Vector Addition — Hiker's Displacement (Worked Example)

Easy

Problem

A hiker walks 3.0 km due east, then turns and walks 4.0 km due north. What is the hiker's resultant displacement from the starting point (magnitude and direction)?

Tip-to-tail addition: the resultant is the hypotenuse.

Given

A = 3.0 km east
B = 4.0 km north
legs are perpendicular

Find

Resultant vector R: magnitude and direction.

Principle

Perpendicular vectors add tip-to-tail; the resultant is the hypotenuse of the right triangle they form.

Formula

R = \sqrt{A^2 + B^2}, \qquad \theta = \tan^{-1}\!\left(\frac{B}{A}\right)

R = √(A² + B²); θ = tan⁻¹(B/A)

Solution

Magnitude: $R = \sqrt{3.0^2 + 4.0^2} = \sqrt{9 + 16}$
Calculate: $R = \sqrt{25} = 5.0\ \mathrm{km}$
Direction: $\theta = \tan^{-1}(4.0/3.0) = \tan^{-1}(1.333)$
Calculate: $\theta = 53.1°$ north of east

5.0 km at 53.1° north of east

Unit check

√(km²) = km; the angle is unitless (degrees). Consistent.

What this means physically

The hiker walked 7.0 km total but is only 5.0 km from camp — displacement cares about the shortcut, not the path. Rescue teams use exactly this math to locate lost hikers.

Common mistake

Adding the distances (3 + 4 = 7 km) and calling it displacement. Distance is the path; displacement is the straight-line vector — direction matters.