Is this right?

Question

vertos: algebraically 4 km east, 6km south, 3km east, 5km north, 10 km west, 8km north, and 3 km south.

I'm trying to find the 5km, 36.87 South of West

Answer 1

If you net the east and west displacements, you get a value of 3 km west.
If you net the north and south displacements, you get a value of 4 km north.

So you have 3 W and 4 N. The resultant will have a magnitude of
5 km ( = sqrt (3^2 + 4^2)).

The direction is north of west by an angle whose sin is 4/5 = 0.8.
sin^-1 (0.8) = 53.13 degrees.

This is similar to the answer shown in your question, but is rotated exactly 90 degrees clockwise. Hope you can figure out why your proposed answer does not match my analysis.