Scalars and vectors

CIE A-Level Physics· 15 min read

1. Scalar vs Vector Quantities★☆☆☆☆⏱ 10 min

All physical quantities measured in physics are categorised as either scalars or vectors, based on whether direction is required to fully define the quantity.

📘 Definition

Scalar Quantity

A physical quantity that has only magnitude (size) and no associated direction.

Example:

Mass = 5 kg, speed = 10 ms⁻¹, energy = 200 J

📘 Definition

Vector Quantity

A physical quantity that has both magnitude and direction, both required to fully define the quantity.

Example:

Force = 10 N downwards, velocity = 20 ms⁻¹ north

Common scalars: mass, time, distance, speed, energy, temperature, power, work
Common vectors: displacement, velocity, acceleration, force, momentum, weight, electric field strength

📐 Worked Example

Classify each of the following as scalar or vector: (a) 25 °C (b) 15 ms⁻¹ upwards (c) 40 J of work

1
(a) 25 °C only has magnitude, no direction, so it is a scalar.
2
(b) 15 ms⁻¹ upwards has both magnitude (15 ms⁻¹) and direction (upwards), so it is a vector.
3
(c) 40 J of work only has magnitude, so it is a scalar.

2. Addition of Vectors★★☆☆☆⏱ 15 min

When combining multiple vectors, the result is called the resultant vector: a single vector that has the same effect as all the individual vectors combined. Vectors add differently to scalars, because their direction affects the final result.

📘 Definition

Resultant Vector

A single vector that produces the same overall effect as the combination of all individual vectors being added.

For parallel vectors, addition is simple: add magnitudes if vectors point in the same direction, subtract magnitudes if they point in opposite directions. For perpendicular vectors, use Pythagoras' theorem to find the resultant magnitude, and trigonometry to find its direction.

📐 Worked Example

Two perpendicular forces of 3.0 N and 4.0 N act at right angles at a point. Find the magnitude and direction of the resultant force.

1
Label the two perpendicular forces: $F_{1} = 3.0$ N, $F_{2} = 4.0$ N. Use Pythagoras' theorem for the resultant magnitude $R$ :
2
$R = \sqrt{F_1^2 + F_2^2} = \sqrt{3.0^2 + 4.0^2} = \sqrt{25} = 5.0 \text{ N}$
3
Find the angle $θ$ between the resultant and the 3.0 N force using the tangent relationship:
4
$\tan\theta = \frac{4.0}{3.0} \implies \theta = \tan^{-1}(1.333) \approx 53^\circ$
5
Final answer: The resultant force is 5.0 N at $5 3^{\circ}$ from the 3.0 N force.

3. Resolution of Vectors★★☆☆☆⏱ 15 min

Resolving a vector is the reverse of adding two perpendicular vectors: we split one vector into two perpendicular components (usually horizontal and vertical, parallel and perpendicular to a surface). This is one of the most frequently used skills in A-Level Physics, used for projectile motion, force equilibrium, and fields.

For a vector of magnitude $V$ at an angle $θ$ to the horizontal axis, the horizontal ( $V_{x}$ ) and vertical ( $V_{y}$ ) components are:

V_x = V \cos\theta, \quad V_y = V \sin\theta

📐 Worked Example

A ball is kicked with an initial velocity of 12 ms⁻¹ at 40° above the horizontal. Find the horizontal and vertical components of the velocity.

1
Identify $V = 12$ ms⁻¹, $θ = 4 0^{\circ}$ to the horizontal. Calculate the horizontal component:
2
$V_x = 12 \times \cos(40^\circ) \approx 12 \times 0.766 = 9.2 \text{ ms}^{-1}$
3
Calculate the vertical component:
4
$V_y = 12 \times \sin(40^\circ) \approx 12 \times 0.643 = 7.7 \text{ ms}^{-1}$
5
Final answer: Horizontal component = 9.2 ms⁻¹, vertical component = 7.7 ms⁻¹, both directed upwards from the horizontal.

4. Common Pitfalls

Wrong move:

Calling speed a vector and velocity a scalar

Why:

Speed only describes how fast something is moving (no direction), while velocity includes direction

Correct move:

Always classify speed as scalar, velocity as vector

Wrong move:

Swapping sine and cosine when resolving vectors

Why:

Incorrectly applying trigonometric ratios to the angle relative to the axis

Correct move:

The component along the axis that $θ$ is measured from is always $V cos θ$ , the perpendicular component is $V sin θ$

Wrong move:

Adding vector magnitudes directly regardless of direction

Why:

Vectors have direction, so magnitudes only add when vectors point the same way

Correct move:

Resolve all vectors to perpendicular components before adding any non-parallel vectors

Wrong move:

Forgetting to state the direction of a final vector answer

Why:

Vectors require both magnitude and direction to be fully defined

Correct move:

Always give the direction (angle relative to a reference axis) for any vector result

5. Quick Reference Cheatsheet

Concept	Key Rule / Formula
Scalar vs Vector	Scalar: magnitude only; Vector: magnitude + direction
Parallel vector addition	Same direction: $R = V_{1} + V_{2}$ ; Opposite: $R = ∣ V_{1} - V_{2} ∣$
Perpendicular resultant	$R = V_{1}^{2} + V_{2}^{2}$ , $θ = tan^{- 1} (\frac{V _{2}}{V _{1}})$
Vector resolution ( $θ$ from x-axis)	$V_{x} = V cos θ$ , $V_{y} = V sin θ$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Classify scalar/vector quantities
2023 · 2
Resolve force components
2021 · 1
Calculate resultant perpendicular force

Going deeper

syllabusOfficial CIE 9702 SyllabusFull unit 1 learning outcomes

What's Next

Scalars and vectors are the foundational skill for almost all topics in CIE A-Level Physics, from two-dimensional motion to force equilibrium and field interactions. Mastering vector addition and resolution lets you break complex multi-directional problems into simple perpendicular components that are easy to solve. This core skill is used repeatedly in mechanics, electromagnetism, and waves, so building fluency now will make all future topics much easier to grasp.