HSC Mathematics Advanced vectors (2024 syllabus, HSC 2027+ guide)

Why vectors are in HSC Mathematics Advanced now

Vectors were previously only in Extension 1. The new 2024 syllabus (first HSC sat in 2027) moves them into Advanced as a core topic. If you are starting Year 11 in 2026 or later, vectors will be a major part of your Mathematics Advanced course and exam.

This is one of the most significant additions in the new syllabus. Around 10-15% of the Advanced exam paper will assess vector content. Getting comfortable with vector arithmetic, geometry, and applications is essential.

What a vector is

A vector has two components: a magnitude (length) and a direction.

In two dimensions, a vector can be written as a column vector:

or in component form using unit vectors:

where $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ are the standard unit vectors.

A vector is just a directed arrow on a page. Two arrows of the same length pointing the same way represent the same vector regardless of where they are drawn.

Vector arithmetic

Addition

Vectors add component-wise:

Geometrically: place the tail of one vector at the head of the other; the sum vector goes from the start to the end. This is the tip-to-tail rule. Equivalently, complete the parallelogram with the two vectors as adjacent sides: the diagonal is their sum.

Scalar multiplication

Multiplying a vector by a scalar $k$ scales it:

If $k > 0$, the direction is unchanged; if $k < 0$, it reverses; if $|k| > 1$, the vector is longer; if $|k| < 1$, shorter.

Magnitude

The magnitude (length) of a vector $\mathbf{u} = (a, b)$ is:

This is just Pythagoras' theorem.

Unit vectors

A unit vector has magnitude 1. To find the unit vector in the direction of $\mathbf{u}$:

The scalar (dot) product

The scalar product of two vectors is a number (not a vector). Two equivalent definitions:

Component form. $\mathbf{u} \cdot \mathbf{v} = ac + bd$ where $\mathbf{u} = (a, b)$ and $\mathbf{v} = (c, d)$.

Geometric form. $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$ where $\theta$ is the angle between the vectors.

These two definitions agree. Setting them equal lets you solve for the angle:

Key properties

• Commutative. $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$.
• Distributive. $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$.
• Self-product. $\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$.
• Perpendicularity. Two vectors are perpendicular if and only if $\mathbf{u} \cdot \mathbf{v} = 0$.

A worked example

Find the angle between $\mathbf{u} = (3, 4)$ and $\mathbf{v} = (1, 2)$.

Step 1: $\mathbf{u} \cdot \mathbf{v} = 3 \cdot 1 + 4 \cdot 2 = 3 + 8 = 11$.

Step 2: $|\mathbf{u}| = \sqrt{9 + 16} = 5$. $|\mathbf{v}| = \sqrt{1 + 4} = \sqrt{5}$.

Step 3: $\cos\theta = \frac{11}{5\sqrt{5}} = \frac{11}{5\sqrt{5}} \approx 0.9839$.

Step 4: $\theta = \cos^{-1}(0.9839) \approx 10.3°$.

Vector projections

The scalar projection of $\mathbf{u}$ onto $\mathbf{v}$ is the component of $\mathbf{u}$ along the direction of $\mathbf{v}$:

This is a number, possibly negative if $\mathbf{u}$ has a component opposite to $\mathbf{v}$.

The vector projection is the actual projected vector:

When projection appears

Projection problems test whether you can decompose a vector into components along and perpendicular to a given direction. Common contexts: physics-style problems (force along a slope), geometry (foot of a perpendicular from a point to a line), and proofs about angles.

Applications

The equation of a line in vector form

A line through the point $\mathbf{a}$ with direction vector $\mathbf{d}$ has parametric vector equation:

where $t$ is a real parameter. As $t$ varies, $\mathbf{r}$ traces out every point on the line.

Geometry with vectors

Vector methods often produce shorter proofs than coordinate geometry. Common patterns:

• Showing a quadrilateral is a parallelogram. Show that opposite sides are equal as vectors.
• Showing two lines are perpendicular. Show their direction vectors satisfy $\mathbf{d}_1 \cdot \mathbf{d}_2 = 0$.
• Finding the midpoint. The position vector of the midpoint of segment from $\mathbf{a}$ to $\mathbf{b}$ is $\frac{1}{2}(\mathbf{a} + \mathbf{b})$.

A worked geometry example

Show that the diagonals of a rhombus are perpendicular.

Let the rhombus have vertices $A$, $B$, $C$, $D$ with position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$, where all sides have equal length.

Diagonal $AC$ has direction $\mathbf{c} - \mathbf{a}$.
Diagonal $BD$ has direction $\mathbf{d} - \mathbf{b}$.

In a rhombus, sides are equal: $|\mathbf{b} - \mathbf{a}| = |\mathbf{d} - \mathbf{a}|$. By the parallelogram law (and using vector addition), one can show:

which means the diagonals are perpendicular. The vector proof is much cleaner than a coordinate proof.

Common vector traps

Confusing scalar and vector results

$\mathbf{u} \cdot \mathbf{v}$ is a number. $k\mathbf{u}$ is a vector. Keep track.

Sign of the scalar product

A negative scalar product means the angle is obtuse (between 90° and 180°). A positive scalar product means acute. Zero means perpendicular.

Magnitude vs vector

When asked for "the projection", check whether the question wants the scalar (a number) or the vector (a vector quantity). They differ by a factor of $\mathbf{v}/|\mathbf{v}|$.

Unit vector calculation

Forgetting to divide by the magnitude is a common slip. Always check that your candidate unit vector has length 1.

Direction vectors of lines vs position vectors of points

A line has a position vector (where it starts) and a direction vector (which way it goes). Mixing them is a 2-mark loss.

How vectors are examined in the new syllabus

In Section II of the HSC Mathematics Advanced paper (2024 syllabus):

• Short questions (2-3 marks). Find the magnitude of a vector. Compute a scalar product. Test perpendicularity.
• Medium questions (5-7 marks). Find the angle between two vectors. Compute a vector projection. Find the equation of a line in vector form.
• Long extended-response questions (8-10 marks). Multi-part geometric problems using vector methods. Often combine vectors with other Advanced topics (e.g. calculus on a parametrised curve).

Practice strategy

For HSC Mathematics Advanced vectors (2024 syllabus students):

• Term 2 of Year 12. Drill component arithmetic until it's automatic.
• Term 3. Move to scalar product, projections, geometric applications.
• Term 4. Past-paper-style practice. Most schools have built up a bank of vector questions in the new syllabus format.

Build the habit of drawing the vectors when you start a problem. Diagrams catch sign errors and direction mistakes that pure algebra hides.