# Scalars and vectors

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u1-scalars-and-vectors/

This sub-topic covers the core distinction between scalar and vector quantities, methods for vector addition, and resolution of vectors into perpendicular components, a foundational skill for all kinematics and dynamics topics in A-Level Physics.

**Prerequisites:** Basic right-angled trigonometry; Pythagoras' theorem for right triangles

## Learning objectives

- Distinguish between scalar and vector physical quantities
- Add vectors using graphical and calculation methods
- Resolve any vector into two perpendicular components
- Apply vector concepts to common physics problems

## Scalar vs Vector Quantities

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

**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

**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.

## Addition of Vectors

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.

**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 $\theta$ 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 $53^\circ$ from the 3.0 N force.

## Resolution of Vectors

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.

> **SOH-CAH-TOA**
>
> Use this mnemonic to remember trigonometric ratios for right triangles: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

For a vector of magnitude $V$ at an angle $\theta$ 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, $\theta = 40^\circ$ 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.

## Common pitfalls

- **Wrong:** Calling speed a vector and velocity a scalar
  - Why it fails: Speed only describes how fast something is moving (no direction), while velocity includes direction
  - Correct: Always classify speed as scalar, velocity as vector
- **Wrong:** Swapping sine and cosine when resolving vectors
  - Why it fails: Incorrectly applying trigonometric ratios to the angle relative to the axis
  - Correct: The component along the axis that $\theta$ is measured from is always $V\cos\theta$, the perpendicular component is $V\sin\theta$
- **Wrong:** Adding vector magnitudes directly regardless of direction
  - Why it fails: Vectors have direction, so magnitudes only add when vectors point the same way
  - Correct: Resolve all vectors to perpendicular components before adding any non-parallel vectors
- **Wrong:** Forgetting to state the direction of a final vector answer
  - Why it fails: Vectors require both magnitude and direction to be fully defined
  - Correct: Always give the direction (angle relative to a reference axis) for any vector result

## 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 | $R = \sqrt{V_1^2 + V_2^2}$, $\theta = \tan^{-1}\left(\frac{V_2}{V_1}\right)$ |
| Vector resolution ($\theta$ from x-axis) | $V_x = V\cos\theta$, $V_y = V\sin\theta$ |

## 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.

- [Projectile Motion](https://www.owlsprep.com/study/cie-9702-u2-projectile-motion/)
- [SI base and derived units](https://www.owlsprep.com/study/cie-9702-u1-si-base-and-derived-units/)
- [Unit prefixes](https://www.owlsprep.com/study/cie-9702-u1-unit-prefixes/)

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