Vectors and scalars

The idea

A scalar quantity has magnitude only, so it is fully described by a number and a unit. Mass, distance, speed, time and energy are scalars: a runner with a speed of 5 m/s needs no direction to make that statement complete. A vector quantity has both magnitude and direction, so the direction is part of the quantity. Force, velocity, displacement, acceleration and momentum are vectors: a velocity of 5 m/s due north differs from 5 m/s due south, and a force of 10 N to the right differs from 10 N to the left. The key difference is therefore the presence of direction, not the size of the number. Two objects can share the same speed but have different velocities because their directions differ, and adding vectors must take direction into account, whereas scalars simply add as numbers.

What earns the marks

• States that a scalar has magnitude only, while a vector has both magnitude and direction.
• Gives a correct scalar example such as mass, speed, distance or time and explains that direction is not needed.
• Gives a correct vector example such as force, velocity, displacement or acceleration and states its direction.
• Explains that two objects with equal speed but opposite directions have different velocities.
• Explains that direction must be considered when combining vectors but not when combining scalars.

Common mistakes and how to put them right

• Treats speed as a vector.

  Put it right: Recognise that speed is a scalar and velocity is the vector that includes direction.

• Treats mass as a vector.

  Put it right: State that mass is a scalar with magnitude only, while weight is a force and so a vector.

• Says a vector is simply a larger quantity.

  Put it right: Explain that the difference is direction, not size.

• Adds velocities as plain numbers while ignoring direction.

  Put it right: Include direction when combining vectors.

Exam tips

• Always pair each definition with a named example and say whether direction is included.
• Use speed versus velocity to show the scalar–vector contrast clearly.
• State the direction explicitly when you give a vector example, such as 10 N downwards.