Physical quantities & measurement
Scalars and vectors
Extended (Supplement)
Scalars and vectors defined
A scalar quantity has magnitude only — a size with a unit but no direction.
A vector quantity has both magnitude and direction.
A vector quantity has both magnitude and direction.
This distinction matters enormously in mechanics: 5 m north and 5 m south are very different, even though their magnitudes are equal.
List of scalar and vector quantities
You are required to know all of the following. They will not all be on the formula sheet.
| Scalar quantities | Vector quantities |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Time | Acceleration |
| Mass | Force |
| Energy | Weight |
| Temperature | Momentum |
| Electric potential | Electric field strength |
| Pressure | Gravitational field strength |
A common exam question: "State whether [quantity] is a scalar or vector, and give a reason." The reason for a vector is always "it has both magnitude and direction." For a scalar: "it has magnitude only."
Representing vectors
Vectors are drawn as arrows:
- The length of the arrow represents the magnitude (drawn to scale).
- The direction of the arrow represents the direction of the vector.
- The arrow is often labelled with its symbol in bold or with a bar/arrow above: \(\vec{F}\) or F.
Scalar vs. vector in context
The distinction between scalars and vectors creates paired quantities in mechanics:
- Distance (scalar) vs. Displacement (vector): distance is how far you travel; displacement is how far you are from the start in a given direction.
- Speed (scalar) vs. Velocity (vector): speed is the rate of change of distance; velocity is the rate of change of displacement.
Exam question — scalar or vector?
"A car travels 500 m due north, then 300 m due south. State (a) the total distance travelled and (b) the total displacement."
(a) Distance = 500 + 300 = 800 m (scalar — only magnitude)
(b) Displacement = 500 − 300 = 200 m north (vector — magnitude AND direction)
Note: the displacement answer must include a direction to be complete.
Many students confuse "distance" with "displacement" in their definitions. Always include "in a specified direction" when defining displacement or velocity.