Size alone or size and direction?
Some quantities are fully described by a number and a unit. Others also need a direction.
| Scalar (magnitude only) | Vector (magnitude and direction) |
|---|---|
| distance: 5 km | displacement: 5 km north |
| speed: 60 km/h | velocity: 60 km/h east |
| mass: 70 kg | weight (force): 686 N downward |
| temperature: 22 °C | force: 30 N at 40° |
| time: 3 s | acceleration: 9.8 m/s² down |
| energy, volume, area | momentum |
Why direction matters
Walking 3 km east then 3 km west covers 6 km of distance (scalar) but has zero displacement (vector): you’re back where you started.
Two people pushing a box with 50 N each:
- Same direction → 100 N total
- Opposite directions → 0 N total
- At right angles → about 70.7 N
Adding vectors depends on direction, not just size.
Describing vectors
A vector has:
- Magnitude: its size (length), never negative
- Direction: an angle, bearing or compass direction
Ways to write direction:
| Method | Example |
|---|---|
| compass | 40 km/h northeast |
| bearing | 300 km on a bearing of 060° |
| angle from a reference | 20 N at 30° above the horizontal |
| quadrant bearing | N30°E (30° east of north) |
Notation
- Bold letter: v
- Arrow over a letter: v⃗
- Arrow over two points: AB⃗ (from A to B)
- Magnitude: |v| or ‖v‖
Equal and opposite vectors
- Equal vectors: same magnitude and direction (position doesn’t matter).
- Opposite vectors: same magnitude, opposite direction: −v.
Speed vs velocity in action
A car drives around a circular track at a constant 100 km/h. Its speed is constant, but its velocity keeps changing, because the direction changes.
Worked examples
Example 1: Scalar or vector: 15 m/s north? Vector (velocity).
Example 2: Scalar or vector: 4 L? Scalar (volume).
Example 3: You walk 400 m east and 300 m west. Distance? Displacement? 700 m; 100 m east.
Common mistakes
- Using speed and velocity as if they’re the same.
- Adding vector magnitudes directly when directions differ.
- Giving a vector without its direction.
Try it yourself
- Classify: time, force, distance, acceleration.
- Run 2 laps of a 400 m track. Distance? Displacement?
- Are 10 N east and 10 N west equal vectors?
- What two parts describe a vector?
Answers: 1) scalar, vector, scalar, vector 2) 800 m; 0 m 3) No: they’re opposite. 4) magnitude and direction
Quick recap
- A scalar has magnitude only; a vector has magnitude and direction.
- Displacement and velocity are vectors; distance and speed are scalars.
- Vectors combine according to direction.