Calculating The Total Displacement (Magnitude and Direction)

Question

Find the magnitude and direction of Mary's total displacement as she travels home.

Her following displacements are:

1) 50 m North

2) 100 m at 45 deg South of East

3) 200 m at 30 deg West of North

Answer 1

Let her home be at the origin of a cartesian coordinate system. Let's break each displacement vector into their x and y components.

50 m north vector: y component 50 m, no x coordinate

45 degree south of east vector: y component 100 sin (-45) deg = -70.7 m , x component 100 cos (-45) deg = 70.7 m

30 degree west of north vector: y component 200 sin(120) deg = 173.2 m , x component 200 cos(120) deg.= -100 m

Add the y components and the x components.

sum y components: 50 + -70.7 + 173.2 = 152.5 m (y component of resultant)

sum x components: 0 + 70.7 - 100 m = -29.3 m (x component of resultant)

Magnitude of resultant: √ (152.5² + -29.3²) = 155.3 m

Direction of resultant: arctan 152.5/-29.3 = -79.1 deg. as the vector is in the second quadrant, the angle is 90 - 79.1 = 10.9 degrees west of north.

Please check calculations.