11–12Mathematics Extension 1 11–12 Syllabus

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Year 11

Year 12

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Introduction to 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
ME1-12-02
operates with 2D and 3D vectors and uses 2D vectors to solve problems involving motion in two dimensions

Introduction to vectors

Vector representation and notation

Define a vector as a quantity having both magnitude and direction
Associate vectors with directed line segments and recognise that a vector may have many directed line segments associated with it
Identify and use notation for vectors in both two dimensions and three dimensions, including $a$ , $\underset{\sim}{a}$ and $\vec{AB}$ , where $\vec{AB}$ is the vector with magnitude and direction those of the directed line segment from $A$ to $B$
Use notations $|a|$ , $|\underset{\sim}{a}|$ and $|\vec{AB}|$ to represent the magnitude of a vector
Describe a position vector as a vector with its tail at the origin
Represent vectors graphically with and without graphing applications
Recognise and use the fact that two vectors are equal if they have the same magnitude and direction to solve problems

Introduction to 2D and 3D vectors

Use Cartesian coordinates to represent points in 2-dimensional (2D) and 3-dimensional (3D) space with and without graphing applications
Use the midpoint and distance formulas in two dimensions and three dimensions
Identify that, in three dimensions, all points on the $xy$ -plane have a $z$ -coordinate of 0, and deduce similar properties for points on the $xz$ and $yz$ -planes and thus the equations of the three coordinate planes
Define the zero vector $0$ , written as $\underset{\sim}{0}$ , as the vector with zero magnitude and no direction
Define unit vectors as vectors of magnitude 1
Recognise and use $\hat{a}$ and $\hat{\underset{\sim}{a}}$ as the notation for the unit vector in the direction of $a$
Define the standard perpendicular unit vectors $i$ and $j$ in two dimensions and $i$ , $j$ and $k$ in three dimensions
Express 2D vectors in component form, $x i + y j$ ; as an ordered pair, $(x, y)$ ; and in column vector notation $(\begin{matrix} x \\ y \end{matrix})$
Recognise $\vec{AB} = (\begin{matrix} u - x \\ v - y \end{matrix})$ as the vector associated with the directed line segment from the point $A (x, y)$ to $B (u, v)$ in two dimensions
Express 3D vectors in component form, $x i + y j + z k$ ; as an ordered triple, $(x, y, z)$ ; and in column vector notation $(\begin{matrix} x \\ y \\ z \end{matrix})$
Recognise $\vec{AB} = (\begin{matrix} u - x \\ v - y \\ w - z \end{matrix})$ as the vector associated with the directed line segment from $A (x, y, z)$ to $B (u, v, w)$ in three dimensions

Operating with vectors

Define a scalar as a real number that is used to multiply a vector
Represent geometrically a scalar multiple of a vector in two dimensions and three dimensions with and without graphing applications
Perform multiplication of a vector by a scalar algebraically in component form
Establish and identify $a = k b$ , for a non-zero scalar $k$ , as a condition for two non-zero vectors $a$ and $b$ to be parallel to each other and determine with justification if two vectors are parallel to one another
Identify $(\begin{matrix} b \\ - a \end{matrix})$ and $(\begin{matrix} - b \\ a \end{matrix})$ as vectors perpendicular to $(\begin{matrix} a \\ b \end{matrix})$ and with equal magnitude
Perform addition and subtraction of vectors algebraically in component form, and verify, with and without graphing applications, that geometrically these are obtained using the triangle law or the parallelogram law
Establish and calculate the magnitude of a vector using $|x i + y j| = \sqrt{x^{2} + y^{2}}$ for 2D vectors and $|x i + y j + z k| = \sqrt{x^{2} + y^{2} + z^{2}}$ for 3D vectors
Use the magnitude of a vector to find the unit vector $\hat{a} = \frac{1}{|a|} a$ in two dimensions and three dimensions

Further operations with vectors

Define $a \cdot b = x_{1} x_{2} + y_{1} y_{2}$ as the scalar (dot) product of vectors $a = x_{1} i + y_{1} j$ and $b = x_{2} i + y_{2} j$ and use the scalar product to solve problems
Define $a \cdot b = x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}$ as the scalar product of vectors $a = x_{1} i + y_{1} j + z_{1} k$ and $b = x_{2} i + y_{2} j + z_{2} k$ and use the scalar product to solve problems
Use $a \cdot b = |a| |b| c o s θ$ as a geometric expression of the scalar product of non-zero vectors $a$ and $b$ in two dimensions and three dimensions, where $θ$ is the angle between the vectors and $0 ° \leq θ \leq 180 °$ .
Verify the equivalence of $a \cdot b = |a| |b| c o s θ$ with the algebraic definition of the scalar product, $a \cdot b = x_{1} x_{2} + y_{1} y_{2}$ for two dimensions and $a \cdot b = x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}$ for three dimensions
Derive and use the property $a \cdot a = {|a|}^{2}$ to establish the scalar product definition of the magnitude of a vector ( $|a| = \sqrt{a \cdot a}$ ) in two dimensions and three dimensions
Calculate the angle between two non-zero vectors $a$ and $b$ , in both two dimensions and three dimensions, using the scalar product by deriving and applying the relationship $\cos θ = \frac{a \cdot b}{|a| |b|} = (\hat{a} \cdot \hat{b})$
Establish $a \cdot b = 0$ as a condition for two non-zero vectors $a$ and $b$ to be perpendicular to each other and use it to determine if two vectors are perpendicular
Establish $a \cdot b = \pm |a| |b|$ as another way to determine if two non-zero vectors $a$ and $b$ are parallel
Define the projection of a vector $a$ onto a vector $b$ , denoted by ${proj}_{b} a$ , to be the vector component of $a$ in the direction of vector $b$
Examine the proof of the formula ${p r o j}_{b} a = (\frac{a \cdot b}{{|b|}^{2}}) b = (a \cdot \hat{b}) \hat{b}$ and use the formula to solve problems
Determine that the component of a vector $a$ perpendicular to another vector $b$ is $a - {proj}_{b} a$

Motion in vector form in two dimensions

Describe the position of an object at a point in 2D space using a vector
Describe the changing positions of an object by expressing its vector as a function of time using $r (t) = x (t) i + y (t) j$ , or $r = x i + y j$ where $x$ and $y$ are functions of time $t$
Recognise that $x (t)$ and $y (t)$ form a pair of parametric equations for the path of the object
Find the Cartesian equation of the path of an object, where the path is a straight line, parabola or circle
Express the change in an object’s position between two points as a displacement vector and recognise the magnitude of the displacement vector as the distance between the two points
Solve motion problems involving constant velocity using vectors
Solve relative velocity problems involving constant crosswind/cross-current using vector diagrams, and describe the direction of a vector where required
Find the velocity vector $v = \dot{x} i + \dot{y} j$ and acceleration vector $a = \ddot{x} i + \ddot{y} j$ of an object using differential calculus
Find the position vector and the velocity vector of an object using integral calculus given its acceleration vector
Solve motion problems involving non-constant velocity using vectors

Projectile motion

Recognise that the gravitational force on a mass may be regarded as a constant acting in a downwards direction when the motion of the object is restricted to a small region near the Earth’s surface
Model and analyse a projectile’s path where the projectile is a point and air resistance is negligible, subject to only acceleration due to gravity, assuming that the projectile is moving close to the Earth’s surface
Represent the motion of a projectile using vectors
Recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors
Derive and use the equations of motion of a projectile in vector form by splitting 2D motion into horizontal and vertical components to solve problems on projectiles
Find the Cartesian equation of the path of a projectile using parametric equations for the horizontal and vertical components of the displacement vector
Determine features of the flight of a projectile, including time of flight, maximum height, range, instantaneous velocity and impact velocity
Solve problems relating to the path of a projectile in which the initial velocity and/or angle of projection may be unknown, in a variety of contexts

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11–12Mathematics Extension 1 11–12 Syllabus

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