What is a Vector?

A vector is a quantity that has both magnitude and direction.

The word magnitude means the size or amount of something. Direction tells us where it is going, such as north, south, east, west, upwards, downwards, left, or right.

For example, a displacement of 8 km north-east is a vector because it tells us both the distance, 8 km, and the direction, north-east.

What you will learn

• Explain what a vector is
• Explain what a scalar is
• Tell the difference between vectors and scalars
• Tell the difference between distance and displacement, and between speed and velocity
• Recognise vector quantities in words
• Use correct vector notation with an arrow above the letters
• Understand how a vector can be drawn as an arrow

Key Ideas

Vector or Scalar?

In Mathematics, it is important to know whether a quantity is a vector or a scalar.

Type of Quantity	Meaning	Examples
Vector	Has magnitude and direction	Velocity, displacement, force
Scalar	Has magnitude only	Mass, time, temperature, speed, distance

Examples of Scalars

A scalar has size only. It does not need a direction.

• 30°C is a scalar because it gives temperature only.
• 10 kg is a scalar because it gives mass only.
• 60 km/h is speed, so it is a scalar because no direction is given.
• A distance of 8 km is a scalar because no direction is given.
• 5 seconds is a scalar because it gives time only.

Examples of Vectors

A vector has size and direction.

• 60 km/h due south is a vector because it gives speed and direction. It is called a velocity.
• A displacement of 8 km north-east is a vector because it gives distance and direction.
• A force of 5 N pushing east is a vector because it gives force and direction.
• A displacement of 3 m upward is a vector because it gives distance and direction.

Drawing a Vector

A vector can be represented by a directed line segment. This means it is drawn as an arrow.

• The starting point is where the vector begins.
• The arrowhead shows where the vector ends.
• The length of the arrow shows the magnitude.
• The direction of the arrow shows the direction of the vector.

Vector Notation

There are different ways to write a vector.

Notation	Meaning
AB	The vector from point A to point B
PQ	The vector from point P to point Q
a or a	A vector named a

When using two letters, the order matters.

Worked Examples

Example 1: State whether each quantity is a vector or a scalar.

(i) A force of 5 N pushing east

1. A force of 5 N pushing east has magnitude, 5 N, and direction, east.
2. So it is a vector.

(ii) A temperature of 30°C

1. A temperature of 30°C has magnitude only.
2. It does not give a direction.
3. So it is a scalar.

Answer: (i) Vector, (ii) Scalar

Example 2: A displacement is described as 8 km north-east. Is this a vector?

1. The magnitude is 8 km.
2. The direction is north-east.
3. Since it has both magnitude and direction, it is a vector.

Answer: Yes, it is a vector.

Example 3: State whether 60 km/h is a vector or scalar.

1. 60 km/h gives speed.
2. It tells us the size of the speed.
3. It does not tell us a direction.
4. Therefore, it is a scalar.

Answer: Scalar

Example 4: State whether 60 km/h due south is a vector or scalar.

1. 60 km/h gives the magnitude.
2. Due south gives the direction.
3. Since it has magnitude and direction, it is a vector. It is called a velocity.

Answer: Vector (a velocity)

Common Mistakes

• Thinking that every number with a unit is a vector.
• Forgetting that a vector must have direction.
• Calling speed a vector when no direction is given.
• Confusing speed and velocity. Speed is scalar, but velocity is vector.
• Confusing distance and displacement. Distance is scalar, but displacement is vector.
• Writing AB and BA as if they mean the same thing.
• Forgetting that the arrowhead shows the direction of the vector.

Exam-style multiple choice

Practice Questions

1. State whether a distance of 12 km is a vector or a scalar.
2. State whether a displacement of 12 km due west is a vector or a scalar.
3. What two features must a vector have?
4. The vector from P to Q is labelled p on a diagram. Write down two ways of denoting this vector.
5. State whether a mass of 15 kg is a vector or scalar.
6. State whether a force of 20 N acting upwards is a vector or scalar.
7. The vector AB points north-east. In which direction does BA point?

Answers

1. Scalar. Distance has magnitude only and no direction is given.
2. Vector. A displacement has magnitude and direction.
3. A vector must have magnitude and direction.
4. PQ or the bold/underlined letter p.
5. Scalar. Mass has magnitude only.
6. Vector. It has magnitude, 20 N, and direction, upwards.
7. South-west. BA has the same length as AB but the opposite direction.