Scalar and vector quantities

1. Definition

A. Scalar quantity

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

B. 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 (and use SI units in calculations unless told otherwise).

3. Detailed Explanations

A. Scalar vs vector (comparison table)

B. 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)

C. Scalars vs vectors you often confuse

Scroll diagram horizontally to read all labels.

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”).
• When explaining, use marking verbs: state, deduce, calculate, compare.
• Keep answers mark-friendly: one marking point per line.

6. Worked Examples

Example 1: Classifying quantitiesCore

Example 1: Classify each quantity as scalar or vector: mass, velocity, temperature, force, distance, displacement.

Show Answer

• Scalar: mass, temperature, distance
• Vector: velocity, force, displacement

Example 2: Fixing an incomplete vector statementCore

Example 2: A student writes: “A force of 10 N acts on the box.” Explain what is missing and give a correct full statement.

Show Answer

• Missing: direction
• Example full statement: “A force of 10 N acts to the right on the box.”

Example 3: Interpreting negative signs (1D)Core

Example 3: Take right as positive. What does a displacement of -3.0 m mean?

Show Answer

• It means a displacement of 3.0 m to the left.

Example 4: Vector-only multiple choiceCore

Example 4: Which option contains only vectors?

1. force, work, energy
2. displacement, momentum, acceleration
3. power, energy, time
4. momentum, mass, velocity

Show Answer

• Vectors: force, displacement, momentum, acceleration, velocity
• Scalars: work, energy, power, time, mass
• Answer: 2

Example 5: Interpreting signs for a velocity statementCore

Example 5: Take north as positive. What does a velocity of -12 m s⁻¹ mean?

Show Answer

• It means a velocity of 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: FR = 12 - 5 = 7 N east

8. Practice, Quiz and Next Step

Close your notes and use Scalar and vector quantities in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: A hiker walks 3 km north then 4 km east in 1.0 h.

1. Retrieve: define scalar and vector classification in your own words, including units, sign or conditions where relevant.
2. Represent: Draw a head-to-tail vector triangle with directions and resultant.
3. Apply: Find distance, displacement magnitude, average speed and average velocity, including direction where required.

Check the response before looking back

• Scalar and vector quantities are distinguished, including direction and sign.
• A graph result is inferred from its labelled axes before a formula is chosen.
• Working uses compatible units and the final statement describes the motion.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theO-Level topic checks orpractice browser for an independent re-test.