A scalar quantity has magnitude (size) only; a vector quantity has both magnitude and direction. This distinction matters enormously in physics: 60 m/s tells you how fast something moves (scalar speed), but 60 m/s due north tells you how fast and where it is heading (vector velocity). Adding, subtracting, and resolving vectors requires careful treatment of direction.
What are scalar quantities?
A scalar is a quantity that is completely described by a magnitude alone — a number with an appropriate unit. You do not need to state a direction.
Common scalar quantities:
| Quantity | Unit | Example value |
|---|---|---|
| Distance | metre (m) | 150 m |
| Speed | metre per second (m/s) | 12 m/s |
| Mass | kilogram (kg) | 70 kg |
| Time | second (s) | 45 s |
| Temperature | °C or kelvin (K) | 37 °C |
| Energy | joule (J) | 4,000 J |
| Power | watt (W) | 100 W |
| Pressure | pascal (Pa) | 101,325 Pa |
Adding scalars is straightforward ordinary arithmetic: 5 kg + 3 kg = 8 kg.
What are vector quantities?
A vector is a quantity that has both magnitude and direction. Stating a direction is essential — without it the quantity is physically incomplete.
Common vector quantities:
| Quantity | Unit | Example value |
|---|---|---|
| Displacement | metre (m) | 150 m due north |
| Velocity | metre per second (m/s) | 12 m/s at 30° north of east |
| Force | newton (N) | 500 N downward |
| Acceleration | metre per second squared (m/s²) | 10 m/s² downward |
| Momentum | kilogram metre per second (kg·m/s) | 840 kg·m/s eastward |
| Weight | newton (N) | 700 N downward |
Vectors are represented as arrows: the length of the arrow is proportional to the magnitude, and the arrowhead shows the direction.
What is the difference between distance and displacement, and speed and velocity?
These paired quantities are the most important scalar–vector pairs in GCSE physics:
Distance vs displacement:
- Distance is a scalar: the total path length travelled (e.g., "the runner ran 400 m").
- Displacement is a vector: the straight-line distance from start to finish point, with direction stated (e.g., "the runner ended 0 m from the start" — if they completed a lap of a track).
Speed vs velocity:
- Speed is a scalar: how fast an object moves, regardless of direction (speed = distance ÷ time).
- Velocity is a vector: speed in a stated direction (velocity = displacement ÷ time).
An object can travel at constant speed while its velocity continuously changes — this happens in circular motion (speed is constant but direction, and therefore velocity, changes continuously).
How do you add vectors?
Adding scalars is ordinary arithmetic. Adding vectors requires accounting for direction.
Case 1 — vectors in the same direction: Two forces act on an object: 40 N east and 25 N east. Resultant = 40 + 25 = 65 N east (same direction, just add magnitudes).
Case 2 — vectors in opposite directions: A force of 50 N east and 30 N west act on the same object. Resultant = 50 − 30 = 20 N east (subtract smaller from larger, keep direction of larger).
Case 3 — vectors at right angles (Pythagoras' theorem): A boat moves at 3 m/s north across a river, while the river current flows at 4 m/s east.
| Component | Value |
|---|---|
| Northward velocity | 3 m/s |
| Eastward velocity | 4 m/s |
| Resultant speed | √(3² + 4²) = √(9+16) = √25 = 5 m/s |
The resultant velocity is 5 m/s at an angle north-east of the river bank. At GCSE you may use Pythagoras' theorem for perpendicular vectors, or a scale drawing for other angles.
Case 4 — scale drawing method (any angle):
- Choose a scale (e.g., 1 cm = 10 N).
- Draw the first vector as an arrow to scale in the correct direction.
- Place the tail of the second vector at the tip of the first.
- Draw the resultant from the tail of the first to the tip of the second.
- Measure the resultant arrow's length and convert back using the scale; measure its angle with a protractor.
Why does direction matter for Newton's second law?
Force is a vector. Newton's second law, F = ma, requires the resultant force — the single vector found by adding all individual forces acting on an object, accounting for direction.
If two equal and opposite forces act (e.g., 10 N left and 10 N right), the resultant is zero, so acceleration is zero — the object remains stationary or moves at constant velocity (Newton's first law). If forces are unequal, there is a net resultant force and the object accelerates in the direction of that resultant.
Frequently asked questions
What is the difference between a scalar and a vector?
A scalar quantity has magnitude only — it tells you "how much" but not "which way". A vector quantity has both magnitude and direction — it tells you "how much" and "which way". For example, speed (scalar) tells you how fast an object moves; velocity (vector) tells you how fast and in which direction. Mass, temperature, and energy are scalars; force, acceleration, displacement, and momentum are vectors.
Why does direction matter when adding forces?
Forces are vectors, so their directions must be taken into account when finding the resultant. A 10 N force north and a 10 N force south cancel out to give a resultant of zero — the object does not accelerate. The same two forces both northward would give a resultant of 20 N north and cause significant acceleration. Simple addition of magnitudes without direction can give a completely wrong answer for the resultant, so always indicate direction when working with vectors.
How do you find the resultant of two perpendicular forces?
For two forces at right angles, use Pythagoras' theorem: resultant² = F₁² + F₂². For example, a 6 N force eastward and an 8 N force northward give a resultant of √(6² + 8²) = √(36 + 64) = √100 = 10 N. The angle of the resultant with respect to one of the original forces can be found using trigonometry (tan θ = opposite ÷ adjacent) or by a scale drawing and protractor.
Can velocity and speed have the same numerical value?
Yes — a car travelling at 30 m/s due east has a speed of 30 m/s (scalar) and a velocity of 30 m/s due east (vector). The numbers are identical; only the vector description adds the directional information. However, they differ in change: if the car turns a corner at the same speed, the speed is unchanged but the velocity has changed (direction changed), meaning the car is accelerating (even though it is not speeding up or slowing down). This is why uniform circular motion involves constant speed but continuously changing velocity, and therefore requires a centripetal force.
For Socratic GCSE physics with Professor Newton — predict the resultant force direction before any calculation, then test your reasoning — visit aitutors.me.