# Vectors and Scalars

> Some quantities only need a size. Others need a size and a direction. Here is how to tell them apart and add them.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/vectors-and-scalars/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The treasure map
Zara finds an old map in her grandfather's shed. It says: "Walk 10 steps. Dig."

She walks 10 steps from the gate and digs. Nothing. She tries 10 steps another way. Still nothing. After an hour of holes, she flips the map over. On the back, in tiny writing: "10 steps **north** of the gate."

She digs once more and hits a tin box.

The first instruction gave only a size. The second gave a size **and** a direction. In physics, that difference has a name.

**In short:** A scalar is a quantity with size only, like distance, speed, mass or time. A vector has size and direction, like displacement, velocity, acceleration or force, and is drawn as an arrow.

## Scalars and vectors

A **scalar** has **size only** (also called magnitude).
A **vector** has **size and direction**.

| Scalars (size only) | Vectors (size and direction) |
|---|---|
| Distance: 5 km | Displacement: 5 km east |
| Speed: 20 m/s | Velocity: 20 m/s north |
| Time: 10 s | Acceleration: 3 m/s² downwards |
| Mass: 60 kg | Force: 600 N downwards |
| Energy: 500 J | Weight: 600 N towards Earth |
| Temperature: 25 °C | Momentum (Year 11) |

### Key idea: A quick test
Ask yourself: "Does it make sense to say **which way**?" A temperature of 25 °C to the left makes no sense, so temperature is a scalar. A push of 10 N to the left makes sense, so force is a vector.

## Drawing vectors

We draw a vector as an **arrow**.
- The **length** of the arrow shows the size.
- The **arrowhead** shows the direction.

*Diagram: Two arrows: a short arrow labelled 3 newtons east and a longer arrow labelled 6 newtons east*
*Longer arrow means bigger size.*

## Adding vectors in a straight line

When vectors point along the same line, choose one direction as **positive**. The opposite direction is **negative**. Then just add.

- 5 m east + 3 m east = **8 m east**
- 5 m east + 3 m west = 5 + (−3) = **2 m east**

The answer you get when you add vectors is called the **resultant**.

### Worked example: The tug of war
Team Red pulls left with 450 N. Team Blue pulls right with 500 N.

Take right as positive: $500 + (-450) = +50$ N.

The resultant is 50 N to the right, so the rope moves towards Team Blue.

## Distance and displacement

- **Distance** = total length of the path. A scalar.
- **Displacement** = straight line from start to finish, with direction. A vector.

Zara walks 4 m east, then 3 m north. Her distance is $4 + 3 = 7$ m. Her displacement is the straight line from start to end. Using Pythagoras:

$$s = \sqrt{4^2 + 3^2} = \sqrt{25} = 5 \text{ m}$$

So she ends up 5 m from the start, in a north-east direction.

## Speed and velocity

- **Speed** = distance ÷ time (scalar)
- **Velocity** = displacement ÷ time (vector)

A car going round a roundabout at a steady 30 km/h has **constant speed** but **changing velocity**, because its direction keeps changing.

**Mind map**

```text
Vectors and scalars
  Scalar
    Size only
    Distance
    Speed
    Mass, time, energy
  Vector
    Size and direction
    Displacement
    Velocity
    Force, acceleration
  Adding
    Same line: use + and −
    Resultant
    Right angles: Pythagoras
```

### Year 11: Vectors at angles
For two vectors at right angles, use Pythagoras for the size and trigonometry for the direction.

Zara's walk: 4 m east then 3 m north.

$$\theta = \tan^{-1}\left(\frac{3}{4}\right) \approx 37^\circ \text{ north of east}$$

You can also **resolve** one vector into two parts at right angles. A vector of size $F$ at angle $\theta$ to the horizontal has:

$$F_x = F\cos\theta \qquad F_y = F\sin\theta$$

This is used everywhere: forces on ramps, projectiles, and anything that moves at an angle.

To **subtract** vectors, add the negative: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. Change in velocity is $\Delta\vec{v} = \vec{v} - \vec{u}$.

### Year 12: Projectile motion
A ball thrown at an angle is just two simple motions happening at once:

- **Horizontal**: constant velocity $u_x = u\cos\theta$ (ignoring air resistance)
- **Vertical**: constant acceleration $g = 9.8 \text{ m/s}^2$ downwards, starting with $u_y = u\sin\theta$

The two parts are **independent**. A bullet fired sideways and a bullet dropped from the same height hit the ground at the same time. Range on flat ground is greatest at $45^\circ$.

## Quick summary

- Scalars have size only. Vectors have size and direction.
- Distance, speed, mass, time, energy are scalars.
- Displacement, velocity, acceleration, force are vectors.
- Draw vectors as arrows. Length shows size, the head shows direction.
- In a straight line, pick a positive direction and add with signs. The answer is the **resultant**.
- At right angles, use Pythagoras for the size.

## Quiz

Try these questions from the chapter quiz.

1. Which of these is a vector?
   - A. Speed
   - B. Mass
   - C. Velocity
   - D. Energy
   - Answer: C. Velocity. Velocity includes a direction. The others are size only.

2. Which of these is a scalar?
   - A. Force
   - B. Displacement
   - C. Acceleration
   - D. Temperature
   - Answer: D. Temperature. Temperature has no direction, so it is a scalar.

3. A person walks 6 m east then 2 m west. What is the displacement?
   - A. 8 m east
   - B. 4 m east
   - C. 4 m west
   - D. 8 m west
   - Answer: B. 4 m east. Take east as positive: 6 + (−2) = +4, so 4 m east.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/vectors-and-scalars/#quiz