Geometry and Measurement · Measurement · Lesson 8
Scalar and Vector Quantities
Some measurements, like the mass of a bag of cement, need just a number and a unit. Others, like the wind blowing over Cape Coast, need a size and a direction. In this lesson you will learn to sort quantities into scalars and vectors, and to draw a vector from its length and bearing.
By the end you will be able to
- ✔Define scalar and vector quantities and give examples of each.
- ✔Sort quantities such as weight, force, velocity, speed, distance, mass, volume, energy, work and momentum into scalars and vectors.
- ✔Describe a vector by its magnitude (length) and its bearing.
- ✔Draw a vector such as (6 km, 245°) and explain what a zero vector is.
Key words
- Scalar
- A quantity that has magnitude (size) only, e.g. mass = 5 kg.
- Vector
- A quantity that has both magnitude and direction, e.g. 6 km on a bearing of 245°.
- Magnitude
- The size or length of a quantity. For a vector it is the distance along the vector.
- Displacement
- The straight-line distance from the start to the finish in a stated direction.
- Zero vector
- A vector with no magnitude and no direction: a movement that ends where it began.
Curriculum reference
Content standard: B7.3.2.3 Demonstrate understanding of bearings, vector and its components using real life cases.
Indicator: Distinguish between scalar and vector quantities.
Starter
Same 5 km, different journeys
Ama and Yaw both walk 5 km. Ama arrives at the Makola market stall where she works. Yaw ends up at the lorry station on the other side of Accra. Both walked '5 km', yet they ended in very different places! Saying '5 km' is not enough to fix where you finish. We also need the direction, such as 'on a bearing of 060°'. That difference is what separates a scalar from a vector.
1. Scalars: size only
A scalar quantity is completely described by a number and a unit. You do not ask 'in which direction?' about it. A 50 kg bag of rice has a mass of 50 kg; it makes no sense to ask which way the mass points. A trotro journey that takes 3 hours: the time has no direction. A bottle of water of volume 1.5 litres: no direction either.
Scalar quantities include distance, speed, mass, volume, time, energy, work, temperature and area. We can add them in the ordinary way: 3 kg + 4 kg = 7 kg.
2. Vectors: size and direction
A vector quantity needs a size and a direction. If a pilot is told 'fly 200 km' she asks 'in which direction?' When a footballer kicks a ball with a force of 50 newtons, it matters whether the force is pushing the ball towards goal or away from it. Force, weight, velocity, momentum and displacement are all vectors.
Look carefully at two pairs that are often confused. Distance is a scalar: it is the total length of the path travelled. Displacement is a vector: it is the straight-line distance from the start to the finish in a particular direction. In the same way, speed (60 km/h) is a scalar, while velocity (60 km/h on a bearing of 090°) is a vector.
▶ Notice the difference between the long bendy road (distance) and the straight arrow (displacement).
| Scalar quantity (size only) | Vector quantity (size and direction) |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Weight (a force towards the Earth) |
| Volume | Force |
| Energy | Momentum |
| Work | Acceleration |
| Time | Position from a fixed point |
3. Describing a vector with a bearing
In this course we describe a vector as a movement along a given bearing. Two things fix it: its magnitude (how far) and its bearing (the three-figure angle clockwise from North). We write it as (distance, bearing), for example (6 km, 245°).
To draw (6 km, 245°) with a scale of 1 cm to 1 km:
- 1Mark the starting point A and draw a North line through it.
- 2Use a protractor to measure 245° clockwise from North (that is 180° to South plus another 65°).
- 3Mark the point and draw a line from A through it.
- 4Measure 6 cm along the line and mark B. Add an arrow head at B.
- 5Write the vector as = (6 km, 245°).
▶ The vector from A to B is drawn at 245°, in the south-west. Its length 6 km is the magnitude.
The magnitude is the distance along the vector: 6 km. The direction is the three-figure clockwise angle from North: 245°. The arrow tells you which end is the start and which is the finish: goes from A to B, and goes the opposite way (with bearing 245° − 180° = 065°).
4. The zero vector
What happens if you walk 3 km East and then 3 km back West? You have walked 6 km, but you finished where you started. Your total movement is called the zero vector, written 0. It has no magnitude and no direction, because there is no net movement.
▶ Distance walked is 6 km, but the vector sum is the zero vector.
A zero vector is like the point on the map where you are already standing. We cannot give it a bearing because it does not point anywhere. This is why the zero vector is special: it is the only vector with no direction.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Scalar or vector?
Show the solution
- Step 1(a) Mass has size only: scalar.
- Step 2(b) Wind speed with a direction is a velocity: vector.
- Step 3(c) Volume has size only: scalar.
Answer: (a) scalar (b) vector (c) scalar
Example 2
Grouping quantities
Show the solution
- Step 1Ask 'Does direction matter?' for each quantity.
- Step 2Direction matters for weight (downwards force), force, velocity and momentum.
- Step 3No direction is needed for time, speed, distance, mass, volume, energy and work.
Answer: Vectors: weight, force, velocity, momentum. Scalars: time, speed, distance, mass, volume, energy, work.
Example 3
Magnitude and bearing
▶ The vector (6 km, 245°) drawn from A.
Show the solution
- Step 1The first number is the distance: the magnitude is 6 km.
- Step 2The second is the three-figure bearing: the direction is 245° clockwise from North.
- Step 3245° is between 180° and 270°, so the vector points South-West.
Answer: Magnitude 6 km, bearing 245° (South-West)
Example 4
Distance and displacement
Show the solution
- Step 1Distance is the total path: 4 km + 3 km = 7 km.
- Step 2Displacement is the straight line from start to finish. It forms the hypotenuse of a right-angled triangle with sides 4 and 3.
- Step 3 km.
- Step 4The displacement (5 km) is smaller than the distance (7 km).
Answer: Distance 7 km; displacement 5 km
Example 5
Zero vector
Show the solution
- Step 1Outward vector: (2 km, 030°). Return vector: (2 km, 210°) because 030° + 180° = 210°.
- Step 2She ends exactly where she began.
- Step 3The displacement has no magnitude and no direction: it is the zero vector.
Answer: The zero vector (displacement 0)
Summary
- A scalar has magnitude only; a vector has magnitude and direction.
- Scalars: distance, speed, mass, volume, time, energy, work. Vectors: displacement, velocity, force, weight, momentum.
- In bearings, a vector is written (distance, bearing), e.g. (6 km, 245°): magnitude 6 km, direction 245° clockwise from North.
- The zero vector has no magnitude and no direction.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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