7. The plane \(\Pi\) has vector equation
$$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
- Find an equation of \(\Pi\) in the form \(\mathbf { r } \cdot \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant.
The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
- Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\).
The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
- Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.