Module 2 · Foundations of Physics

Scalars and Vectors

Specification: OCR A H556 | Section: 2.3.1 Scalars and vectors | Focus: scalar vs vector quantities, vector magnitude and direction, addition, subtraction, vector triangles and resolving into perpendicular components

By the end of this topic you should be able to…

tell the difference between a scalar quantity and a vector quantity
give clear examples of each and explain why they belong in that category
add and subtract vectors using diagrams and components
resolve a vector into perpendicular components using sine and cosine correctly
spot common OCR exam traps involving direction, angle choice and signs

Big idea: a scalar tells you how much. A vector tells you how much and which way. That extra direction changes how the maths works.

Part 1

Scalar or vector?

A scalar has magnitude only. A vector has magnitude and direction.

Scalars

Examples include distance, speed, mass, time, energy, temperature and charge.

Vectors

Examples include displacement, velocity, acceleration, force, momentum and electric field strength.

Why it matters

Scalars add normally. Vectors must be combined by considering direction as well as size.

Common misconception

Speed is a scalar, but velocity is a vector. Distance is a scalar, but displacement is a vector. These pairs are not interchangeable.

Useful visuals

What vectors look like

Magnitude and direction

A vector is defined by both its size and its direction: the arrow length represents magnitude, and the arrowhead plus angle show which way it acts.

Resultant by vector triangle

Place vectors head-to-tail. The resultant goes from the start of the first vector to the end of the last.

Part 2

Adding and subtracting vectors

To add vectors, place them head-to-tail. The resultant is the single vector with the same overall effect.

To subtract a vector, add the opposite vector instead. In other words, A − B means A + (−B).

Key ideaA − B = A + (−B)

OCR exam tip

If the question asks for a resultant of two coplanar vectors, you may need either a scale drawing or a component method. Both depend on direction being handled correctly.

Exam trap

Do not subtract magnitudes unless the vectors are already along the same straight line and in clearly opposite directions.

Part 3

Resolving a vector into perpendicular components

Resolving means splitting one vector into two perpendicular parts, usually horizontal and vertical.

For a vector F at angle θ to the horizontalF_x = F cos θ and F_y = F sin θ

This is the OCR A notation given in the specification: F_x = F cos i and F_y = F sin i.

Resolving a force

Here θ is measured from the horizontal, so the horizontal component is adjacent to θ and the vertical component is opposite to θ.

When angles cause trouble

If the angle is measured from the horizontal, use cos for the horizontal component. If the angle is measured from the vertical, use cos for the vertical component.

Interactive

Vector addition and component explorer

Vector A magnitude8.0 N

Vector A angle30°

Vector B magnitude6.0 N

Vector B angle110°

A components: A_x = 6.9, A_y = 4.0

B components: B_x = −2.1, B_y = 5.6

Resultant: R = 10.8 N at 58°

This is most useful for checking how direction changes the resultant. Try two large vectors pointing in nearly opposite directions.

Part 4

Common misconceptions and exam traps

“Velocity is just speed”

No. Velocity includes direction, so it is a vector.

“Resultant means add the numbers”

Only if the vectors are in the same direction. Usually you must consider angles.

“Cos is always horizontal”

Only if the angle is measured from the horizontal side you are using.

“A negative component is wrong”

No. A negative component just means the vector points opposite to your chosen positive axis.

High-value OCR habit

Sketch the vector first, label the angle clearly, and only then choose the trig function. This avoids many sine/cosine mistakes.

Worked Examples

Worked examples

Worked example 1

State whether each quantity is scalar or vector: a) distance b) displacement c) speed d) velocity.

distance is a scalar

displacement is a vector

speed is a scalar

velocity is a vector

Worked example 2

A force of 20 N acts at 35° above the horizontal. Resolve it into horizontal and vertical components.

F_x = 20 cos 35° = 16.4 N

F_y = 20 sin 35° = 11.5 N

Worked example 3

Two forces act along the same line: 12 N east and 5 N west. Find the resultant.

Take east as positive

Resultant = 12 + (−5) = 7 N east

Worked example 4

A boat travels 3.0 km east and then 4.0 km north. Find the magnitude of the displacement.

These are perpendicular components of displacement

Magnitude = √(3.0² + 4.0²)

Magnitude = 5.0 km

Worked example 5

A force of 10 N is at 60° to the horizontal. Find the components.

F_x = 10 cos 60° = 5.0 N

F_y = 10 sin 60° = 8.66 N

Worked example 6

A student wants A − B. Explain a good vector method.

Reverse the direction of B to make −B

Then add A and −B head-to-tail

The resultant from the start to the finish is A − B

Worked example 7

A force has components 6 N east and 8 N north. Find the magnitude of the resultant and its angle above east.

Magnitude = √(6² + 8²) = 10 N

tan θ = 8 / 6

θ = 53° above east

Worked example 8

A 15 N force makes an angle of 20° with the vertical. Find the horizontal and vertical components.

The angle is from the vertical, so the vertical component is adjacent

Vertical component = 15 cos 20° = 14.1 N

Horizontal component = 15 sin 20° = 5.13 N

This is a classic angle-choice trap

Knowledge

Knowledge Check

What is the difference between a scalar and a vector?

2 marks

Scalar has magnitude only
Vector has magnitude and direction

State whether force is scalar or vector.

1 mark

Vector

What does resolving a vector mean?

1 mark

Splitting a vector into components, usually perpendicular components

How would you treat A − B using a vector diagram?

1 mark

Add A to the opposite of B / add A and −B

Exam Practice

Exam-Style Questions

State one scalar quantity and one vector quantity.

2 marks

Any correct scalar, for example mass or speed
Any correct vector, for example force or velocity

A force of 18 N acts at 25° above the horizontal.

a) Calculate the horizontal component. [2 marks]
b) Calculate the vertical component. [2 marks]

4 marks

a) F_x = 18 cos 25° = 16.3 N
b) F_y = 18 sin 25° = 7.61 N

Explain why speed is scalar but velocity is vector.

2 marks

Speed has magnitude only
Velocity has magnitude and direction

Two perpendicular forces act on a point: 9 N east and 12 N north.

a) Find the magnitude of the resultant force. [2 marks]
b) Find the direction of the resultant force above east. [2 marks]

4 marks

a) Resultant = √(9² + 12²) = 15 N
b) tan θ = 12 / 9, so θ = 53° above east

A force F acts at an angle i to the horizontal.

a) Write expressions for F_x and F_y. [2 marks]
b) State one condition under which these expressions would need to be changed. [1 mark]

3 marks

a) F_x = F cos i and F_y = F sin i
b) If the angle is measured from the vertical / a different axis, the sine and cosine choice changes

A student says, “To find the resultant of any two vectors, just add their magnitudes.” Explain why this is wrong.

3 marks

Vectors have direction as well as magnitude
The angle between the vectors affects the resultant
Only vectors in the same direction can be added by adding magnitudes directly

A 24 N force acts at 30° to the vertical.

a) Calculate the vertical component. [2 marks]
b) Calculate the horizontal component. [2 marks]
c) Explain one common mistake in this question. [1 mark]

5 marks

a) Vertical component = 24 cos 30° = 20.8 N
b) Horizontal component = 24 sin 30° = 12.0 N
c) Using the wrong trig function because the angle is from the vertical, not the horizontal

A hiker walks 5.0 km east and then 12.0 km north.

a) Find the magnitude of the displacement. [2 marks]
b) Find the direction of the displacement above east. [2 marks]
c) State the total distance travelled. [1 mark]

5 marks

a) Magnitude = √(5.0² + 12.0²) = 13.0 km
b) tan θ = 12.0 / 5.0, so θ = 67° above east
c) Distance = 17.0 km

Topic Summary

Scalars

Scalars have magnitude only, so they add using ordinary arithmetic.

Vectors

Vectors have magnitude and direction, so direction must always be included in the maths.

Components

Resolve vectors carefully and choose sine or cosine by looking at the angle actually given.

Resultants

Use head-to-tail diagrams, scale drawings or components to combine vectors properly.

F_x = F cos i

F_y = F sin i

A − B = A + (−B)

Resultant depends on angle