A scalar quantity has size only. A vector quantity has size and direction, so it is incomplete without a direction.
This lesson is part of SPM Physics measurement. It builds on the units from converting SI prefixes and units.
What is the direction test?
Ask: “Is this quantity complete without a direction?” If yes, it is a scalar. If no, it is a vector.
| Quantity | Type | Reason |
|---|---|---|
| Distance | Scalar | “5 m” is complete |
| Displacement | Vector | “5 m north” needs a direction |
| Speed | Scalar | “20 m/s” is complete |
| Velocity | Vector | “20 m/s east” needs a direction |
| Mass, time, energy | Scalar | Complete without a direction |
| Force, weight, acceleration | Vector | Each acts in a direction |
Worked example: a right-angled path
An original question: a student walks 6 m east and then 8 m north.
Distance = 6 + 8 = 14 m, the total length of the path.
Displacement is the straight line from start to finish. Using Pythagoras, √(6² + 8²) = √100 = 10 m. Its direction is tan⁻¹(8 ÷ 6) = 53° north of east.
The distance is larger than the displacement because the path bends.
A full lap
A runner completes one lap of a 400 m track and stops at the start line. The distance is 400 m. The displacement is 0 m, because the final position is the same as the starting position.
If the lap takes 80 s, the average speed is 400 ÷ 80 = 5 m/s, but the average velocity is 0 ÷ 80 = 0.
The mistake that costs marks
The common slip is to use the words speed and velocity as if they were the same. The lap shows that the two values can differ completely.
| Statement | Wrong | Right |
|---|---|---|
| Average velocity over a full lap | 5 m/s | 0 m/s |
| Displacement after 6 m east and 8 m north | 14 m | 10 m at 53° north of east |
Always give a direction for a vector answer.
Check yourself
A car travels 30 km east and then 40 km north in 1 hour. Find the distance, displacement, average speed and average velocity.
Answer
Distance = 30 + 40 = 70 km.
Displacement = √(30² + 40²) = 50 km, at tan⁻¹(40 ÷ 30) = 53° north of east.
Average speed = 70 ÷ 1 = 70 km/h.
Average velocity = 50 ÷ 1 = 50 km/h at 53° north of east.
What to study next
Continue with reading measuring instruments and uncertainty. The measurement practice set includes classification questions.
The units and significant-figure checker supports answer presentation. If you want a teacher to test the direction check on new paths, see online one-to-one Physics tuition.