Question

A ship leaves the island of Guam and sails a distance 300 km at an angle 41.0 ∘ north of west

A)

In which direction must it now head so that its resultant displacement will be 105 km directly east of Guam? (Express your answer as an angle measured south of east

B)

How far must it sail so that its resultant displacement will be 105 km directly east of Guam?

Answer 1

Rewrite this in a vector form in cartesian coords
A = 300 km at 41deg North of West = 300 km * (cos(105) i + sin(105) j) = -77.65 i + 289.78 j km
C = 105 km East of guam = 105 i + 0 j km

A+B = C ==> B = C - A
B = (105 + 77.65) i + (0 - 289.78) j km = 182.65 i - 289.78 j km

B is the vector that gets us there.
Angle = atan(y/x) = atan(-289.78/182.65) = -57.78 deg

This angle is measured from the +x (East) axis. So this would be called 57.78 deg South of East.

The magnitude of the B vector is the distance it must sail
||B|| = sqrt(182.65^2 + 289.78^2) km = 342.54 km