N2017 P1 Q10

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 \(\left( \begin{matrix}3 \\1 \\-2 \end{matrix} \right)\). A new cable is installed which passes through points \(P\,(1,2-1)\) and \(Q\,(5,7,a)\).

1. Find the value of \(a\) for which \(C\) and the new cable will meet.[4]

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 \(C\).

2. 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.[4]
3. 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.[5]