In a high-precision manufacturing bay, a robotic arm maps the positions of three docking pins relative to a fixed calibration origin $O$. The pins have the following position vectors:

$\bullet$ Pin $P$ has position vector $4\mathbf{i}-\mathbf{j}+2\mathbf{k}$ $\bullet$ Pin $Q$ has position vector $6\mathbf{i}+3\mathbf{j}-2\mathbf{k}$ $\bullet$ Pin $R$ has position vector $k\mathbf{i}+3\mathbf{j}+5\mathbf{k}$

where $k$ is a constant.

The guide rail $L$ is a straight line passing through the centers of pins $P$ and $Q$.