8. The line \(l\) has equation
$$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$
and the plane \(\Pi\) has equation
$$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$
- Find the coordinates of the point of intersection of \(l\) and \(\Pi\).
The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
- Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\).
The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
- Find the coordinates of the point \(A ^ { \prime }\).
- Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form
$$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$
where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.