# How do displacement and distance differ?

Feb 8, 2015

Displacement, as a vector, depends on how far a moving object is in relation to its initial position, versus distance, as a scalar, is the amount traveled regardless of where it starts.

Think of displacement as having to do with magnitude and direction. If I were to move right from the origin at 8 meters (m), then move left 7 m, and then move right 2 m, my final displacement will be 3 m. This is dependent on the coordinate system and where you set your signs for direction. In my example, I had my x-direction being positive in the right and negative to the left.

Thus, I took the 8 m, subtract it by 7 m, and then add 2 m to get the overall displacement. Now if they are asking for just the first two movements, then the displacement will be 1 m.

For distance, I do not have to worry about direction and just add the absolute values of each movement. For the example above, I would just add the 8 m, 7 m, and 2 m to get the total amount traveled, which is 17 m.

Displacement can also be expressed as $\Delta d = {d}_{f} - {d}_{i}$, where ${d}_{f}$ is the final position and ${d}_{i}$ is the initial position. This can correlate with distance in another example below.

Starting at 5 m right from the origin, I walk at a distance of 40 m to the left, and then 20 m to the right. The total distance is 60 meters, but my overall displacement is $\Delta d = - 15 m - 5 m = - 20 m$. My final position is -15 m since I took the +5 m (the initial), subtract it by 40 m (moving to left), and then add 20 m.

Here is a cool video on how both components work. Coordinate systems can also be drawn with compasses, designating east as positive and west as negative.

Doing pictures is always the best option, so hopefully the differences of distance and displacement will make more sense as you do representations of similar examples.