# Introduction to Vectors

> Quantities with size and direction. Vector notation, components, magnitude, adding, subtracting, scaling and unit vectors.

From HSC Maths Extension 1 on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-extension-1/introduction-to-vectors/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The pilot's real path
A pilot points her plane due east at 300 km/h. But a wind blows from the south at 40 km/h. The plane does not travel due east. It drifts a little north, and its true speed is a little more than 300 km/h.

To work out where the plane really goes, she needs numbers that carry a **direction** as well as a size. These are **vectors**, and adding them is how pilots, sailors and game designers predict real motion.

**In short:** A vector has a size (magnitude) and a direction, and is written as $\overrightarrow{AB}$, $\mathbf{a}$ or $x\mathbf{i} + y\mathbf{j}$. Vectors add and subtract component by component, and the magnitude of $x\mathbf{i} + y\mathbf{j}$ is $\sqrt{x^2 + y^2}$.

## Starting point: what you already know

- **Pythagoras** and **SOH CAH TOA**.
- Coordinates: the point $(3, 4)$ is 3 right and 4 up from the origin.
- **True bearings** and the difference between distance (how far) and displacement (how far and which way), from science.
- Gradient: lines with the same gradient are parallel.

## Scalars and vectors

A **scalar** has size only: mass 5 kg, time 10 s, temperature $20^\circ\text{C}$.

A **vector** has size **and** direction: a displacement of 5 km north, a velocity of 30 m/s east, a force of 10 N downward.

We draw a vector as an arrow. Its length shows the size, and the arrowhead shows the direction.

## Writing vectors

- As a **directed line segment** from $A$ to $B$: $\overrightarrow{AB}$.
- As a bold letter: $\mathbf{a}$. By hand, write $\underset{\sim}{a}$ with a squiggle under it.
- In **component form**: $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ or $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}$.

Here $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is one unit right and $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ is one unit up.

For points $A(x_1, y_1)$ and $B(x_2, y_2)$:
$$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}$$
"Where you finish minus where you start."

## Magnitude and direction

The **magnitude** (length) of $\mathbf{a} = x\mathbf{i} + y\mathbf{j}$ is

$$|\mathbf{a}| = \sqrt{x^2 + y^2}$$

and its direction angle $\theta$ from the positive x-axis satisfies $\tan\theta = \dfrac{y}{x}$. Check the quadrant from a sketch.

### Worked example: Magnitude and direction
$\mathbf{a} = 3\mathbf{i} - 4\mathbf{j}$. Then $|\mathbf{a}| = \sqrt{9 + 16} = 5$. It points right and down, at $\tan^{-1}\dfrac{4}{3} \approx 53^\circ$ below the positive x-axis.

### Calculator tip: Pol and Rec
`Pol(3, 4)` turns components into magnitude and direction: $r = 5$ and $\theta = 53.13^\circ$ from the positive $x$-axis (in degree mode). `Rec(10, 120)` does the reverse: $x = -5$, $y = 8.66$. They are usually `SHIFT +` and `SHIFT -`.

## Adding and subtracting

**Add** vectors head to tail: start the second where the first ends. In components, just add.

$$\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}$$

%%FIG:vecadd%%

**Subtract:** $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$, where $-\mathbf{b}$ has the same length as $\mathbf{b}$ but points the opposite way.

### Worked example: The pilot
Plane: $300\mathbf{i}$. Wind: $40\mathbf{j}$. True velocity: $300\mathbf{i} + 40\mathbf{j}$.

Speed $= \sqrt{300^2 + 40^2} \approx 302.7$ km/h. Direction: $\tan^{-1}\dfrac{40}{300} \approx 7.6^\circ$ north of east, a bearing of about $082^\circ$.

## Scalar multiples and unit vectors

$k\mathbf{a}$ stretches $\mathbf{a}$ by a factor $k$. If $k < 0$ it also flips the direction. Two vectors are **parallel** when one is a scalar multiple of the other.

A **unit vector** has length 1. The unit vector in the direction of $\mathbf{a}$ is

$$\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$$

### Worked example: A unit vector
$\mathbf{a} = 3\mathbf{i} - 4\mathbf{j}$ has length 5, so $\hat{\mathbf{a}} = \dfrac{3}{5}\mathbf{i} - \dfrac{4}{5}\mathbf{j}$.

**Mind map**

```text
Vectors
  Meaning
    Size and direction
    Scalars have size only
  Notation
    AB with an arrow
    Bold a
    Column or i, j form
  Size
    |a| = √(x² + y²)
    Direction tan θ = y/x
  Operations
    Add head to tail
    Subtract adds the opposite
    ka scales
  Special
    Unit vector a/|a|
    Parallel means a = kb
    Position vectors from O
```

### Year 11: Vectors from bearings
A hiker walks 6 km on a bearing of $030^\circ$. The angle from the positive x-axis (east) is $90^\circ - 30^\circ = 60^\circ$, so
$$\mathbf{d} = 6\cos 60^\circ\,\mathbf{i} + 6\sin 60^\circ\,\mathbf{j} = 3\mathbf{i} + 3\sqrt{3}\,\mathbf{j}$$
In general, a vector of length $r$ at angle $\theta$ to the x-axis is $r\cos\theta\,\mathbf{i} + r\sin\theta\,\mathbf{j}$.

### Year 12: Position vectors and midpoints
The **position vector** of a point $P$ is $\overrightarrow{OP}$, from the origin. If $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$, then
$$\overrightarrow{AB} = \mathbf{b} - \mathbf{a}, \qquad \text{midpoint } M: \ \overrightarrow{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$$
**Example.** Show $A(1, 2)$, $B(3, 5)$ and $C(7, 11)$ lie on one line. $\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j}$ and $\overrightarrow{AC} = 6\mathbf{i} + 9\mathbf{j} = 3\overrightarrow{AB}$. They are parallel and share point $A$, so the points are collinear.

**Dividing in a ratio.** If $P$ divides $AB$ in the ratio $m : n$, then $\overrightarrow{OP} = \mathbf{a} + \dfrac{m}{m + n}(\mathbf{b} - \mathbf{a}) = \dfrac{n\mathbf{a} + m\mathbf{b}}{m + n}$. For $AP : PB = 1 : 3$, $\overrightarrow{OP} = \frac{3\mathbf{a} + \mathbf{b}}{4}$.

**Parallelograms.** In $ABCD$, $\overrightarrow{AD} = \overrightarrow{BC}$, so $D = A + (C - B)$.

**Forces.** The **resultant** is the sum of the forces. A particle is in **equilibrium** (stays still) when the forces add to the zero vector, so a missing force is minus the sum of the others. Find a bearing from the components: $3\mathbf{i} + 4\mathbf{j}$ is $\tan^{-1}\frac{3}{4} \approx 37^\circ$ east of north, a bearing of $037^\circ$.

**A short proof.** In triangle $OAB$ with midpoints $M$ of $OA$ and $N$ of $OB$: $\overrightarrow{MN} = \frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a} = \frac{1}{2}\overrightarrow{AB}$, so $MN$ is parallel to $AB$ and half as long.

**Length of a sum.** $|\mathbf{p} + \mathbf{q}| \leq |\mathbf{p}| + |\mathbf{q}|$ (one side of a triangle is shorter than the other two together). They are equal only when $\mathbf{p}$ and $\mathbf{q}$ point the same way.

## Quick summary

- A vector has magnitude and direction. Write it as $\overrightarrow{AB}$, $\mathbf{a}$, a column, or $x\mathbf{i} + y\mathbf{j}$.
- $\overrightarrow{AB}$ = end point minus start point.
- $|x\mathbf{i} + y\mathbf{j}| = \sqrt{x^2 + y^2}$.
- Add and subtract component by component. $k\mathbf{a}$ is parallel to $\mathbf{a}$.
- Unit vector: $\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}$.

## Quiz

Try these questions from the chapter quiz.

1. Which is a vector?
   - A. a mass of 3 kg
   - B. a time of 5 s
   - C. a velocity of 20 m/s north
   - D. a speed of 20 m/s
   - Answer: C. a velocity of 20 m/s north. It has a size and a direction.

2. Find $|6\mathbf{i} + 8\mathbf{j}|$.
   - A. 14
   - B. 10
   - C. 100
   - D. 2
   - Answer: B. 10. $\sqrt{36 + 64} = 10$.

3. $A = (1, 2)$ and $B = (4, 6)$. Find $\overrightarrow{AB}$.
   - A. $3\mathbf{i} + 4\mathbf{j}$
   - B. $5\mathbf{i} + 8\mathbf{j}$
   - C. $-3\mathbf{i} - 4\mathbf{j}$
   - D. $4\mathbf{i} + 6\mathbf{j}$
   - Answer: A. $3\mathbf{i} + 4\mathbf{j}$. End minus start: $(4 - 1, 6 - 2)$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-extension-1/introduction-to-vectors/#quiz