N2026 P1 Q12

A mining company has identified a mineral layer below ground. The mining company intends to drill down to the mineral layer.

The ground is modelled as a plane with equation \(z=0\), with respect to an origin \(O\).

The mineral layer is modelled as part of a plane that contains the points \(A(10,15,-60)\), \(B(5,8,-40)\) and \(C(-5,10,-50)\) where the units are metres.

1. Find a cartesian equation of the plane containing \(A\), \(B\) and \(C\).[5]
2. Find the acute angle between the ground and the mineral layer.[2]

The mining company plans to drill vertically from the point \(P(-17,188,0)\) on the ground to reach the mineral layer.

3. Find the vertical distance from \(P\) to the plane containing the mineral layer.[2]
4. Find the shortest distance from \(P\) to the plane containing the mineral layer.[6]