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 the position vectors of \(P\) and \(Q\).

Let’s tackle the last part first.

Can we express the position vectors of \(P\) and \(Q\) in terms of parameters?

How can we write down the fact that \(PQ\) is perpendicular to each of the lines?