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Scalar and vector quantities

Key idea: Scalars vs vectors for O Level Physics: definitions, how to state direction, common exam pairs (distance/displacement, speed/velocity), and practice questions.

  • SEC G3 Physics 2027
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Learning objectives

  • Represent a physical quantity with a numerical magnitude and unit
  • Recall the six prescribed SI base quantities and their units
  • Use the prescribed SI prefixes from nano to tera
  • Compare orders of magnitude from a typical atom to the Earth
  • Select and justify measuring instruments by range and precision
  • Distinguish scalar and vector quantities and give examples
  • Add two vectors graphically to determine a resultant

1. Definitions

Scalar quantity

A scalar is a physical quantity with magnitude only (no direction). It is fully described by a number and a unit.

Vector quantity

A vector is a physical quantity with magnitude and direction. It is fully described only when both are stated (e.g. 10 N to the right).

2. Key Ideas

  • Scalars: magnitude only (e.g. mass, time, speed).
  • Vectors: magnitude + direction (e.g. force, velocity, displacement).
  • A vector answer is incomplete without a direction.
  • A vector’s magnitude is never negative. A minus sign (in 1D) indicates direction relative to your chosen positive direction.
  • Always include units. Keep calculation units mutually consistent and report the final answer in the requested unit.

3. Detailed Explanations

Scalar and vector comparison

FeatureScalarVector
Definitionmagnitude onlymagnitude + direction
Fully described bynumber + unitnumber + unit + direction
Example statement5.0 m5.0 m east
Examples (with units)mass m (kg), time t (s), speed v (m s⁻¹), temperature T (K)displacement (m) with direction, velocity (m s⁻¹) with direction, acceleration a (m s⁻²) with direction, force F (N) with direction

Stating direction clearly

You can state direction using:

  • compass directions (north, south, east, west)
  • left/right, up/down
  • “towards A / away from A” (when the diagram defines A)

Scalar–vector pairs that are often confused

PairScalarVector
distance vs displacementdistance (how far), unit mdisplacement (change in position), unit m + direction
speed vs velocityspeed, unit m s⁻¹velocity, unit m s⁻¹ + direction
Distance and displacement for the same journeyA winding route from point A to point B is labelled distance travelled. A straight arrow from A to B is labelled displacement and points north-east.One journey, two quantitiesA · startB · finishdistance travelledlength of the complete routedisplacementstraight arrow, north-east
Scroll diagram horizontally to read all labels.
Distance follows the whole route and has magnitude only. Displacement joins the start to the finish and requires both magnitude and direction.

More practice on adding vectors: Vector Addition (Graphical Method).

4. Common Mistakes

  • Giving a vector without a direction (e.g. “10 N” with no direction stated).
  • Mixing up distance/displacement or speed/velocity (same unit, different meaning).
  • Saying a vector has “negative magnitude” (magnitude is always positive).
  • Using a negative sign without stating the sign convention (what is positive?).
  • Forgetting units (especially m s⁻¹ and m s⁻²).

5. Exam Tips

  • If the question says “vector”, include a direction word (east, left, upward, etc.).
  • For 1D questions, write a sign convention first (e.g. “take right as positive”).
  • Match the command word: state a vector directly, show working for a calculation, and compare the same feature on both sides.
  • Keep answers mark-friendly: one marking point per line.

6. Worked Examples

Modelled example 1

Classifying quantities

Core

Problem

Classify mass, velocity, temperature, force, distance and displacement as scalar or vector quantities.
Study the worked solution

  1. Test whether direction is required

    Method

    Ask whether a number and unit alone fully describe each quantity.

    Reason

    Scalars need magnitude only; vectors need magnitude and direction.

    Working

    Mass, temperature and distance do not require direction.

  2. Classify the quantities

    Method

    Place velocity, force and displacement in the vector group.

    Reason

    Each changes meaning when its direction changes.

    Working

    Scalars: mass, temperature, distance. Vectors: velocity, force, displacement.

Guided practice 2

Interpreting negative signs (1D)

About 4 min

Problem

Take right as positive. Interpret a displacement of -3.0 m using a positive magnitude and a direction word.

Translate the sign convention

Displacement statement

Hints

Hint 1: use the convention
The negative direction is opposite to right.
Hint 2: separate magnitude and direction
Write the magnitude as 3.0 m, then add “left”.
View solution step by step

  1. Reverse the positive direction

    Method

    Interpret the negative sign as left.

    Reason

    Right was defined as the positive direction.

    Working

    -3.0 m = 3.0 m left

Common misconception 3

Fixing an incomplete vector statement

Find and correct the mistake

Learner response

A student writes: “A force of 10 N acts on the box, so the vector is fully stated.” Locate the missing information and repair the statement.

Identify the missing vector feature

Missing information

View solution step by step

  1. Check the three vector parts

    Method

    Verify number, unit and direction.

    Reason

    A vector is fully specified only by magnitude and direction.

    Working

    The statement has 10 and N but no direction.

  2. Add a defined direction

    Method

    Write, for example, “10 N to the right on the box”.

    Reason

    The direction completes the force vector.

    Working

    vector F = 10 N right

Examiner practice 4

Vector-only multiple choice

1 mark

Examination question

Which option contains only vectors? (1) force, work, energy; (2) displacement, momentum, acceleration; (3) power, energy, time; (4) momentum, mass, velocity. [1 mark]

Eliminate any option containing a scalar

Correct option

View solution step by step

  1. Test each list

    1 mark

    Method

    Select option 2.

    Reason

    Work, energy, power, time and mass are scalars, so they eliminate options 1, 3 and 4.

    Working

    Vector-only set: displacement, momentum, acceleration.

Challenge 5

Interpreting signs for a velocity statement

Minimal support

Axis transfer

A tracking system defines north as positive and reports a vehicle’s velocity as -12 m s⁻¹. State the velocity using a positive magnitude and compass direction.

Translate signed component into a vector statement

Hints

Hint 1: reverse the defined positive direction
Opposite to north is south.
Hint 2: make magnitude positive
Use 12 m s⁻¹ as the magnitude.
View solution step by step

  1. Interpret the sign convention

    Method

    Map the negative sign to south.

    Reason

    North was chosen as positive.

    Working

    -12 m s⁻¹ = 12 m s⁻¹ south

7. Mind Stretchers

Mind stretcher 1: Distance vs displacement in one tripExtension

Problem 1: A student walks 30 m east, then 20 m west. State the distance travelled and the displacement.

Show Answer
  • Distance travelled (scalar): 30 + 20 = 50 m
  • Displacement (vector): 30 - 20 = 10 m east

Mind stretcher 2: Resultant force (direction matters)Extension

Problem 2: Two horizontal forces act on a block: 12 N east and 5 N west. Find the resultant force (magnitude and direction).

Show Answer
  • Take east as positive.
  • Resultant force: F_R = 12 - 5 = 7 N east

Continue with the next resource in this course.

Course and syllabus information
Course
SEC G3 Physics
Edition
SEC G3 Physics 2027