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.

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°