# Types of physical quantity

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u1-types-of-physical-quantity/

This sub-topic introduces the core classification of physical quantities into scalars and vectors, and covers key skills of vector addition and resolution. This foundational knowledge is required for all mechanics and electromagnetism topics in CIE A-Level Physics.

**Prerequisites:** Basic arithmetic and right-angled triangle trigonometry

## Learning objectives

- Distinguish between scalar and vector physical quantities
- Identify common examples of scalar and vector quantities
- Calculate the resultant of two perpendicular vectors
- Resolve a vector into two perpendicular components

## Classification of Scalars and Vectors

**Scalar quantity** — A physical quantity that has only magnitude (size) and no direction. Fully described by a single number and unit.

*Notation:* Written as plain symbol, e.g. $m$ for mass

*Example:* Mass, temperature, energy, distance, speed, time

**Vector quantity** — A physical quantity that requires both magnitude and direction to be fully described.

*Notation:* Written with an arrow over the symbol, e.g. $\vec{v}$ for velocity

*Example:* Displacement, velocity, acceleration, force, momentum

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

> **Common exam distinction**
>
> Speed is a scalar (only magnitude) but velocity is a vector (speed + direction). This is tested very frequently in multiple choice questions.

**Worked example:** Classify each of the following as scalar or vector: (a) kinetic energy, (b) acceleration, (c) weight, (d) density

1. Check if the quantity requires direction to be fully described.
2. (a) Kinetic energy only has size, no direction: **scalar**.
3. (b) Acceleration needs magnitude and direction of change of velocity: **vector**.
4. (c) Weight is a force acting towards the centre of the Earth, so has direction: **vector**.
5. (d) Density is mass per unit volume, only has size: **scalar**.

## Adding Vectors to Find the Resultant

When two or more vectors act at the same point, their combined effect is called the **resultant vector**. For parallel vectors, add magnitudes if they act in the same direction, subtract magnitudes if opposite. For non-parallel vectors, we use geometry to find the resultant.

**Resultant vector** — A single vector that produces the same effect as the combination of all original vectors.

**Worked example:** A hiker walks 4.0 km east, then 3.0 km north. Calculate the resultant displacement from the starting point.

1. The two displacement vectors are perpendicular, so use Pythagoras' theorem for magnitude.
2. $$|\vec{s}| = \sqrt{(4.0)^2 + (3.0)^2} = \sqrt{25} = 5.0 \text{ km}$$
3. Calculate direction, measured from east towards north:
4. $$\theta = \tan^{-1}\left(\frac{3.0}{4.0}\right) = 37^\circ$$
5. Final resultant displacement is **5.0 km at 37° north of east**.

## Resolving Vectors into Perpendicular Components

Any vector can be split into two independent, perpendicular components. This process (called resolution) is the most used vector skill in A-Level mechanics, used for projectile motion, inclined planes, and equilibrium.

> **tip**
>
> For any vector of magnitude $V$ at an angle $\theta$ to the x-axis, the components are always:

$$V_x = V \cos\theta \\ V_y = V \sin\theta$$

**Worked example:** A force of 30 N acts at 40° above the horizontal. Find the horizontal and vertical components of the force.

1. Identify $F = 30$ N, $\theta = 40^\circ$ from the horizontal axis.
2. Calculate the horizontal (x) component:
3. $$F_x = 30 \times \cos(40^\circ) = 30 \times 0.766 = 23 \text{ N (2 s.f.)}$$
4. Calculate the vertical (y) component:
5. $$F_y = 30 \times \sin(40^\circ) = 30 \times 0.643 = 19 \text{ N (2 s.f.)}$$
6. Final components: 23 N horizontal, 19 N vertical.

## Common pitfalls

- **Wrong:** Confusing speed (scalar) and velocity (vector) as the same quantity type
  - Why it fails: This is a common multiple-choice trap set by examiners
  - Correct: Always remember: speed is scalar (only magnitude), velocity is vector (speed + direction)
- **Wrong:** Swapping sine and cosine when resolving vectors
  - Why it fails: Misremembering which side of the triangle matches the angle
  - Correct: Always label the triangle: the component along the angle direction uses cosine, perpendicular uses sine
- **Wrong:** Adding scalar magnitudes of non-parallel vectors to get the resultant
  - Why it fails: Direction changes the total magnitude, so simple addition only works for parallel vectors
  - Correct: Use Pythagoras for perpendicular vectors, or trigonometry on a vector diagram for non-parallel vectors
- **Wrong:** Forgetting to state the direction of a resultant vector
  - Why it fails: Examiners require full description of vectors, and will deduct marks for missing direction
  - Correct: Always add direction (e.g. 25° north of east) when giving a final vector answer
- **Wrong:** Resolving vectors into non-perpendicular components for calculations
  - Why it fails: Non-perpendicular components are not independent, leading to wrong results
  - Correct: Always resolve into two perpendicular components for all standard A-Level problems

## Cheatsheet

| Concept | Key Rule |
| --- | --- |
| Scalar quantity | Only magnitude, no direction |
| Vector quantity | Both magnitude and direction |
| Resultant of 2 perpendicular vectors | $R = \sqrt{A^2 + B^2}$ |
| Resolve $V$ at angle $\theta$ to x-axis | x: $V\cos\theta$, y: $V\sin\theta$ |
| Parallel vectors same direction | Add magnitudes |
| Parallel vectors opposite direction | Subtract magnitudes |

## What's next

Understanding the classification of physical quantities and the core skills of vector addition and resolution is the foundation for almost all topics in CIE A-Level Physics. You will use vector resolution constantly in upcoming topics, including forces on inclined planes, static equilibrium, projectile motion, and even analysing electromagnetic fields. Mastering this early sub-topic eliminates common confusion later, and builds a strong problem-solving framework for all mechanics questions, which make up a large portion of your exam marks. Next, you will learn about SI units and measurement uncertainty, before applying vector skills to forces and motion.

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

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