HSC Maths Extension 1 vectors: deep dive (2026 guide)

What this guide covers

Vectors were introduced into HSC Mathematics Extension 1 in the 2017 syllabus and are now examined every year. This guide covers:

• Vector arithmetic, magnitude and unit vectors.
• The scalar (dot) product, the angle between vectors, orthogonality.
• Scalar and vector projection.
• Parametric vector equations of lines and intersection problems.
• Geometric proofs using vectors.

Vector arithmetic

A two-dimensional vector $\mathbf{a} = (a_1, a_2)$ can be written in column form, component form, or unit-vector form ($a_1 \mathbf{i} + a_2 \mathbf{j}$ with $\mathbf{i} = (1, 0)$, $\mathbf{j} = (0, 1)$).

Addition, subtraction and scalar multiplication

Geometrically: $\mathbf{a} + \mathbf{b}$ is the parallelogram-rule diagonal; $\lambda \mathbf{a}$ scales magnitude by $|\lambda|$ and may reverse direction if $\lambda < 0$.

Magnitude and unit vector

$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}$, and the unit vector $\hat{\mathbf{a}} = \frac{1}{|\mathbf{a}|} \mathbf{a}$.

Position vector versus displacement vector

If $A = (a_1, a_2)$ is a point, $\mathbf{OA} = \mathbf{a}$ is its position vector (origin to $A$). The displacement from $A$ to $B$ is $\mathbf{AB} = \mathbf{b} - \mathbf{a}$.

The scalar (dot) product

Angle between two vectors

For non-zero vectors, $\theta \in [0, \pi]$.

Orthogonality test

$\mathbf{a} \perp \mathbf{b}$ iff $\mathbf{a} \cdot \mathbf{b} = 0$.

This is the most common HSC use of the scalar product.

Properties

The dot product is commutative, bilinear, and $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$.

This is the vector form of the cosine rule.

Projections

Scalar projection

Signed length: positive if $\mathbf{a}$ has a component in the direction of $\mathbf{b}$, negative if opposite.

Vector projection

This is the scalar projection times the unit vector $\hat{\mathbf{b}}$.

Decomposition

Any vector $\mathbf{a}$ can be split into a component parallel to $\mathbf{b}$ (the vector projection) and a component perpendicular to $\mathbf{b}$.

By construction $\mathbf{a}_{\perp} \cdot \mathbf{b} = 0$.

Parametric vector equations of lines

Point-direction form

A line through $A$ (position vector $\mathbf{a}$) with direction $\mathbf{d}$:

Line through two points

Direction $\mathbf{AB} = \mathbf{b} - \mathbf{a}$, so $\mathbf{r} = \mathbf{a} + \lambda (\mathbf{b} - \mathbf{a}) = (1 - \lambda) \mathbf{a} + \lambda \mathbf{b}$.

At $\lambda = 0$: at $A$. At $\lambda = 1$: at $B$.

Convert to Cartesian

If $\mathbf{d} = (d_1, d_2)$ with $d_1 \neq 0$, $\lambda = \frac{x - a_1}{d_1}$, so $y = a_2 + \frac{x - a_1}{d_1} d_2$. Slope: $\frac{d_2}{d_1}$.

Intersection of two lines

Set parametric equations equal:

Solve for $\lambda$ and $\mu$. If a unique solution, lines meet at a point. If the system is inconsistent, lines are parallel and disjoint.

Geometric proofs using vectors

The recipe

1. Assign position vectors to labelled points.
2. Express the relevant displacement vectors algebraically.
3. Compute the geometric relation (equality, scalar multiple, dot product zero, etc.).
4. Conclude the geometric statement.

Worked example: midpoint connector

In triangle $ABC$, let $P$ and $Q$ be midpoints of $AB$ and $AC$. Show $PQ$ is parallel to $BC$ and half its length.

Position vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$.

$P = \frac{\mathbf{a} + \mathbf{b}}{2}$, $Q = \frac{\mathbf{a} + \mathbf{c}}{2}$.

$\mathbf{PQ} = Q - P = \frac{\mathbf{c} - \mathbf{b}}{2} = \frac{1}{2} \mathbf{BC}$.

So $\mathbf{PQ}$ is parallel to $\mathbf{BC}$ (same direction) and half the magnitude.

Worked example: parallelogram diagonals

$ABCD$ is a parallelogram, so $\mathbf{AB} = \mathbf{DC}$, i.e., $\mathbf{a} + \mathbf{c} = \mathbf{b} + \mathbf{d}$.

Midpoint of $AC$ is $\frac{\mathbf{a} + \mathbf{c}}{2}$; midpoint of $BD$ is $\frac{\mathbf{b} + \mathbf{d}}{2}$.

These are equal, so the diagonals bisect each other.

Worked example: perpendicular diagonals of a rhombus

If a parallelogram has perpendicular diagonals, the sides are equal in length (rhombus).

Let $\mathbf{AB} = \mathbf{u}$ and $\mathbf{AD} = \mathbf{v}$. Diagonals: $\mathbf{AC} = \mathbf{u} + \mathbf{v}$, $\mathbf{BD} = \mathbf{v} - \mathbf{u}$.

Perpendicular: $(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{v} - \mathbf{u}) = |\mathbf{v}|^2 - |\mathbf{u}|^2 = 0$, so $|\mathbf{u}| = |\mathbf{v}|$.

Common exam questions

Question type A: angle and orthogonality

Find the angle between two given vectors. Find $k$ so that two vectors are perpendicular. These are 2-3 mark items.

Question type B: projections

Find the scalar or vector projection. Decompose a vector into parallel and perpendicular components. These are 3-4 mark items.

Question type C: parametric vector equation of a line

Write the vector equation, convert to Cartesian, or find an intersection. 3-4 marks.

Question type D: geometric proof using vectors

Show two lines are parallel, two are perpendicular, or a figure is a parallelogram. 4-5 marks.

Question type E: applications

Resolve a force into components along a given direction. Find collision conditions for two moving particles. 4-5 marks.

Common traps

Trap: $\mathbf{PQ} = Q - P$, not IMATH_80

The displacement from $P$ to $Q$ is the position vector of $Q$ minus the position vector of $P$. Reversing gives the opposite vector.

Trap: Variance vs. standard deviation in dot-product squaring

$|\mathbf{a}|^2 = \mathbf{a} \cdot \mathbf{a}$, so $|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + 2 \mathbf{a} \cdot \mathbf{b} + |\mathbf{b}|^2$. This expansion is essential for the vector cosine rule.

Trap: $|\mathbf{b}|$ vs IMATH_88

Scalar projection has $|\mathbf{b}|$ in the denominator. Vector projection has $|\mathbf{b}|^2$.

Trap: Parallel parameters

Two lines with the same direction vector are parallel. They may or may not coincide. Check if a point on one is also on the other.

Trap: Different parameters for different lines

When intersecting two lines, use $\lambda$ for one and $\mu$ for another. Reusing the same parameter creates confusion.

Exam strategy

Vector questions typically run 2-5 marks each. Allocate $3-6$ minutes per question. Steps:

1. Read carefully: scalar or vector projection? parallel or equal?
2. Write down position or displacement vectors.
3. Compute the relevant quantity algebraically.
4. State the answer with units (if appropriate) and direction (positive, negative, parallel, perpendicular).

For geometric proofs, work the algebra one line at a time and cite each step. Markers reward clear structure.

Linked dot points

For more detail, see the dot point pages at vector arithmetic and magnitude, scalar product, vector projection, parametric vector equations of lines, and geometric proofs with vectors.

For NESA past papers and marking guidelines, refer to educationstandards.nsw.edu.au.