Calculating Distance Between Two Joggers on a Run

[cvc121]

Homework Statement

Two joggers start their run at the same starting point. One jogger runs 2.0km north and then turns to the east and runs an additional 3.0km. The second jogger initially heads south and runs 4.0km before turning to the west and runs another 1.0km. How far apart are the two joggers once they are done?

Homework Equations

The Attempt at a Solution

My attempt at the solution is below. Can anyone verify my work to see if it is correct? If not, where did I go wrong?

 

[fresh_42]

Try to draw all jogging paths in one graphic and look whether you can produce symmetries. It'll shorten your calculation to 2 lines.

 

[gneill]

Your result is correct. But if I may suggest, you've taken a rather complicated way to get there by going through conversions to polar form and all the trig it entails.

If you stick to Cartesian representation of the vectors for the calculations it would go much easier. Then you can add or subtract the components directly, and at the end find the magnitude. For example, if you let a vector r = (x,y) represent a jogger's position when he finishes his trek, where x is the east-west direction component and y the north-south direction component, both in km, then for the two joggers:

r1 = (3, 2) which represents {3 km east, 2 km north}
r2 = (-1, -4) which represents {1 km west, 4 km south}

It's then a simple matter to form r = r1 - r2, a vector from jogger 2 to jogger 1, and find the magnitude of r.

 

[PhanthomJay]

Looks good, but you went through a lot of steps when instead you just needed the last four steps because the x and y components of A and B can be determined without those earlier steps.