Further Work with Vectors

vectors year12 ext2

Further work with vectors | NSW Curriculum Website

• ME2-12-02 uses vectors to model lines and curves, solve problems, and prove results involving geometry and coordinate geometry

📖 Prior Knowledge

Content	Prerequisite relationships
Linear Relationships A	- Find the gradient of an interval → Gradient and direction vector
Equations C	- Solve simultaneous equations algebraically → Intersection of lines in 2D
Non-Linear Relationships C	- Graph circles with centre (a,b) → Vector equation of a circle
Circle Geometry	- Prove and apply tangent and secant properties → Solve circle and sphere problems
Further Work with Functions	- Understand parametric equations → Express a curve as a vector equation - Convert between parametric and Cartesian form → Cartesian equation from a vector equation - Graph curves in parametric form → Identify curves from vector equations
Introduction to Vectors	- Points in three dimensions → Vector equation of a sphere - Vector joining two points → Direction vector of a line - Vector joining two points → Triangle centres in vectors - Test for parallel vectors → Parallel lines in vector form - Test for parallel vectors → Test collinearity with vectors - Find the magnitude of a vector → Vector equation of a circle - Dot product from components → Properties of the dot product - Dot product from components → Dot products of unit vectors - Geometric form of the dot product → Prove the Cauchy–Schwarz inequality - Test for perpendicular vectors → Triangle centres in vectors - Position as a vector function of time → Vector equation of a line
The Nature of Proof	- Prove inequalities using squares → Prove the Cauchy–Schwarz inequality

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 $\overrightarrow{PQ}$ 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 $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$ as the vector equation of a straight line in two dimensions and three dimensions to solve problems, where $\mathbf{r}$ is the position vector of a point on the line, $\mathbf{a}$ is the position vector of a particular point on the line, $\mathbf{b}$ is a direction vector of the line and $\lambda$ is a scalar parameter
• Identify $\mathbf{r}=\mathbf{p}+\lambda (\mathbf{q}-\mathbf{p})$ as one possible vector equation for the straight line through points $P$ and $Q$, where $\mathbf{p}$ and $\mathbf{q}$ are the position vectors of $P$ and $Q$ respectively, noting its equivalence with the form $\mathbf{r}=\lambda \mathbf{q}+(1-\lambda )\mathbf{p}$, and establish the correspondence between the value of $\lambda$ 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 ${\mathbf{r}}_1={\mathbf{a}}_1+{\lambda }_1{\mathbf{b}}_1$ and ${\mathbf{r}}_2={\mathbf{a}}_2+{\lambda }_2{\mathbf{b}}_2$ in three dimensions intersect, and if so, find the unique value of either ${\lambda }_1$ or ${\lambda }_2$ 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: $\mathbf{r}=\left(\begin{array}{c}x(t)\\ y(t)\end{array}\right)$ or $\mathbf{r}=\left(\begin{array}{c}x(t)\\ y(t)\\ z(t)\end{array}\right)$, 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 $\mathbf{r}=r\cos (\theta )\mathbf{i}+r\sin (\theta )\mathbf{j}$ and $|\mathbf{r}|=r$ as vector equations of a circle in two dimensions with radius $R$ centred at the origin
• Recognise $\mathbf{r}=(r\cos (\theta )+x_C)\mathbf{i}+(r\sin (\theta )+y_C)\mathbf{j}$ and $|\mathbf{r}-\mathbf{c}|=r$ as vector equations of a circle in two dimensions with radius $r$ and centre $C$ with position vector $\mathbf{c}=\left(\begin{array}{c}{x}_c\\ y_c\end{array}\right)$
• Recognise $|\mathbf{r}-\mathbf{c}|=r$ as the vector equation of a sphere in three dimensions with radius $r$ and centre $C$ with position vector $\mathbf{c}$
• 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 $(\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a})$, distributivity $(\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c})$ and scalar multiplication $(k(\mathbf{a}\cdot \mathbf{b})=(k\mathbf{a})\cdot \mathbf{b}=\mathbf{a}\cdot (k\mathbf{b}))$
• Establish and use the results $\mathbf{i}\cdot \mathbf{i}=\mathbf{j}\cdot \mathbf{j}=\mathbf{k}\cdot \mathbf{k}=1$ and $\mathbf{i}\cdot \mathbf{j}=\mathbf{j}\cdot \mathbf{k}=\mathbf{k}\cdot \mathbf{i}=0$
• Prove and use the Cauchy–Schwarz inequality for vectors: $|\mathbf{a}\cdot \mathbf{b}|\le |\mathbf{a}||\mathbf{b}|$
• 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