VCE Specialist Mathematics (Year 11) · Algebra and Structure · 20m read

Vectors — Introduction and Operations

Ever wondered how video game characters move smoothly across the screen, or how engineers design bridges that won't collapse? That's vectors at work. Vectors are quantities with both size and direction (unlike regular numbers), and you'll learn to represent them as arrows on a graph and as coordinate lists. Understanding vector operations like addition and finding magnitude unlocks how movement, forces, and changes work in physics, engineering, and computer graphics — essential skills for VCE Maths and beyond.

Learning objective: Represent vectors geometrically and algebraically; perform vector addition, subtraction and scalar multiplication; find the magnitude and direction of a vector.

By the end of this lesson you can:

I represent vectors in component form $\vec{v} = x\hat{i} + y\hat{j}$ and as column vectors.
I perform addition, subtraction and scalar multiplication geometrically and algebraically.
I find the magnitude $|\vec{v}| = \sqrt{x^2 + y^2}$ and direction $\theta = \tan^{-1}(y/x)$.
I determine whether two vectors are parallel or collinear.

Vectors are directions with distance. "Walk 5 km north-east" is a vector — it has size and direction. Adding vectors is following one set of directions then another.

Vectors — quantities with magnitude and direction

A vector is a quantity possessing both magnitude and direction. Contrast with a scalar, which has only magnitude.

| Scalars | Vectors |
|---------|---------|
| Distance $d$ | Displacement $\vec{d}$ |
| Speed $|v|$ | Velocity $\vec{v}$ |
| Mass $m$ | Force $\vec{F}$ |
| Temperature $T$ | Electric field $\vec{E}$ |

### Notation

Vectors are written with an arrow ($\vec{v}$), bold face (v) or as column vectors:

$$\vec{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix} = v_x \hat{i} + v_y \hat{j}$$

where $\hat{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\hat{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ are unit vectors along the $x$ and $y$ axes.

### Position vectors

The position vector of a point $A(a_1, a_2)$ from the origin is $\vec{OA} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$.

The vector from $A$ to $B$ is:
$$\vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} b_1 - a_1 \\ b_2 - a_2 \end{pmatrix}$$

### Equal vectors

Two vectors are equal if they have the same magnitude AND direction — they don't need to start at the same point.

Vector operations — addition, subtraction, scalar multiplication

### Vector addition

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

Geometrically: triangle law (place tip of $\vec{a}$ at tail of $\vec{b}$, result goes from original tail to final tip) or parallelogram law.

### Vector subtraction

$$\vec{a} - \vec{b} = \vec{a} + (-\vec{b}) = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}$$

The zero vector: $\vec{0} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$. Note $\vec{a} - \vec{a} = \vec{0}$.

### Scalar multiplication

For scalar $\lambda \in \mathbb{R}$:
$$\lambda \vec{a} = \begin{pmatrix} \lambda a_1 \\ \lambda a_2 \end{pmatrix}$$

If $\lambda > 0$: same direction, magnitude scaled by $\lambda$.
If $\lambda < 0$: opposite direction, magnitude scaled by $|\lambda|$.
If $\lambda = 0$: zero vector.

### Magnitude (length)

$$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$$

In 3D: $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

### Unit vector

$$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$

This has magnitude 1 and points in the same direction as $\vec{v}$.

### Parallel vectors

$\vec{a}$ and $\vec{b}$ are parallel if $\vec{a} = \lambda \vec{b}$ for some scalar $\lambda \neq 0$.

Collinear points: A, B, C are collinear if $\vec{AB} = \lambda \vec{AC}$ for some scalar.

Vector Formula Sheet

Addition: \vec{a} + \vec{b} = \begin{pmatrix} a_1+b_1 \\ a_2+b_2 \end{pmatrix} (Component-wise)
Magnitude: |\vec{v}| = \sqrt{v_x^2 + v_y^2} (Pythagoras in component form)
Unit vector: \hat{v} = \dfrac{\vec{v}}{|\vec{v}|} (Magnitude = 1)
Direction angle: \theta = \tan^{-1}\!\left(\dfrac{v_y}{v_x}\right) (From positive $x$-axis; use quadrant to adjust)
Vector $A \to B$: \vec{AB} = \vec{OB} - \vec{OA} ("Tip minus tail" of position vectors)
Parallel test: \vec{a} = \lambda \vec{b} (For some real $\lambda$; same or opposite direction)

Vector problem protocol

Write in component form: Convert position vectors to $\begin{pmatrix}x\\y\end{pmatrix}$ before computing.
AB = OB − OA: "Tip minus tail": position of B minus position of A.
Direction angle: Use $\tan^{-1}(y/x)$ then adjust for quadrant (add/subtract 180° if in Q2/Q3).
Parallelism: If $\vec{a} = \lambda \vec{b}$, vectors are parallel — ratio of components must be equal.
Unit vector: Divide by magnitude: $\hat{v} = \vec{v}/|\vec{v}|$.

Worked example (Easy)

Let $\vec{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$. Find: (a) $\vec{a} + \vec{b}$, (b) $2\vec{a} - \vec{b}$, (c) $|\vec{a}|$.

(a) Addition. \vec{a} + \vec{b} = \begin{pmatrix} 3-1 \\ -4+2 \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}
(b) $2\vec{a} - \vec{b}$. 2\vec{a} - \vec{b} = \begin{pmatrix} 6 \\ -8 \end{pmatrix} - \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 7 \\ -10 \end{pmatrix}
(c) Magnitude of $\vec{a}$. |\vec{a}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Answer: (a) $\begin{pmatrix} 2 \\ -2 \end{pmatrix}$; (b) $\begin{pmatrix} 7 \\ -10 \end{pmatrix}$; (c) $|\vec{a}| = 5$.

Worked example (Medium)

A is at $(2, -1)$ and B is at $(6, 5)$. Find $\vec{AB}$, $|\vec{AB}|$, and the unit vector in the direction of $\vec{AB}$.

Find $\vec{AB}$. $\vec{AB} = \vec{OB} - \vec{OA}$: \vec{AB} = \begin{pmatrix} 6-2 \\ 5-(-1) \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}
Find magnitude. |\vec{AB}| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
Unit vector. \hat{u} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{1}{2\sqrt{13}}\begin{pmatrix} 4 \\ 6 \end{pmatrix} = \begin{pmatrix} \frac{2}{\sqrt{13}} \\ \frac{3}{\sqrt{13}} \end{pmatrix}

Answer: $\vec{AB} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$, $|\vec{AB}| = 2\sqrt{13}$, $\hat{u} = \frac{1}{\sqrt{13}}\begin{pmatrix} 2 \\ 3 \end{pmatrix}$.

Worked example (Hard)

Show that points $A(1, 2)$, $B(4, 6)$ and $C(7, 10)$ are collinear.

Find $\vec{AB}$ and $\vec{AC}$. \vec{AB} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \qquad \vec{AC} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}
Check for parallelism. Is $\vec{AC} = \lambda \vec{AB}$ for some scalar $\lambda$? \begin{pmatrix} 6 \\ 8 \end{pmatrix} = 2 \begin{pmatrix} 3 \\ 4 \end{pmatrix} \checkmark
Conclude. $\vec{AC} = 2\vec{AB}$, so $\vec{AB}$ and $\vec{AC}$ are parallel. Since they share the common point A, A, B and C are collinear.

Answer: $\vec{AC} = 2\vec{AB}$, so A, B, C are collinear (B is the midpoint of AC).

Common mistakes

$\vec{AB} = \vec{OA} - \vec{OB}$ (wrong order) A common confusion. $\vec{AB}$ points FROM A TO B, so it equals $\vec{OB} - \vec{OA}$ ("where you end up minus where you start"). Fix: Mnemonic: "Tip minus tail." The tip is B (where the arrow points), the tail is A. So $\vec{AB} = \vec{B} - \vec{A}$.
Not adjusting angle for quadrant in direction calculations $\tan^{-1}(y/x)$ gives an angle in $(-90°, 90°)$. If the vector is in Q2 or Q3, the calculator gives the wrong quadrant. Fix: Draw the vector, identify the quadrant. Q1: angle is fine. Q2: add 180°. Q3: add 180°. Q4: angle is fine (negative).

Vectors in physics — why direction matters as much as magnitude

Vectors are essential in physics because most physical quantities have both magnitude and direction. Force, velocity, acceleration, displacement and momentum are all vectors. Scalars (temperature, mass, speed, energy) have magnitude only.

The power of vector addition becomes clear with forces: if two people push a box with forces $\vec{F_1}$ and $\vec{F_2}$ at different angles, the net force is the vector sum $\vec{F_1} + \vec{F_2}$ (found using the parallelogram rule or component addition). This explains why pulling a sled at an angle is less effective than pulling horizontally — only the horizontal component of the force contributes to forward motion.

In Specialist Maths, you work with 2D vectors using column notation. In Units 3 & 4, you'll extend to 3D vectors and learn the dot product ($\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$) and cross product ($\vec{a} \times \vec{b}$), which are foundational tools in physics, engineering and computer graphics. The dot product tells you how much two vectors are aligned; the cross product gives a vector perpendicular to both — used to calculate torque, angular momentum and surface normals in 3D rendering.

Practice questions

If $\vec{v} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}$, then $|\vec{v}|$ equals:
1. $13$
2. $17$
3. $7$
4. $\sqrt{119}$
(1 marks)
Find a unit vector in the direction of $\vec{u} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
(2 marks)
Given $\vec{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -6 \\ 3 \end{pmatrix}$, show that $\vec{a}$ and $\vec{b}$ are parallel and state their relationship.
(3 marks)

Summary

$\vec{AB} = \vec{OB} - \vec{OA}$ — "tip minus tail."
$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$ — Pythagorean magnitude.
$\hat{v} = \vec{v}/|\vec{v}|$ — unit vector (magnitude 1, same direction).
Parallel: $\vec{a} = \lambda \vec{b}$. Collinear: parallel + common point.
Operations are component-wise: add, subtract, scalar multiply each component separately.