7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.
- Find a vector equation for \(l _ { 1 }\).
The line \(l _ { 2 }\) has vector equation
$$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$
where \(\mu\) is a scalar parameter.
- Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
- Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
- Find the position vector of \(C\).