Complex Addition/Examples/Travel 1/Proof 1

Proof
Let the route of the traveller be embedded in the complex plane.


 * Travel Example.png

Let $P$ be the final location of the traveller.

The given directions can be translated into absolute arguments thus:


 * Northeast is $45 \degrees$ north of east, and therefore an argument of $45 \degrees$


 * $30 \degrees$ west of north is $30 + 90 \degrees$ north of east, and therefore an argument of $120 \degrees$


 * $60 \degrees$ south of west is $60 + 180 \degrees$ north of east, and therefore an argument of $240 \degrees$.

Thus the problem reduces to finding the sum:

Then: