Displacement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Displacement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Displacement is the straight arrow from where you began to where you ended, ignoring the wandering path between.

Common stuck point: 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.

Sense of Study hint: Ask: Do I want the straight start-to-end change, ignoring the wandering path in between?

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.

Example 4

medium
A point moves along the sides of a rectangle with vertices (0,0),(6,0),(6,4),(0,4)(0,0), (6,0), (6,4), (0,4), going around once. Find both the displacement and the total distance traveled.

Example 5

challenge
A particle moves along r⃗(t)=(cos⁡t,sin⁡t)\vec r(t) = (\cos t, \sin t) from t=0t = 0 to t=2πt = 2\pi. Compare the total distance traveled to the displacement.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A car drives 3 km north, then 4 km east. What is the magnitude of the total displacement?

Example 2

hard
A particle undergoes three displacements: d1⃗=(2,−1)\vec{d_1} = (2, -1), d2⃗=(−3,4)\vec{d_2} = (-3, 4), d3⃗=(1,2)\vec{d_3} = (1, 2). Find the net displacement vector and its magnitude.

Example 3

easy
You walk 3 blocks east then 4 blocks north. What is your displacement magnitude?

Example 4

easy
You walk all the way around a circular track and return to start. What is your displacement?

Example 5

easy
A point moves from (1,2)(1, 2) to (4,6)(4, 6). Write the displacement as a vector.

Example 6

easy
Is displacement a vector or a scalar?

Example 7

easy
You walk 5 m east, then 5 m west. What is your displacement?

Example 8

easy
A car drives 10 km north. What is the direction of its displacement?

Example 9

easy
Distance traveled is 12 m but displacement is 8 m. Is this possible?

Example 10

easy
A point starts at (2,3)(2, 3) and has displacement ⟨5,−1⟩\langle 5, -1 \rangle. Find its ending position.

Example 11

medium
You walk 6 m east, 3 m north, then 2 m west. Find your displacement vector.

Example 12

medium
After walking 6 m east, 3 m north, 2 m west, find the magnitude of your displacement.

Example 13

medium
A drone flies from (0,0)(0,0) to (3,4)(3,4) to (6,0)(6,0). Find its total displacement from start.

Example 14

medium
Two trips: A) 10 m straight east; B) 6 m east, then 8 m east. Which has greater displacement?

Example 15

medium
A hiker's displacement is ⟨−3,−4⟩\langle -3, -4 \rangle km. How far and roughly which way from start are they?

Example 16

medium
Why can two people who travel very different distances end up with the same displacement?

Example 17

medium
A robot's displacement over a trip is ⟨0,0⟩\langle 0, 0 \rangle, but it traveled 20 m. What kind of path did it take?

Example 18

medium
Displacement from AA to BB is ⟨3,5⟩\langle 3, 5 \rangle. What is the displacement from BB to AA?

Example 19

challenge
A ship sails 12 km on a bearing of 90∘90^\circ (east), then 12 km on a bearing of 180∘180^\circ (south). Find its displacement magnitude and rough direction.

Example 20

challenge
A particle moves around three sides of a square of side 44 (start at a corner, traverse 3 sides). Find its displacement magnitude and the distance traveled.

Example 21

challenge
A plane flies 100 km north, then 100 km east, then 100 km south. Find the magnitude of its displacement.

Example 22

challenge
Explain why average velocity uses displacement, not distance, and what that implies for a round trip.

Example 23

easy
A point moves from (0,0)(0,0) to (8,6)(8,6). Find its displacement magnitude.

Example 24

easy
A drone moves from (2,1)(2,1) to (5,5)(5,5). Write the displacement as a vector.

Example 25

easy
A particle is displaced by ⟨−4,3⟩\langle -4, 3 \rangle. Find its displacement magnitude.

Example 26

easy
Starting at (1,4)(1, 4) and ending at (1,−2)(1, -2), find the displacement vector.

Example 27

easy
A point starts at (0,0)(0,0) and is displaced by ⟨7,−24⟩\langle 7, -24 \rangle. Find its ending position and displacement magnitude.

Example 28

medium
A bird flies from (2,3)(2, 3) to (10,9)(10, 9), then to (2,9)(2, 9). What is its displacement from start to end?

Example 29

medium
Two displacements u⃗=⟨3,5⟩\vec u = \langle 3, 5 \rangle and v⃗=⟨−1,2⟩\vec v = \langle -1, 2 \rangle are applied in sequence. Find the net displacement.

Example 30

medium
A car travels 2020 km east and then 1515 km north. Compute the displacement magnitude.

Example 31

medium
From point AA to point BB, the displacement is ⟨6,8⟩\langle 6, 8 \rangle. A different person walks 2020 m getting from AA to BB. Explain why the magnitudes differ.

Example 32

medium
A particle's displacement is ⟨9,−12⟩\langle 9, -12 \rangle. Find its magnitude and write a unit vector in the same direction.

Example 33

medium
Why must displacement satisfy ∣Δr⃗∣≤total distance traveled|\Delta \vec r| \le \text{total distance traveled}?

Example 34

medium
A particle moves ⟨2,3⟩\langle 2, 3 \rangle, then ⟨4,−1⟩\langle 4, -1 \rangle, then ⟨−6,2⟩\langle -6, 2 \rangle. Find the net displacement.

Example 35

medium
A point's displacement from AA to BB is ⟨5,12⟩\langle 5, 12 \rangle and from BB to CC is ⟨−5,−12⟩\langle -5, -12 \rangle. Where is CC relative to AA?

Example 36

hard
A boat heads 30° east of north for 10 km. Find the displacement's east and north components.

Example 37

hard
A particle's displacement vector ⟨a,b⟩\langle a, b \rangle has magnitude 1313 and points along ⟨5,12⟩\langle 5, 12 \rangle direction. Find aa and bb.

Example 38

hard
A particle's position at time tt is r⃗(t)=(t,t2)\vec r(t) = (t, t^2). Find its displacement from t=1t = 1 to t=3t = 3.

Example 39

hard
Average velocity over an interval is displacement divided by time. A car's position changes by ⟨60,80⟩\langle 60, 80 \rangle km in 22 hours. Find its average velocity vector and its speed.

Example 40

hard
A particle moves around an equilateral triangle of side 66 starting from one corner and traversing two sides. Find the displacement magnitude and the distance traveled.

Example 41

hard
If displacement from AA to BB is u⃗\vec u and from BB to CC is v⃗\vec v, what is the displacement from AA to CC? From CC to AA?

Example 42

hard
A rocket's displacement in 3D is ⟨6,−8,24⟩\langle 6, -8, 24 \rangle km. Find its magnitude.

Example 43

challenge
A point moves on a circle of radius rr from angle θ1=0\theta_1 = 0 to θ2=π/2\theta_2 = \pi/2 (counterclockwise). Find its displacement magnitude in terms of rr.
← Back to Displacement Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

vector intuition