Scalars and Vectors

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus
  2. Add and subtract coplanar vectors
  3. Represent a vector as two perpendicular components

1. What are Scalars and Vectors?

Understanding the Difference

In physics, every measurement we make falls into one of two categories: scalars or vectors.

Scalar quantities have magnitude only. Magnitude simply means size or amount. When you measure a scalar, you only need to state how much of it there is. For example, if someone asks "How far did you walk?", you might say "5 kilometres". That's all the information needed – just the distance.

Vector quantities have both magnitude and direction. When you measure a vector, you need to state both how much of it there is AND which way it's pointing. For example, if someone asks "What is your displacement?", you need to say something like "5 kilometres north-east". The direction matters just as much as the distance.

Why Direction Matters

Here's a simple way to tell if something is a vector or scalar: Can two equal amounts cancel each other out?

  • If you add 3 minutes to 3 minutes, you always get 6 minutes. Direction doesn't matter. Time is a scalar.
  • If you walk 3 metres north and then 3 metres south, you end up back where you started. The two movements cancel out because they're in opposite directions. Displacement is a vector.

Distance vs Displacement: A Key Example

Let's look at the most important example to understand this concept:

Distance is the total length of the path you actually travelled. It doesn't matter which direction you went – you just add up all the metres or kilometres you covered. Distance is a scalar.

Displacement is the straight-line distance from your starting point to your finishing point, measured in a specific direction. It's like drawing an arrow directly from start to finish. Displacement is a vector.

Example: Imagine walkers who travel from town A to town B (8 km), then from town B to town C (7 km). Their winding route means they covered a total distance of 15 km. However, if you draw a straight line from town A to town C, it's only 10 km. Their displacement is 10 km in the direction from A to C (let's say 30° east of north).

Examples of Scalars and Vectors in Your Syllabus

You need to know which quantities are scalars and which are vectors:

Scalar quantities (magnitude only):

  • Distance
  • Speed
  • Mass
  • Time
  • Energy
  • Temperature
  • Density
  • Pressure
  • Electric charge
  • Work

Vector quantities (magnitude and direction):

  • Displacement
  • Velocity
  • Acceleration
  • Force
  • Momentum
  • Weight (which is a type of force)

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