11–12Mathematics Extension 2 11–12 Syllabus

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Implementation from 2026

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Overview Rationale Aim Outcomes Content Assessment Glossary Teaching and learning support

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Year 12

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Examples

Further work with vectors

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME2-12-02
uses vectors to model lines and curves, solve problems, and prove results involving geometry and coordinate geometry

Further work with vectors

Vector equations of lines and curves

Define the direction vector of a straight line and identify that a straight line through two points $P$ and $Q$ has $\vec{PQ}$ as a possible direction vector in both two dimensions and three dimensions
Establish the relationship between the gradient, $m$ , of a straight line in two dimensions and its direction vector
Examine and use $r = a + λ b$ as the vector equation of a straight line in two dimensions and three dimensions to solve problems, where $r$ is the position vector of a point on the line, $a$ is the position vector of a particular point on the line, $b$ is a direction vector of the line and $λ$ is a scalar parameter
Identify $r = p + λ (q - p)$ as one possible vector equation for the straight line through points $P$ and $Q$ , where $p$ and $q$ are the position vectors of $P$ and $Q$ respectively, noting its equivalence with the form $r = λ q + (1 - λ) p$ , and establish the correspondence between the value of $λ$ and the position of the point specified by it along the line
Express a line in two dimensions given in gradient–intercept form ( $y = mx + c$ ) as a vector equation, and vice versa
Determine when two lines in vector form are parallel in two dimensions and three dimensions
Determine whether a given point lies on a line in vector form
Determine using vector methods whether three points are collinear in two dimensions and three dimensions
Determine the point of intersection of two non-parallel lines expressed as vector equations in two dimensions
Determine whether two distinct lines $r_{1} = a_{1} + λ_{1} b_{1}$ and $r_{2} = a_{2} + λ_{2} b_{2}$ in three dimensions intersect, and if so, find the unique value of either $λ_{1}$ or $λ_{2}$ corresponding to the point of intersection and determine its coordinates
Define skew lines in three dimensions and apply vector methods to determine whether two lines in three dimensions are skew
Recognise that a parametrically defined curve in two dimensions or three dimensions can be expressed as a vector equation: $r = (\begin{matrix} x (t) \\ y (t) \end{matrix})$ or $r = (\begin{matrix} x (t) \\ y (t) \\ z (t) \end{matrix})$ , where $t$ is a parameter
Examine and identify curves given as vector equations in two dimensions and three dimensions using graphing applications
Identify the vector equation of a curve given its graph
Recognise $r = r \cos (θ) i + r \sin (θ) j$ and $|r| = r$ as vector equations of a circle in two dimensions with radius $r$ centred at the origin
Recognise $r = (r \cos (θ) + x_{C}) i + (r \sin (θ) + y_{C}) j$ and $|r - c| = r$ as vector equations of a circle in two dimensions with radius $r$ and centre $C$ with position vector $c = (\begin{matrix} x_{C} \\ y_{C} \end{matrix})$
Recognise $|r - c| = r$ as the vector equation of a sphere in three dimensions with radius $r$ and centre $C$ with position vector $c$
Use the circle equations and sphere equations to solve problems
Determine the Cartesian equation of a curve in two dimensions given its vector equation and graph the curve, where the curve is within the scope of this syllabus

Vectors and geometry

Examine and use properties of the scalar (dot) product, including commutativity ( $a \cdot b = b \cdot a$ ), distributivity $(a \cdot (b + c) = a \cdot b + a \cdot c$ ) and scalar multiplication ( $k (a \cdot b) = (k a) \cdot b = a \cdot (k b)$ )
Establish and use the results $i \cdot i = j \cdot j = k \cdot k = 1$ and $i \cdot j = j \cdot k = k \cdot i = 0$
Prove and use the Cauchy–Schwarz inequality for vectors: $|a \cdot b| \leq |a| |b|$
Define the medians, altitudes, perpendicular bisectors and angle bisectors of a triangle and recognise these definitions in vector proofs
Solve problems and prove geometric results in two dimensions and three dimensions using vectors

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11–12Mathematics Extension 2 11–12 Syllabus

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