Part of the mains water system for a housing estate consists of water pipes buried beneath the ground surface. The water pipes are modelled as straight line segments. One water pipe, \(W\), is buried beneath a particular road. With respect to a fixed origin \(O\), the road surface is modelled as a plane with equation \(3 x - 5 y - 18 z = 7\), and \(W\) passes through the points \(A ( - 1 , - 1 , - 3 )\) and \(B ( 1,2 , - 3 )\). The units are in metres.
Use the model to calculate the acute angle between \(W\) and the road surface.
A point \(C ( - 1 , - 2,0 )\) lies on the road. A section of water pipe needs to be connected to \(W\) from \(C\).
Using the model, find, to the nearest cm, the shortest length of pipe needed to connect \(C\) to \(W\).