Displacement Formula

The Formula

\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

\Delta \vec{r} or \vec{d} for displacement; |\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

\Delta\vec{r} = \vec{r}_{\text{final}} - \vec{r}_{\text{initial}} \in \mathbb{R}^n; |\Delta\vec{r}| = \|\vec{r}_{\text{final}} - \vec{r}_{\text{initial}}\| \leq \int_\gamma |d\vec{r}| (distance traveled along path \gamma)

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Represent east as positive x and west as negative x.
  2. 2
    Step 2: Displacement = +5 + (-5) = 0 m.
  3. 3
    Step 3: The person is back at their starting point — zero displacement.

Answer

Displacement = 0 m (back to start).
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) to point B(4, 6). Find the displacement vector and its magnitude.

Common Mistakes

  • Confusing displacement with distance traveled — walking in a circle covers a large distance but has zero displacement
  • Forgetting that displacement is a vector (has both magnitude and direction), not just a number
  • Computing displacement as the sum of all path lengths instead of the straight-line change from start to end

Why This Formula Matters

Distinguishes 'net change in position' from 'total path length.'

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?

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

Why is the Displacement formula important in Math?

Distinguishes 'net change in position' from 'total path length.'

What do students get wrong about Displacement?

Walking in a complete circle covers a large distance traveled, but your net displacement is exactly zero.

What should I learn before the Displacement formula?

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

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