7 Three planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in a line \(l\); planes \(\Pi _ { 2 }\) and \(\Pi _ { 3 }\) intersect in a line \(m\).

1. Show that \(l\) and \(m\) are in the same direction.
2. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
3. By considering the cartesian equations of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\), or otherwise, determine whether or not the three planes have a common line of intersection.