Question

Suppose you first walk 28.6 m in a direction 20 degrees west of north and then 22 m in a direction 40 degrees south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position?

Answer 1

29.5 m 64° west of north

Explanation:

A suitable vector calculator can add the vectors for you. (See attached.) Here, we have used North as the 0° reference and positive angles in the clockwise direction (as bearings are measured).

___

You can also use a triangle solver (provided by many graphing calculators and stand-alone apps). For this, and for manual calculation (below) it is useful to realize the angle difference between the travel directions is 70°.

___

Using the Law of Cosines to find the distance from start (d), we have (in meters) ...

d² = 28.6² + 22² -2·28.6·22·cos(70°) ≈ 871.561

d ≈ √871.561 ≈ 29.52 . . . . meters

The internal angle between the initial travel direction and the direction to the end point is found using the Law of Sines:

sin(angle)/22 = sin(70°)/29.52

angle = arcsin(22/29.52×sin(70°)) ≈ 44.44°

This angle is the additional angle the destination is west of the initial travel direction, so is ...

20° west of north + 44.44° farther west of north = 64.44° west of north

__

In the second attachment, North is to the right, and West is down. This is essentially a reflection across the line y=x of the usual map directions and angles. Reflection doesn't change lengths or angles, so the computations are valid regardless of how you assign map directions to x-y coordinates.