2024 HCI Promo Q11

A construction company is installing cables on the side of a building. Points \((x, y, z)\) are defined relative to a main anchor point \(O\) located at \((0, 0, 0)\), where units are in metres. Cables are laid in straight lines and the widths of the cables can be neglected.

An existing cable \(L\) starts at the main anchor point \(O\) and goes in the direction of \(\begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}\). A new cable \(M\) passing through points \(P(1, -1, 1)\) and \(Q(3, 0, 0)\) is installed.

1. Using a non-calculator method, show that the cables \(L\) and \(M\) do not meet.[4]

The construction company wishes to connect the point \(P\) to an inclined rectangular roof extension that is modelled by the plane with equation \(2x + 3y + z = 28\). It was found that the best way is to use a cable to connect the point \(P\) to the point \(F\) on the rectangular roof extension that is closest to \(P\).

2. Find the coordinates of the point \(F\).[3]
3. Hence find the exact length, in metres, of the cable needed to connect point \(P\) to the rectangular roof extension.[2]