K223 / K224 / 2027
Physical quantities and measurement overview

Lesson 6 of 6

Scalars and vectors

  • I can define scalar and vector quantities and give examples.Syllabus

    Syllabus K223 / K224, 1(f). Define scalar and vector quantities and give examples.

Make a guess. You are not marked.

You walk 3 m east, then 4 m north. What is your displacement from where you started?

  1. 5 m, at an angle north of east.
  2. 7 m, adding the two distances walked.
  3. 1 m, the difference between the two walks.
Show the answer

5 m, at an angle north of east.

Displacement is a vector: the straight line from start to finish. The two walks form a right angle, so the size is sqrt(32 + 42) = 5 m, pointing north of east.

In short

  • A scalar has size only, like mass, distance or time.
  • A vector has size and direction, like force, velocity or displacement.
  • Two vectors in the same direction add; in opposite directions, subtract.

A scalar has magnitude only. A vector has both magnitude and direction.

Magnitude means size, given by a numerical value and its unit where applicable. A speed of 4 m/s tells us how fast an object moves. A velocity of 4 m/s east also tells us its direction.

Equal magnitudes with opposite directionsTwo equally long velocity arrows represent 4 m/s east and 4 m/s west. Their magnitudes are equal but their directions differ, so they are different vectors.Velocity: 4 m/s eastVelocity: 4 m/s westSame magnitude; different directions
Arrow length represents magnitude using a chosen scale; the arrowhead shows direction. These two velocities have the same magnitude, but they are different vectors.
Scalar and vector examples
ScalarRelated vector
Distance: total length of the path travelled.Displacement: change in position from start to finish, including direction.
Speed: distance travelled per unit time.Velocity: rate of change of displacement, including direction.
Mass: a property measured in kilograms.Weight: the gravitational force on an object, measured in newtons (N). Near Earth it acts downwards.

Time, temperature and energy are also scalars. Force and acceleration are vectors. Acceleration describes a change in velocity per unit time, so the direction of that change matters.

Compare the descriptions

Same magnitude does not mean the same vector

Two cyclists each have a speed of 4 m/s. One travels east and one west. Their speeds are equal, but their velocities differ because they point in opposite directions.

If east is chosen as positive along a straight line, the velocities can be written as +4 m/s and -4 m/s. The signs now encode the chosen directions; both magnitudes are 4 m/s.

A minus sign alone does not make a quantity a vector. A temperature of -5 °C is still a scalar. The value is below zero on the Celsius scale; it does not point in a spatial direction.

In a vector answer, state the direction as well as the magnitude, such as 6 N upwards. For signed values, say which direction you have taken as positive.

Check your understanding

Which of these describes a vector quantity?

  1. An energy of 40 J stored in a stretched spring.
  2. A velocity of 4 m/s west.
  3. A temperature of -5 degrees Celsius.
Show the answer

A velocity of 4 m/s west.

Velocity needs both a magnitude (4 m/s) and a direction (west). A velocity of 4 m/s east would be a different vector.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

State the difference between a scalar quantity and a vector quantity. Then sort these quantities into scalars and vectors: mass, velocity, distance, weight. [3]

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