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8.1 Vectors and Scalars

GCSE Physics · AQA · Chapter 8: Forces in Balance

A free GCSE Physics lesson for AQA (8463) from OBA School: distance and displacement, vector and scalar quantities, and drawing vectors as arrows to scale. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

  • Explain the difference between distance and displacement
  • Define vector and scalar quantities and sort quantities into each group
  • Represent a vector quantity with an arrow whose length shows its magnitude and whose direction shows its direction
  • Draw and use scale diagrams, always stating the scale

Key words

  • Magnitude: The size of a quantity, e.g. 25 N or 300 m. It does not include a direction.
  • Scalar: A quantity that has magnitude only, e.g. mass, distance, speed, time, energy.
  • Vector: A quantity that has magnitude and direction, e.g. force, displacement, velocity.
  • Distance: How far something travels along its path. A scalar.
  • Displacement: The straight-line distance from start to finish in a stated direction. A vector.
  • Scale diagram: An accurate drawing in which a stated length represents a stated amount, e.g. 1 cm ≡ 10 N.

Distance and displacement

If you walk along a winding path, the distance you travel is the total length of the path. Your displacement is how far you end up from where you started, measured in a straight line, together with the direction. Displacement is distance in a given direction.

Worked example: a walk (shown in the figure (drawn in the printable lesson notes))

A walker sets off from a car park and goes 600 m due north, then 800 m due east.
Distance = 600 m + 800 m = 1400 m (no direction needed).
The displacement is the straight arrow from start to finish. On the grid it is 10 squares long and each square is 100 m, so its magnitude is 1000 m.
You can check with Pythagoras because the two legs are at right angles: √(600² + 800²) = √1 000 000 = 1000 m.

Answer: distance 1400 m; displacement 1000 m at about 53° east of north

Key fact: If you finish where you started, your displacement is zero, however far you travelled. A runner who completes a lap of a track has a displacement of 0 m.

Vectors and scalars

Every physical quantity has a magnitude (size). Some quantities also have a direction, and the direction matters: a 20 N push to the left has a very different effect from a 20 N push to the right.

  • displacement; Scalars (magnitude only) distance
  • velocity; Scalars (magnitude only) speed
  • force (including weight, friction, tension); Scalars (magnitude only) mass
  • acceleration; Scalars (magnitude only) time
  • momentum; Scalars (magnitude only) energy
  • gravitational field strength; Scalars (magnitude only) power, temperature

Some quantities come in pairs. Speed is a scalar; velocity is speed in a given direction, so it is a vector. Mass (in kg) is a scalar, but weight is a force (in N) that always acts downwards towards the centre of the Earth, so weight is a vector.

Exam tip: Ask yourself: “Does it make sense to say which way it points?” If yes, it is a vector. You can say which way a force pushes, but not which way a mass or an energy points.

Representing a vector with an arrow

A vector can be shown as an arrow:

  • the direction of the arrow shows the direction of the vector
  • the length of the arrow is proportional to the magnitude of the vector.

On a force diagram, label every arrow with the name of the force and its size. If the diagram is drawn to scale, write the scale on it, e.g. “1 cm ≡ 10 N” (≡ means “represents”).

Worked example: arrow lengths

Scale: 1 cm ≡ 10 N.
Pull of 45 N: length = 45 ÷ 10 = 4.5 cm. Friction of 15 N: length = 15 ÷ 10 = 1.5 cm.
Reading back: an arrow 3.2 cm long represents 3.2 × 10 = 32 N.

Answer: length (cm) = force ÷ (newtons per cm); force = length × (newtons per cm)

Scale diagrams of two vectors

To show two vectors on one scale diagram, choose a scale that makes the drawing a sensible size (roughly 5–15 cm for the longest arrow). Draw the first arrow, then draw the second one at the correct angle using a protractor. Use a sharp pencil and a ruler.

Worked example: drawing the walk to scale

Choose 1 cm ≡ 200 m. The 600 m leg is 3.0 cm long (pointing north) and the 800 m leg is 4.0 cm long (pointing east).
Join the start to the finish with a straight arrow. It measures 5.0 cm, so the displacement is 5.0 × 200 = 1000 m.
Measure the angle at the start with a protractor: about 53° east of north.

Answer: displacement = 1000 m at 53° east of north

Key fact: A scale diagram is only useful if the reader knows the scale. Always write it on the diagram, and measure lengths to the nearest millimetre.

Common mistakes

  • Giving a displacement or a velocity with no direction. Without a direction it is just a distance or a speed.
  • Adding the legs of a journey to get the displacement. Adding gives the distance; the displacement is the straight line from start to finish.
  • Calling mass a vector. Mass is a scalar; weight is the force (a vector) caused by gravity acting on that mass.
  • Drawing force arrows that are not to scale, or forgetting to state the scale on a scale diagram.

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these quantities is a vector? (Choose from: mass, energy, weight, speed.)

Answer: weight

Working: Weight is a force, and it always acts towards the centre of the Earth, so it has a direction: it is a vector. Mass, energy and speed have size only (scalars).

Question 2. A student uses the scale 1 cm ≡ 25 N. How long should she draw the arrow for a force of 150 N?

Answer: 6 cm

Working: 150 ÷ 25 = 6, so the arrow is 6.0 cm long.

Question 3. True or false? Two cyclists riding at 8 m/s in opposite directions have the same velocity.

Answer: False

Working: False. They have the same speed (8 m/s), but velocity is a vector and their directions are opposite, so their velocities are different.

Related lessons

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This lesson is section 8.1 (AGCPH0801) of our free GCSE Physics course for AQA. In the full course you get every video for this section, a self-marking practice quiz, a printable worksheet with a mark scheme, chapter tests, and your progress saved. Go to the full GCSE Physics course.

 

 

 

 

 

 

 

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