Scalars and Vectors
A car driving at 60 km/h and a car driving at 60 km/h north — is there a difference? Yes: one is a scalar (speed), the other a vector (velocity). Vectors carry direction as well as magnitude, and that extra piece of information changes everything about how they combine. This lesson builds the foundation you need for all of Module 6.
You walk 3 km east, then 3 km west. Without using any formulas — what is your total distance travelled, and what is your displacement from the start? Why might these two answers be different?
Every physical quantity is either a scalar (magnitude only) or a vector (magnitude and direction). This distinction determines how quantities combine.
Scalars follow ordinary arithmetic: 3 kg + 4 kg = 7 kg always. Vectors do not: 3 km east + 4 km north $\neq$ 7 km, because direction matters. We need the Pythagorean theorem (or vector addition rules from Lesson 4) to find the combined effect.
Vector resultant: $\sqrt{3^2+4^2} = 5$ km (north-east)
Key facts
- A scalar has magnitude only; a vector has magnitude and direction
- Scalar examples: distance, speed, mass, time, temperature, energy
- Vector examples: displacement, velocity, force, acceleration, momentum
- Geometrically, a vector is a directed line segment (arrow)
Concepts
- Why scalars combine by ordinary arithmetic but vectors do not
- The physical meaning of displacement vs distance
- Why equal vectors require the same magnitude and direction
Skills
- Classify any physical quantity as scalar or vector with justification
- Describe a vector geometrically using a directed line segment
- Identify equal vectors from a diagram
A scalar is any quantity that can be completely described by a single number (and an appropriate unit). Direction plays no role.
Common scalar quantities:
- Distance: 5 km (just a length — no direction)
- Speed: 60 km/h (how fast, but not which way)
- Mass: 70 kg
- Time: 10 seconds
- Temperature: 25°C
- Energy: 200 J
Scalars obey ordinary arithmetic. If you walk 3 km then walk a further 4 km, your total distance is always 7 km — no matter which direction you walked.
A scalar is any quantity that can be completely described by a single number (and an appropriate unit). Direction plays no role.
Pause — copy the definition: a scalar is a quantity described by magnitude (number + unit) only; include five examples (mass, temperature, speed, time, energy) into your book.
Quick check: Which of the following is a scalar quantity?
We just saw that scalars are completely described by a single number and unit (e.g.\ $m=5$ kg, $T=300$ K). That raises a question: some physical quantities are not fully described by a number alone — which quantities require a direction as well, and why does direction change their mathematical behaviour fundamentally? This card answers it → vectors like velocity, force, and displacement require magnitude AND direction; $5$ m north $\neq$ $5$ m east even though both have magnitude 5.
A vector is a quantity that requires both magnitude and direction to be fully described. A single number is insufficient.
Common vector quantities:
- Displacement: 5 km north
- Velocity: 60 km/h east
- Force: 50 N downward
- Acceleration: 9.8 m/s² toward Earth's centre
- Momentum: $p$ kg m/s (in the direction of motion)
Two vectors are equal if and only if they have the same magnitude and the same direction. Saying "50 N" is not enough — "50 N downward" and "50 N upward" are completely different vectors (they are, in fact, negatives of each other).
A vector is a quantity that requires both magnitude and direction to be fully described. A single number is insufficient.
Pause — copy the definition: a vector is a quantity described by magnitude AND direction; include five examples (displacement, velocity, force, acceleration, momentum) into your book.
Did you get this? True or false: Two arrows of the same length pointing in opposite directions represent equal vectors.
Worked examples · 3 in a row, reveal as you go
Classify each quantity as a scalar or a vector and justify your answer.
(a) Temperature 25°C (b) Weight 700 N (c) Volume 2 L (d) Momentum $p$
A student walks 3 km east then 3 km west. Find: (a) the total distance travelled, (b) the displacement from the starting point.
In the diagram, $ABCD$ is a parallelogram. Which of the following pairs are equal vectors: $\vec{AB}$ and $\vec{DC}$; $\vec{AB}$ and $\vec{CD}$?
Fill the gap: A car travels 5 km north then 5 km south. Its total distance is km and its displacement is km.
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: In parallelogram $ABCD$, $\vec{AB} = \vec{CD}$.
Activities · practice with the ideas
Classify each as scalar or vector with a brief justification: (a) kinetic energy (b) gravitational force (c) electric potential (d) wind velocity.
A runner completes a 400 m track in a full circle and returns to the start. State the distance and the displacement.
Draw two arrows on a grid that represent equal vectors, where neither arrow starts at the origin. Describe what makes them equal.
In square $PQRS$ (vertices in order), write $\vec{PQ}$ in terms of $\vec{SR}$. Are they equal vectors? Explain.
A pilot flies 200 km due north. Express this as a displacement vector and state its magnitude and direction.
Odd one out: Three of these are vector quantities. Which one is NOT?
Earlier you estimated the distance and displacement for the 3 km east / 3 km west walk.
The answer: distance = 6 km (scalar, adds normally) and displacement = 0 km (vector, opposite directions cancel). This captures the key distinction between scalars and vectors — scalars just count total, while vectors track net change including direction.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. State whether speed is a scalar or a vector quantity. Justify your answer. (1 mark)
Q2. A dog runs 8 m north then 6 m east. Find the magnitude of the dog's displacement from its starting point. (2 marks)
Q3. In parallelogram $ABCD$ (with vertices labelled in order), explain why $\vec{AB} = \vec{DC}$ but $\vec{AB} \neq \vec{CD}$. (2 marks)
Comprehensive answers (click to reveal)
Activity answers: 1. (a) kinetic energy — scalar (no direction) · (b) gravitational force — vector (acts downward) · (c) electric potential — scalar (no direction) · (d) wind velocity — vector (speed + direction). · 2. Distance = 400 m, Displacement = 0 (back at start). · 4. $\vec{SR}$ goes from $S$ to $R$ (same direction as $P$ to $Q$), so $\vec{PQ} = \vec{SR}$. · 5. Magnitude = 200 km, direction = due north.
Q1 (1 mark): Speed is a scalar [1] — it has magnitude only (how fast an object moves) and requires no direction to be fully described.
Q2 (2 marks): The displacement forms the hypotenuse of a right triangle with legs 8 m and 6 m [1]. Magnitude $= \sqrt{8^2 + 6^2} = \sqrt{64+36} = \sqrt{100} = \mathbf{10}$ m [1].
Q3 (2 marks): In parallelogram $ABCD$, $AB \parallel DC$ and $|AB| = |DC|$; both $\vec{AB}$ and $\vec{DC}$ travel in the same direction (from left vertex to right vertex on their respective sides) [1]. However, $\vec{CD}$ travels from $C$ back to $D$, which is the opposite direction to $\vec{AB}$, so $\vec{CD} = -\vec{AB} \neq \vec{AB}$ [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering scalar and vector questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.