Aperson starts walking from home and walks: 4 miles east 3 miles southeast 3 miles south 4 miles southwest 3 miles east find total displacement

A geometry program can show you the total displacement is about 10.14 miles.

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You can use a vector calculator to find the solution, too. Using bearing angles, the sum is
   4∠90° + 3∠135° + 3∠180° + 4∠225° + 3∠90° ≈ 10.1390∠141.6354°

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If you want to do it by hand, you can recognize the sum will be
   7 miles east + 3 miles south + 3 miles southeast + 4 miles southwest

Distances that are not in the direction of one of the coordinate axes can be translated to rectangular coordinates by 
   displacement*(cos(angle), sin(angle))
Angles can be measured in the conventional way—from the positive x-axis. A direction of southeast will be +315° or -45°. A direction of southwest will be +225° or -135°.

Then the sum of the displacements in rectangular coordinates is ...
   = (7, -3) + 3*(cos(-45°), sin(-45°)) + 4*(cos(-135°), sin(-135°))
   = (7, -3) + ((√2)/2)*((3, -3) + (-4, -4))
   = (7, -3) + 0.707107*(-1, -7)
   = (6.2929, -7.9497)
Then the Pythagorean theorem is used to find the direct distance from home to this displaced location.
   d = √(6.2929² +(-7.9497)²) ≈ √102.7990
   d ≈ 10.1390 . . . . miles