Pre calc question relating to vectors

Question

 

A person bikes from home and bikes 3 miles east, 2 miles southeast, 8 miles south, 2 miles southwest, and 5 miles east. If they went straight home, how far (s) would they have to bike?

s=. miles

Answer 1

Decompose the 2 miles southwest to a move sqrt(2) south and sqrt(2) west (the vectors make a 45-45-90 triangle. Aso decompose the 2 southeast to sqrt(2) south and sqrt(2) east)

Now add up east-west with east positive: 3 + sqrt(2) - sqrt(2) + 5 = +8

add up the north-south with north positive: -sqrt(2) - 8 - sqrt(2) = -8-2sqrt(2)

Use Pythagorean theorem on the two net displacement vectors in order to find the distance straight home (displacement equivalent to net motion from starting point.)