The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$ respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Show that the position vector of \(P\) is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and find the position vector of \(Q\). Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), an equation of the plane \(\Pi\) which passes through \(P\) and is perpendicular to \(l _ { 1 }\). The plane \(\Pi\) meets the plane \(\mathbf { r } = p \mathbf { i } + q \mathbf { j }\) in the line \(l _ { 3 }\). Find a vector equation of \(l _ { 3 }\).