Distance vs. Displacement: How Direction Changes the Result

Distance counts the path

Distance and displacement both use units of length, but they answer different questions. Distance is the total length of the route traveled. It is a scalar, so it has magnitude but no direction. If a route contains several segments, add the length of every segment.

Displacement compares final and initial position

Displacement is the change in position:

$$\Delta \mathbf r=\mathbf r_f-\mathbf r_i.$$

It is a vector, so direction matters. In one dimension, a chosen positive axis lets direction be represented with positive and negative signs. A negative displacement does not mean a negative distance; it means the final position lies in the negative direction from the start.

A two-dimensional example

Suppose a hiker walks 3 km east and then 4 km north. The total distance is

$$3+4=7\,\text{km}.$$

The displacement is the vector from the starting point to the ending point. Its magnitude is

$$|\Delta \mathbf r|=\sqrt{3^2+4^2}=5\,\text{km},$$

with direction

$$\theta=\tan^{-1}\!\left(\frac{4}{3}\right)\approx53.1^\circ$$

north of east.

A quick check

Distance can equal the magnitude of displacement only when the traveled path is effectively straight from start to finish without reversing or detouring. In every case,

$$|\text{displacement}|\le\text{distance}.$$

That inequality is a useful way to catch sign or path-length mistakes.