"Walk 5 km" tells your friend how far, but not where to end up. "Walk 5 km north" does both. That single difference, between a number with direction and a number without, is the foundation of all of mechanics. Get the scalar–vector distinction right now and every motion problem in this course becomes a matter of bookkeeping the directions correctly.
You walk 3 km east, then turn around and walk 3 km west, ending up exactly where you started.
Before reading: what total distance did you walk, and what was your overall displacement? Write both numbers and one sentence on why they differ.
Every physical quantity is either "just a size" or "a size and a direction"
A scalar is a plain number with a unit. A vector is that same number plus an arrow telling you which way it points, and that extra information changes how you add things together.
When you say "the mass is $2\ \text{kg}$" or "the trip took $3\ \text{hours}$", direction is meaningless, these are scalars. But "the wind blows at $20\ \text{m s}^{-1}$ from the south" only makes sense with a direction, that is a vector.
A scalar has magnitude only (distance, speed, time, mass). A vector has both magnitude and direction (displacement, velocity, acceleration, force). Vectors are drawn as arrows, length = magnitude, arrowhead = direction.
Pause, copy the highlighted statement into your book before moving on.
Did you get this? Which of the following is a vector quantity?
The clearest example of a scalar and its vector partner
We just saw that some quantities carry direction and some do not. That raises a question: what actually changes when the same trip is measured as a scalar versus a vector? This card answers it → by comparing distance (scalar) with displacement (vector) on the same journey.
Distance is how far you actually travelled along your path; displacement is how far you ended up from where you started, measured in a straight line, with direction.
Distance is a scalar, it just keeps adding up no matter which way you go. Displacement is a vector, it points straight from the start to the finish, so if you double back, your displacement shrinks while your distance keeps growing.
A student walks $4\ \text{m}$ east, then $3\ \text{m}$ north. Find (a) the total distance walked and (b) the magnitude of the displacement.
Distance (scalar) is the total path length; displacement (vector) is the straight-line change in position with direction. Distance ≥ magnitude of displacement, and they are equal only for straight-line motion with no reversal.
Add the highlighted comparison to your notes before the check below.
Quick check: True or false: a runner who completes one full lap of a track has a displacement of zero but a non-zero distance.
Pattern A, Sort then reason
In your book, draw a two-column table headed Scalar and Vector. Place each of these in the correct column: distance, displacement, time, speed, velocity, acceleration, mass, force. Then write one sentence explaining the rule you used to decide.
Two truths, one lie: three of these classifications are correct. Which one is the lie?
Dot point: Classify distance, displacement, time, speed, velocity and acceleration as either scalar or vector quantities
Why direction makes velocity and acceleration vectors
We just learned to separate distance from displacement. That raises a question: does the same scalar–vector split apply to how fast things move? This card answers it → speed is the scalar, velocity and acceleration are the vectors.
Speed tells you only how fast; velocity tells you how fast and which way; acceleration tells you how the velocity is changing, so velocity and acceleration both carry direction.
Speed is the rate of covering distance, a scalar. Velocity is the rate of change of displacement, a vector, so it has a direction. Acceleration is the rate of change of velocity, also a vector. This is why a car going around a roundabout at a steady $40\ \text{km h}^{-1}$ is accelerating: its speed is constant but its velocity direction keeps changing.
Mapping the six core motion quantities: distance and time are scalars; speed is a scalar; displacement, velocity and acceleration are vectors.
Distance, time and speed are scalars; displacement, velocity and acceleration are vectors. Because velocity is a vector, a change in direction alone counts as a change in velocity, so an object moving at constant speed around a curve is accelerating.
Pause, copy the highlighted classification into your book before the check below.
Last check: Decide if the statement is true or false.
A car driving around a circular track at a constant speed of $40\ \text{km h}^{-1}$ is accelerating because its velocity direction is changing.
Odd one out: three of these are vector quantities. Which one is not?
Dot point: Classify distance, displacement, time, speed, velocity and acceleration as either scalar or vector quantities
A fresh set drawn from this lesson's question bank, feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence, that tells the system what to drill next.
UnderstandBand 4(4 marks) 1. A jogger runs $200\ \text{m}$ east, then $200\ \text{m}$ west back to the start. Distinguish between distance and displacement, and use this trip to calculate each.
1 mark: distance is a scalar (total path) · 1 mark: displacement is a vector (straight-line change with direction) · 1 mark: distance $= 400\ \text{m}$ · 1 mark: displacement $= 0\ \text{m}$
ApplyBand 5(3 marks) 2. A car travels around a roundabout at a constant $40\ \text{km h}^{-1}$. Classify its speed and its velocity as scalar or vector, and explain why the car is accelerating even though the speedometer reading never changes.
1 mark: speed is a scalar, velocity is a vector · 1 mark: velocity direction is continuously changing · 1 mark: acceleration is the rate of change of velocity, so a direction change alone is an acceleration
Dot point: Classify distance, displacement, time, speed, velocity and acceleration as either scalar or vector quantities
AnalyseBand 5(3 marks) 3. A student writes: "My displacement for the trip was $12\ \text{km}$." Explain why this statement is incomplete, and state what a complete answer would look like. Contrast it with a correctly reported distance.
1 mark: displacement is a vector so a magnitude alone does not specify it · 1 mark: a direction (or a sign under a stated convention) must be given · 1 mark: distance is a scalar, so $12\ \text{km}$ alone IS complete for distance
Dot point: Classify distance, displacement, time, speed, velocity and acceleration as either scalar or vector quantities
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set from the lesson bank.
Q1 (4 marks): Distance is a scalar quantity, it measures the total length of the path travelled and ignores direction. Displacement is a vector quantity, it measures the straight-line change in position from start to finish and includes direction. For this trip, the jogger covers $200\ \text{m} + 200\ \text{m} = 400\ \text{m}$, so the distance is $400\ \text{m}$. However, the jogger finishes exactly where they started, so the straight-line change in position is zero: the displacement is $0\ \text{m}$. The two differ because the path doubled back on itself.
Q2 (3 marks): Speed is a scalar: it records only how fast the car is going, so at a constant $40\ \text{km h}^{-1}$ the speed is unchanging. Velocity is a vector: it records how fast and in which direction. As the car follows the curve its direction of travel changes continuously, so its velocity is changing even though its speed is not. Acceleration is defined as the rate of change of velocity, not of speed, so a change of direction alone is enough to produce an acceleration. The car is therefore accelerating the whole way round.
Q3 (3 marks): Displacement is a vector quantity, so it is only fully specified when both a magnitude and a direction are given. Stating $12\ \text{km}$ supplies the magnitude and leaves the direction out, so the statement is incomplete. A complete answer would read $12\ \text{km}$ east, or $-12\ \text{km}$ under a stated sign convention such as "east is positive". Distance behaves differently: it is a scalar, carrying magnitude only, so "the distance for the trip was $12\ \text{km}$" is already a complete statement and needs no direction.
You can now classify any motion quantity as a scalar or a vector and explain why distance and displacement can differ. Next lesson uses vectors as arrows to add and subtract motion in two dimensions.
↩ Back to focus-area hubAt the start you walked 3 km east then 3 km west. The answer is a distance of 6 km but a displacement of zero: distance adds the whole path, while displacement is the straight-line change in position, which is nothing once you return to the start. If you wrote 6 km for both, revisit Card 2.
A short spaced-retrieval set, mixing this lesson with earlier ones. It gets harder to guess and easier to remember.
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