Question

Bob walks 370 m south, then jogs 410 m southwest, then walks 370 m in a direction 28 degrees east of north.

Answer 1

Consider east-west direction along x-axis with north being in positive x-direction.

Consider north-south direction along x-axis with north being in positive x-direction.

A = magnitude of displacement in south direction = 370 m

$A_{x}$ = x-component of A = 0 m

$A_{y}$ = y-component of A = - 370 m

B = magnitude of displacement in south-west direction = 410 m

$B_{x}$ = x-component of B = - 410 Cos45 = - 289.91 m

$B_{y}$ = y-component of B = - 410 Sin45 = - 289.91 m

C = magnitude of displacement in 28 deg east of north direction = 370 m

$C_{x}$ = x-component of C = 370 Sin28 = 173.7 m

$C_{y}$ = y-component of C = 370 Cos28 =  326.7 m

$D_{x}$ = x-component of D = ?

$D_{y}$ = y-component of D = ?

To return to starting position, the sum of individual displacements alng each axis must be zero.

$A_{x}$ + $B_{x}$ + $C_{x}$ + $D_{x}$ = 0

0 -  289.91 + 173.7 + $D_{x}$ = 0

$D_{x}$ = 116.21 m

$A_{y}$ + $B_{y}$ + $C_{y}$ + $D_{y}$ = 0

- 370 - 289.91 + 326.7 + $D_{y}$ = 0

$D_{y}$ = 333.21 m

magnitude of D is given using Pythagorean theorem as

magnitude = sqrt(($D_{x}$)² + ($D_{y}$)²)

magnitude = sqrt((116.21)² + (333.21)²) = 352.89 m

direction : tan⁻¹(333.21/116.21) = 70.8 deg

Answer 2

We will measure all angles from West, the negative x-axis and divide the journey into 3 parts:
P1 = 370y
P2 = 410cos(45)x + 410sin(45)y = 290x + 290y
P3 = 370cos(270 - 28)x + 370sin(270 - 28) = -174x - 327y

Overall displacement:
x = 290 - 174 = 116 m
y = 370 + 290 - 327 = 333 m

displacement = √(116² + 333²)
= 353 m

Direction:
tan(∅) = y/x
∅ = tan⁻¹ (333 / 116)
∅ = 70.8° from West.