Revision notes for AQA GCSE Physics Scalar and vector quantities. Open the guide for explanations and worked examples. Written against the AQA GCSE Physics (8463) specification, so the content matches what's examinable rather than general Physics background.
Scalar and vector quantities
What you'll learn
- What a physical quantity is, and what we mean by its magnitude.
- The difference between scalar quantities and vector quantities.
- How to represent a vector using an arrow.
- Common pairs that students mix up, such as speed and velocity.
Starting point: physical quantities
In Physics, a physical quantity is something that can be measured or calculated, usually with a number and a unit.
Examples include:
- time, measured in seconds (s)
- mass, measured in kilograms (kg)
- force, measured in newtons (N)
- speed, measured in metres per second (m/s)
To describe some quantities fully, a number and unit is enough. For others, you also need to say which way the quantity acts.
Definition
Magnitude
The magnitude of a quantity is its size or amount. For a measured quantity, this usually means the number together with its unit, such as 8 m, 12 N, or 3 m/s.
Scalars: magnitude only
A scalar quantity has magnitude only.
That means a scalar is fully described by its size and unit. It does not need a direction.
Definition
Scalar quantity
A scalar quantity has magnitude only. For example, a mass of 5 kg is complete information because mass does not point in a direction.
Common scalar quantities include:
- distance
- speed
- time
- mass
- energy
- temperature
If you say “the journey took 40 s”, that is complete. There is no direction for time.
Key Idea
Scalars are just size
If changing direction would not change the meaning of the quantity, it is probably a scalar.
Vectors: magnitude and direction
A vector quantity has magnitude and an associated direction.
For example, saying “a force of 10 N” is not always enough. You may need to say whether the force acts to the left, to the right, upwards, downwards, north, or at some angle.
Definition
Vector quantity
A vector quantity has both magnitude and direction. For example, “10 N to the right” gives the size of the force and the direction it acts.
Common vector quantities include:
- force
- weight
- displacement
- velocity
- acceleration
The diagram below shows the key difference: a scalar has magnitude only, while a vector needs both magnitude and direction.
Tip
Quick test for a vector
Ask: “Would I need to say which way?” If yes, it is likely to be a vector. Forces, velocities and displacements all need direction.
Example
Classifying quantities
A question gives these measurements: 25 s, 8 m/s north, 12 N downwards, 4 kg, and 30 m. Decide whether each is scalar or vector.
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Check whether a direction is needed or included. The quantities “8 m/s north” and “12 N downwards” both include a direction.
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Classify the quantities with direction as vectors. So 8 m/s north is a velocity, and 12 N downwards is a force. Both are vector quantities.
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Classify the quantities with magnitude only as scalars. Time of 25 s, mass of 4 kg, and distance of 30 m are all scalar quantities.
Similar-looking quantities: don’t mix them up
Some quantities sound very similar, but one is scalar and one is vector.
Distance and displacement
Distance is the total length of the path travelled. It is a scalar.
Displacement is the straight-line distance from the starting point to the finishing point, in a stated direction. It is a vector.
So if you walk around a room and end up back where you started, your distance travelled is not zero, but your displacement is zero.
Example
Distance and displacement on a straight line
You walk 4 m east, then 3 m west. Find your distance travelled and your displacement.
-
Add the actual path lengths to find distance. You walked 4 m and then 3 m, so the distance travelled is 7 m.
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Compare the directions to find displacement. Taking east as positive, the displacement is 4 m−3 m=1 m4\ \text{m} - 3\ \text{m} = 1\ \text{m}4 m−3 m=1 m.
-
Give the direction for the vector answer. The positive result means the final position is 1 m east of the starting point, so the displacement is 1 m east.
Speed and velocity
Speed is how fast something is moving. It is a scalar.
Velocity is speed in a stated direction. It is a vector.
For example:
- “12 m/s” is a speed.
- “12 m/s north” is a velocity.
Common Mistake
Speed and velocity are not identical
In everyday speech, people often use “speed” and “velocity” as if they mean the same thing. In Physics, velocity must include direction, so it is a vector.
Mass and weight
Mass is the amount of matter in an object. It is measured in kilograms (kg) and is a scalar.
Weight is the force of gravity acting on an object. It is measured in newtons (N) and is a vector because it acts towards the centre of the Earth.
Representing vectors with arrows
A vector quantity may be represented by an arrow.
The arrow shows two things:
- The length of the arrow represents the magnitude.
- The direction of the arrow shows the direction of the vector.
For example, a long arrow to the right could represent a larger force to the right. A shorter arrow to the right would represent a smaller force in the same direction.
Key Idea
Vector arrows
For a vector arrow, length shows magnitude and the arrowhead shows direction.
If a scale is given, you can draw the vector accurately. For example, a scale might say that 1 cm represents 5 N.
Example
Drawing a force vector to scale
Draw a force of 15 N acting east, using the scale 1 cm represents 5 N.
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Use the scale to calculate the arrow length. The arrow length is 15 N5 N/cm=3 cm\frac{15\ \text{N}}{5\ \text{N/cm}} = 3\ \text{cm}5 N/cm15 N=3 cm.
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Use the stated direction to choose the arrow direction. East is usually drawn to the right, so the arrow should point right.
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Combine magnitude and direction in the labelled arrow. Draw a 3 cm arrow pointing right and label it “15 N east”.
Common Mistake
Drawing every arrow the same length
If vector arrows are meant to compare magnitudes, larger vectors should be drawn longer. If one force is twice as large as another, its arrow should be twice as long when drawn to scale.
Direction matters
A vector can change even if its magnitude stays the same.
For example, a car going round a bend at a steady speed still changes velocity, because its direction is changing. Speed is scalar, so it only depends on how fast the car is moving. Velocity is vector, so direction matters too.
Common Mistake
Negative signs in vector quantities
Sometimes a negative sign is used to show direction, such as a velocity of −3 m/s-3\ \text{m/s}−3 m/s along a chosen line. The magnitude is still 3 m/s; the negative sign tells you the direction.
Why this matters for forces
This topic sits inside forces and their interactions because forces are vectors.
When you draw forces on an object, you normally use arrows to show:
- where the force acts
- which direction it acts in
- how large it is compared with other forces
This becomes very important later when you study balanced and unbalanced forces.
Exam technique
In the exam
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Check whether the answer needs a direction. If it does, you are dealing with a vector.
-
For vector arrows, make sure the arrowhead points the correct way and the length matches the magnitude if a scale is given.
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Do not confuse common pairs: distance is scalar, displacement is vector; speed is scalar, velocity is vector; mass is scalar, weight is vector.
Self review
Check yourself
- What is the difference between magnitude and direction?
- Is “20 N upwards” a scalar or a vector quantity?
- Why is velocity a vector but speed is a scalar?
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