The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{a} + s\mathbf{b}\) and \(\mathbf{r} = \mathbf{c} + t\mathbf{d}\), where \(s\) and \(t\) are real numbers and

\[\mathbf{a} = 5\mathbf{i}+ 6\mathbf{j}- \mathbf{k}, \quad\mathbf{b} = \mathbf{i} +\mathbf{j}+ \mathbf{k}, \quad\mathbf{c} = \mathbf{i}+ 13\mathbf{j} +7\mathbf{k}, \quad\mathbf{d} = \mathbf{i} + 6\mathbf{j} + 3\mathbf{k}.\]

Also, the point $P $ on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2.\) In any order:

  1. show that \(PQ = \sqrt{38},\)

  2. find the position vectors of \(P\) and \(Q,\)

If \(P\) is on \(l_1\), what form must its position vector \(\mathbf{p}\) take? How about \(Q\) with position vector \(\mathbf{q}\)?

If \(PQ\) is perpendicular to both \(l_1\) and \(l_2\), what can we write down?

  1. obtain the Cartesian equation of the plane containing \(l_1\) and \(PQ\).

Can we find a vector that is normal to the required plane?