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
2
Step 2: Displacement =+5+(−5)=0 m.
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) to point B(4,6). Find the displacement vector and its magnitude.Displacement vector from A(1, 2) to B(4, 6)
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), going around once. Find both the displacement and the total distance traveled.
Example 5
challenge
A particle moves along r(t)=(cost,sint) from t=0 to t=2π. 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?Two perpendicular legs forming a right triangle; hypotenuse = displacement
Example 2
hard
A particle undergoes three displacements: d1=(2,−1), d2=(−3,4), d3=(1,2). Find the net displacement vector and its magnitude.Vector chain: d₁ + d₂ + d₃
Example 3
easy
You walk 3 blocks east then 4 blocks north. What is your displacement magnitude?Find the straight-line 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) to (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) and has displacement ⟨5,−1⟩. 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.Net displacement has magnitude 5 m
Example 13
medium
A drone flies from (0,0) to (3,4) to (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⟩ 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⟩, but it traveled 20 m. What kind of path did it take?
Example 18
medium
Displacement from A to B is ⟨3,5⟩. What is the displacement from B to A?
Example 19
challenge
A ship sails 12 km on a bearing of 90∘ (east), then 12 km on a bearing of 180∘ (south). Find its displacement magnitude and rough direction.
Example 20
challenge
A particle moves around three sides of a square of side 4 (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) to (8,6). Find its displacement magnitude.
Example 24
easy
A drone moves from (2,1) to (5,5). Write the displacement as a vector.
Example 25
easy
A particle is displaced by ⟨−4,3⟩. Find its displacement magnitude.
Example 26
easy
Starting at (1,4) and ending at (1,−2), find the displacement vector.
Example 27
easy
A point starts at (0,0) and is displaced by ⟨7,−24⟩. Find its ending position and displacement magnitude.
Example 28
medium
A bird flies from (2,3) to (10,9), then to (2,9). What is its displacement from start to end?
Example 29
medium
Two displacements u=⟨3,5⟩ and v=⟨−1,2⟩ are applied in sequence. Find the net displacement.
Example 30
medium
A car travels 20 km east and then 15 km north. Compute the displacement magnitude.
Example 31
medium
From point A to point B, the displacement is ⟨6,8⟩. A different person walks 20 m getting from A to B. Explain why the magnitudes differ.
Example 32
medium
A particle's displacement is ⟨9,−12⟩. Find its magnitude and write a unit vector in the same direction.
Example 33
medium
Why must displacement satisfy ∣Δr∣≤total distance traveled?
Example 34
medium
A particle moves ⟨2,3⟩, then ⟨4,−1⟩, then ⟨−6,2⟩. Find the net displacement.
Example 35
medium
A point's displacement from A to B is ⟨5,12⟩ and from B to C is ⟨−5,−12⟩. Where is C relative to A?
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⟩ has magnitude 13 and points along ⟨5,12⟩ direction. Find a and b.
Example 38
hard
A particle's position at time t is r(t)=(t,t2). Find its displacement from t=1 to t=3.
Example 39
hard
Average velocity over an interval is displacement divided by time. A car's position changes by ⟨60,80⟩ km in 2 hours. Find its average velocity vector and its speed.
Example 40
hard
A particle moves around an equilateral triangle of side 6 starting from one corner and traversing two sides. Find the displacement magnitude and the distance traveled.
Example 41
hard
If displacement from A to B is u and from B to C is v, what is the displacement from A to C? From C to A?
Example 42
hard
A rocket's displacement in 3D is ⟨6,−8,24⟩ km. Find its magnitude.
Example 43
challenge
A point moves on a circle of radius r from angle θ1=0 to θ2=π/2 (counterclockwise). Find its displacement magnitude in terms of r.