Complex Addition/Examples/Travel 2/Proof 1

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


 * Travel Example 2.png

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

The given directions can be translated into absolute arguments thus:


 * Southeast is $45 \degrees$ south of east, and therefore an argument of $315 \degrees$


 * Due west is an argument of $180 \degrees$


 * $30 \degrees$ north of east is an argument of $30 \degrees$


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

Thus the problem reduces to finding the sum:

Then: