Section 1 of 5Scalars and Vectors
Quantities in physics are either fully described by a size alone, or need a direction as well.
Scalar
- A quantity with magnitude only, such as mass, time, temperature, energy or speed.
Vector
- A quantity with both magnitude and direction, such as force, velocity, displacement or acceleration.
Notation
- A vector can be drawn as an arrow whose length shows the magnitude and whose head shows the direction.
Signs
- In one dimension, opposite directions are given by opposite signs, so a velocity of −5 m s⁻¹ is the reverse of +5 m s⁻¹.
Section 2 of 5Distance, Displacement, Speed and Velocity
Distance, Displacement, Speed and Velocity
Distance
- The total length of the path travelled; it is always positive and is a scalar.
Displacement
- The straight-line distance from the starting point to the finishing point, in a stated direction;
- it is a vector.
Speed
- The distance travelled per unit time, which is a scalar.
Velocity
- The rate of change of displacement, which is a vector pointing in the direction of motion.
| Scalar | Vector |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Weight |
| Energy | Force |
Section 3 of 5Why the Difference Matters
1
Scalar quantities are added by ordinary arithmetic because direction does not matter.
2
Vector quantities must be added taking their directions into account, using a scale drawing or trigonometry.
3
An object moving in a circle at constant speed has a changing velocity, because the direction keeps changing.
4
Returning to the start gives a displacement of zero even though the distance travelled can be large.
5
Equations of motion require vectors, so a consistent choice of positive direction is essential.
Section 4 of 5Worked Example
Question
- A runner completes one lap of a 400 m track in 80 s and finishes at the starting point.
- Calculate the average speed and the average velocity for the lap.
- Average speed = distance ÷ time = 400 ÷ 80 = 5.0 m s⁻¹.
- Displacement for a full lap = 0 m, since the runner ends where she started.
- Average velocity = displacement ÷ time = 0 ÷ 80 = 0 m s⁻¹.
Answer The average speed is 5.0 m s⁻¹ but the average velocity is zero.
Section 5 of 5Exam Checklist & Tips
2.3.1 – Key Points for the Exam
- Scalar A quantity that has magnitude but no direction.
- Vector A quantity that has both magnitude and direction.
- Displacement The distance from the start to the finish in a given direction.
- Velocity The rate of change of displacement.
- Magnitude The size of a quantity.
Common Mistakes
- Calling weight a scalar because it is often quoted as a number in newtons; it is a force and so a vector.
- Using distance in place of displacement in equations of motion.
- Saying velocity and speed are the same thing because they share a unit.
- Forgetting to specify the direction when quoting a vector.