Question

A water main is to be constructed with a 20​% grade in the north direction and a 25​% grade in the east direction. determine the angle theta required in the water main for the turn from north to east.

Answer 1

Let n = North
and e = east

using vectors,
n =  j + 0.20 k
e = i + 0.25 k

n · e = 1x0 + 0x1 + 0.20×0.25 = 0.05
|n|·|e| = i(1² + 0.20²)·i(1²+0.25²) = 1.105

E° = arccos((n · e)/|n|·|e|)
E° = arccos(0.05/1.105)
E° = arccos (0.0452)
E° = 87.40°