Scalars and Vectors

This lesson explains the difference between scalar and vector quantities, including how to add vectors and find the resultant force, as required by the Edexcel GCSE Combined Science specification (1SC0). Understanding scalars and vectors is fundamental to describing motion and forces correctly.

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Scalar Quantities

A scalar quantity has magnitude (size) only. It does not have a direction.

Scalar Quantity	SI Unit
Distance	m
Speed	m/s
Mass	kg
Time	s
Energy	J
Temperature	K or °C
Power	W

Examples

• A car has a speed of 30 m/s. This tells you how fast it is going, but not which way.
• A bag of sugar has a mass of 1 kg. Mass does not depend on direction.

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Vector Quantities

A vector quantity has both magnitude and direction.

Vector Quantity	SI Unit
Displacement	m
Velocity	m/s
Acceleration	m/s²
Force	N
Momentum	kg m/s
Weight	N

Examples

• A car has a velocity of 30 m/s due north. This tells you how fast and which way.
• The force of gravity on a ball is 5 N downwards. Both the size and direction matter.

Exam Tip: A common exam question asks you to classify quantities as scalar or vector. Remember: if direction matters, it is a vector. Key pairs to know are distance/displacement and speed/velocity.

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Distance vs Displacement

• Distance is the total length of the path travelled. It is a scalar.
• Displacement is the straight-line distance from start to finish in a stated direction. It is a vector.

Worked Example

A runner completes one full lap of a 400 m track, returning to the starting point.

• Distance travelled = 400 m
• Displacement = 0 m (they are back where they started)

A walker goes 3 km north then 4 km east.

• Distance = 3 + 4 = 7 km
• Displacement = straight-line distance = $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ km}$32+42​=9+16​=25​=5 km

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Representing Vectors

Vectors are represented by arrows in diagrams:

• The length of the arrow represents the magnitude.
• The direction of the arrow represents the direction of the vector.

graph LR
    A["Start"] -- "8 N" --> B["End"]

A longer arrow means a larger force or velocity.

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Adding Vectors Along the Same Line

When two vectors act along the same line, you add or subtract them depending on direction.

Same direction

If two forces act in the same direction, add them:

$F_{resultant} = F_1 + F_2$Fresultant​=F1​+F2​

Example: 5 N right + 3 N right = 8 N right

Opposite directions

If two forces act in opposite directions, subtract the smaller from the larger:

$F_{resultant} = F_1 - F_2$Fresultant​=F1​−F2​

Example: 8 N right + 3 N left = 5 N right

Worked Example

A toy car has a driving force of 6 N forward and friction of 2 N backward. What is the resultant force?

$F_{resultant} = 6 - 2 = 4 \text{ N forward}$Fresultant​=6−2=4 N forward

Exam Tip: Always state both the magnitude and the direction when giving a resultant vector. Writing just "5 N" without a direction will cost you a mark.

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Adding Vectors at Right Angles

When two vectors act at right angles, use Pythagoras' theorem to find the magnitude of the resultant:

$R = \sqrt{F_1^2 + F_2^2}$R=F12​+F22​​

Worked Example

A boat travels 30 m east then 40 m north. What is the magnitude of the resultant displacement?

$R = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ m}$R=302+402​=900+1600​=2500​=50 m

graph TB
    A["Start"] -- "30 m east" --> B["Turn"]
    B -- "40 m north" --> C["End"]
    A -. "R = 50 m" .-> C

You can also find the angle using trigonometry:

$\tan\theta = \frac{40}{30} = 1.333...$tanθ=3040​=1.333... $\theta = \tan^{-1}(1.333) = 53.1°$θ=tan−1(1.333)=53.1°

The resultant is 50 m at 53.1° north of east.

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Resultant Force

The resultant force is the single force that has the same effect as all the individual forces acting on an object combined.

• If the resultant force is zero, forces are balanced and the object stays at rest or moves at constant velocity.
• If the resultant force is non-zero, the object will accelerate in the direction of the resultant force.

Worked Example

Three forces act on an object: 10 N right, 4 N left, and 2 N left. Find the resultant.

$F_{resultant} = 10 - 4 - 2 = 4 \text{ N right}$Fresultant​=10−4−2=4 N right

Exam Tip: When adding vectors at right angles, always draw a diagram first. Label the sides, then apply Pythagoras. This helps avoid sign errors and makes your working clear.

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Summary

• Scalar quantities have magnitude only (e.g. distance, speed, mass, energy, time).
• Vector quantities have magnitude and direction (e.g. displacement, velocity, force, acceleration, momentum).
• Distance is total path length (scalar); displacement is straight-line distance in a given direction (vector).
• Vectors are shown as arrows — length = magnitude, arrow direction = vector direction.
• Vectors in the same line: add (same direction) or subtract (opposite directions).
• Vectors at right angles: use Pythagoras' theorem to find the resultant magnitude.
• The resultant force is the single force equivalent to all forces combined.
• Always give both magnitude and direction for vector answers.

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Comparison Table: Scalars vs Vectors at GCSE