Displacement Formula

Displacement is the straight-line change in position from start to end, with both a distance and a direction.

The Formula

Δr⃗=r⃗final−r⃗initial\Delta \vec{r} = \vec{r}_{\text{final}} - \vec{r}_{\text{initial}}

When to use: Where you ended up relative to where you started—direction and distance combined.

Quick Example

Walk 3 blocks east then 4 blocks north: displacement is 5 blocks northeast.

Notation

Δr⃗\Delta \vec{r} or d⃗\vec{d} for displacement; ∣Δr⃗∣|\Delta \vec{r}| for its magnitude

What This Formula Means

The straight-line change in position from start to end, with both a distance and a direction.

Where you ended up relative to where you started—direction and distance combined.

Formal View

Δr⃗=r⃗final−r⃗initial∈Rn\Delta\vec{r} = \vec{r}_{\text{final}} - \vec{r}_{\text{initial}} \in \mathbb{R}^n; ∣Δr⃗∣=∥r⃗final−r⃗initial∥≤∫γ∣dr⃗∣|\Delta\vec{r}| = \|\vec{r}_{\text{final}} - \vec{r}_{\text{initial}}\| \leq \int_\gamma |d\vec{r}| (distance traveled along path γ\gamma)

Worked Examples

Example 1

easy
A person walks 5 m east and then 5 m west. What is their displacement?

Answer

Displacement = 0 m (back to start).

First step

1
Step 1: Represent east as positive x and west as negative x.

Full solution

  1. 2
    Step 2: Displacement =+5+(−5)=0= +5 + (-5) = 0 m.
  2. 3
    Step 3: The person is back at their starting point — zero displacement.
Displacement is the straight-line vector from the starting point to the ending point. It depends only on start and end positions, not the path taken. Total distance walked was 10 m, but displacement was 0 — an important distinction.

Example 2

medium
A robot moves from point A(1,2)A(1, 2) to point B(4,6)B(4, 6). Find the displacement vector and its magnitude.

Example 3

medium
You walk 8 m east, 6 m north, then 8 m west. Find your displacement vector and its magnitude.

Common Mistakes

  • Summing the path length instead of the start-to-end arrow — displacement ignores the route taken.
  • Dropping the direction and reporting only a magnitude — displacement carries a direction too.
  • Using initial minus final instead of final minus initial — displacement is r⃗final−r⃗initial\vec{r}_{\text{final}}-\vec{r}_{\text{initial}}.

Why This Formula Matters

Displacement separates 'how far you traveled' from 'how far you got,' which is the first big idea in motion. A runner doing a 400 m lap travels 400 m but is displaced 0 m — getting this distinction wrong wrecks every later velocity and vector problem. Recognizing it by "Do I want the straight start-to-end change, ignoring the wandering path in between?" — rather than by familiar numbers — is what lets a student tell it apart from distance traveled and position and velocity in a mixed problem set.

Frequently Asked Questions

What is the Displacement formula?

The straight-line change in position from start to end, with both a distance and a direction.

How do you use the Displacement formula?

Where you ended up relative to where you started—direction and distance combined.

What do the symbols mean in the Displacement formula?

Δr⃗\Delta \vec{r} or d⃗\vec{d} for displacement; ∣Δr⃗∣|\Delta \vec{r}| for its magnitude

Why is the Displacement formula important in Math?

Displacement separates 'how far you traveled' from 'how far you got,' which is the first big idea in motion. A runner doing a 400 m lap travels 400 m but is displaced 0 m — getting this distinction wrong wrecks every later velocity and vector problem. Recognizing it by "Do I want the straight start-to-end change, ignoring the wandering path in between?" — rather than by familiar numbers — is what lets a student tell it apart from distance traveled and position and velocity in a mixed problem set.

What do students get wrong about Displacement?

The procedure for displacement is the easy part; the trap is summing the path length instead of the start-to-end arrow. Asking "Do I want the straight start-to-end change, ignoring the wandering path in between?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Displacement formula?

Before studying the Displacement formula, you should understand: vector intuition.

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