The equations of the lines \(l_1\) and \(l_2\) are \(\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}\) and \(\mathbf{r}=\mathbf{c}+\mu \mathbf{d}\) respectively, where \[\mathbf{a}=-6\mathbf{i}+3\mathbf{j}+15\mathbf{k}, \hspace{0.5cm} \mathbf{b}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}, \hspace{0.5cm} \mathbf{c}=6\mathbf{i}+15\mathbf{j}+39\mathbf{k}, \hspace{0.5cm} \mathbf{d}=2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\]

and where \(\lambda\) and \(\mu\) are scalar parameters. The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\). In any order:

  1. find a vector which is parallel to \(\overrightarrow{PQ}\), giving your answer in the form \(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\);
  2. show that \(PQ=10\sqrt{6}\);
  3. find the position vectors of \(P\) and \(Q\).