N2022 P1 Q11

A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, \(P\) and \(Q\).

Points are defined relative to an origin \(\left( 0,0,0 \right)\) at the gas field. The \(x\)-, \(y\)- and \(z\)- axes are in the directions east, north and vertically upwards respectively, with units in metres. \(P\) has coordinates \(\left( 1136,\,\,92,\,\,p \right)\) and \(Q\) has coordinates \(\left( 200,\,\,20,\,\,-15 \right)\).

1. The length of the pipeline \(PQ\) is \(939\) m. Given that the level of \(P\) is below that of \(Q\), find the value of \(p\).[3]

A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are \(\left( 400,\,600,\,-20 \right)\), \(\left( 500,\,200,\,-70 \right)\) and \(\left( 600,\,-340,\,-50 \right)\).

2. Find the cartesian equation of this plane.Find the cartesian equation of this plane.[4]
3. Hence find the coordinates of the point where the pipeline meets the rock.[4]
4. Find the angle that the pipeline between the points \(P\) and \(Q\) makes with the horizontal.[2]