HSC Maths Vectors: Complete Guide for Extension 1

Last updated: 24 July 2026

Vectors is the biggest topic in Extension 1 — roughly 15-20% of the paper. It's also the one students find hardest because it asks you to think geometrically and algebraically at the same time.

Once the notation clicks though, vectors are surprisingly mechanical. Same playbook every time.

What a vector actually is

A vector has direction and magnitude (length). That's it. An arrow from A to B. If you've ever said "go 5 km north," you've used a vector. The 5 is the magnitude, the north is the direction.

In writing, vectors get a little arrow: $\\vec{AB}$ means from A to B. In component form, you'll see $(3, -2)$ or $3\\mathbf{i} - 2\\mathbf{j}$.

Unit vectors i and j

$\\mathbf{i}$ is the unit vector in the x-direction (1, 0). $\\mathbf{j}$ is the unit vector in the y-direction (0, 1). Any 2D vector can be written as a combination of these. So $(3, -2)$ = $3\\mathbf{i} - 2\\mathbf{j}$.

Magnitude (length)

The magnitude uses Pythagoras — same idea as finding the distance between two points.

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

A vector of length 1 is a unit vector. To turn any vector into a unit vector in the same direction, divide by its magnitude.

Dot product — the most useful tool

The dot product of two vectors gives you a single number (a scalar). Formula:

$\\mathbf{a} \\cdot \\mathbf{b} = x_1x_2 + y_1y_2 = |\\mathbf{a}||\\mathbf{b}|\\cos\\theta$

Solve for $\\cos\\theta$ and you've got the angle. Whenever NESA asks "find the angle," reach for the dot product.

Perpendicular vectors: dot product = 0. This shows up constantly — in proofs, in geometry, in projectile motion.

Parallel vectors: one is a scalar multiple of the other. $\\mathbf{a} = k\\mathbf{b}$.

Projections

A projection is "how much of vector a points in the direction of vector b." Think of a shadow falling on a line.

$\\text{proj}_{\\mathbf{b}}\\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\mathbf{b} \\cdot \\mathbf{b}}\\mathbf{b}$

The scalar projection (just the length, without direction) is $\\frac{\\mathbf{a} \\cdot \\mathbf{b}}{|\\mathbf{b}|}$.

Vector proofs in geometry

These are the "hard" vector questions, but they follow a pattern:

1. Express everything in terms of two base vectors
2. Use dot product to test perpendicularity
3. Use magnitude to test equal lengths

A classic one: "Prove the diagonals of a rhombus are perpendicular." You set up two adjacent sides as vectors, express the diagonals in terms of them, and dot them. Zero pops out.

How to practise

Vectors rewards doing mixed questions — one dot product, one projection, one proof. You need to be able to reach for whichever tool fits, and that only comes from seeing enough variations.

Practice HSC vectors with instant marking and worked solutions on PracticePapers.io, free during beta.

The four formulas that do everything

• $\\mathbf{a} \\cdot \\mathbf{b} = x_1x_2 + y_1y_2 = |\\mathbf{a}||\\mathbf{b}|\\cos\\theta$
• Perpendicular: $\\mathbf{a} \\cdot \\mathbf{b} = 0$
• Parallel: $\\mathbf{a} = k\\mathbf{b}$
• Projection: $\\text{proj}_{\\mathbf{b}}\\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\mathbf{b} \\cdot \\mathbf{b}}\\mathbf{b}$

That's it. Four formulas. Every vector question in the HSC uses one of them.