- The drainage system for a sports field consists of underground pipes.
This situation is modelled with respect to a fixed origin \(O\).
According to the model,
- the surface of the sports field is a plane with equation \(z = 0\)
- the pipes are straight lines
- one of the pipes, \(P _ { 1 }\), passes through the points \(A ( 3,4 , - 2 )\) and \(B ( - 2 , - 8 , - 3 )\)
- a different pipe, \(P _ { 2 }\), has equation \(\frac { x - 1 } { 2 } = \frac { y - 3 } { 4 } = \frac { z + 1 } { - 2 }\)
- the units are metres
- Determine a vector equation of the line representing the pipe \(P _ { 1 }\)
- Determine the coordinates of the point at which the pipe \(P _ { 1 }\) meets the surface of the playing field, according to the model.
Determine, according to the model,
the acute angle between pipes \(P _ { 1 }\) and \(P _ { 2 }\), giving your answer in degrees to 3 significant figures,the shortest distance between pipes \(P _ { 1 }\) and \(P _ { 2 }\)