Model cable intersections and minimise drilling distance

Electrical engineers are installing electricity cables on a building site. Points $(x,y,z)$ are defined relative to a main switching site at $(0,0,0)$, where units are metres. Cables are laid in straight lines and the widths of cables can be neglected.

An existing cable $C$ starts at the main switching site and goes in the direction $(3,1,-2)$.

A new cable is installed which passes through points $P(1,2,-1)$ and $Q(5,7,a)$.

(i) Find the value of $a$ for which $C$ and the new cable will meet.

To ensure that the cables do not meet, the engineers use $a=-3$. The engineers wish to connect each of the points $P$ and $Q$ to a point $R$ on $C$.

(ii) The engineers wish to reduce the length of cable required and believe in order to do this that angle $PRQ$ should be $90^\circ$. Show that this is not possible.

(iii) The engineers discover that the ground between $P$ and $R$ is difficult to drill through and now decide to make the length of $PR$ as small as possible. Find the coordinates of $R$ in this case and the exact minimum length.