Introduction to Vectors

vectors year12 ext1

Introduction to vectors | NSW Curriculum Website

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

📖 Prior Knowledge

Content	Prior knowledge	Used for
Linear Relationships C	- midpoint and distance formulas - translation vectors	- midpoint and distance in 3D, magnitude of a vector - vectors in general
Trigonometry and Measure of Angles	- cosine rule - trig ratios	- dot product - projectile motion
Further Work with Functions	- parametric equations	- parametric equations of motion
Applications of Calculus	- velocity and acceleration using calculus	- velocity and acceleration vectors, projectile motion

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 $\mathbf{a}$, $\underset{\sim }{a}$ and $\overrightarrow{AB}$, where $\overrightarrow{AB}$ is the vector with magnitude and direction those of the directed line segment from $A$ to $B$
• Use notations $|\mathbf{a}|$, $|\underset{\sim }{a}|$ and $|\overrightarrow{AB}|$ 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 $0\sim$, as the vector with zero magnitude and no direction
• Define unit vectors as vectors of magnitude 1
• Recognise and use $\hat{\mathbf{a}}$ and $\widehat{\underset{\sim }{a}}$ as the notation for the unit vector in the direction of $\mathbf{a}$
• Define the standard perpendicular unit vectors $\mathbf{i}$ and $\mathbf{j}$ in two dimensions and $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ in three dimensions
• Express 2D vectors in component form, $x\mathbf{i}+y\mathbf{j}$; as an ordered pair, $(x,y)$; and in column vector notation $\left(\begin{array}{c}x\\ y\end{array}\right)$
• Recognise $\overrightarrow{AB}=\left(\begin{array}{c}u-x\\ v-y\end{array}\right)$ 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, $x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$; as an ordered triple, $(x,y,z)$; and in column vector notation $\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$
• Recognise $\overrightarrow{AB}=\left(\begin{array}{c}u-x\\ v-y\\ w-z\end{array}\right)$ 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 $\mathbf{a}=k\mathbf{b}$, for a non-zero scalar $k$, as a condition for two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ to be parallel to each other and determine with justification if two vectors are parallel to one another
• Identify $\left(\begin{array}{c}a\\ -b\end{array}\right)$ and $\left(\begin{array}{c}-a\\ b\end{array}\right)$ as vectors perpendicular to $\left(\begin{array}{c}a\\ b\end{array}\right)$ 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 $|x\mathbf{i}+y\mathbf{j}|=\sqrt{x^2+y^2}$ for 2D vectors and $|x\mathbf{i}+y\mathbf{j}+z\mathbf{k}|=\sqrt{x^2+y^2+z^2}$ for 3D vectors
• Use the magnitude of a vector to find the unit vector $\hat{\mathbf{a}}=\frac{1}{|\mathbf{a}|}\mathbf{a}$ in two dimensions and three dimensions

Further operations with vectors

• Define $\mathbf{a}\cdot \mathbf{b}=x_1{x}_2+y_1{y}_2$ as the scalar (dot) product of vectors $\mathbf{a}=x_1\mathbf{i}+y_1\mathbf{j}$ and $\mathbf{b}=x_2\mathbf{i}+y_2\mathbf{j}$ and use the scalar product to solve problems
• Define $\mathbf{a}\cdot \mathbf{b}=x_1{x}_2+y_1{y}_2+z_1{z}_2$ as the scalar product of vectors $\mathbf{a}=x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k}$ and $\mathbf{b}=x_2\mathbf{i}+y_2\mathbf{j}+z_2\mathbf{k}$ and use the scalar product to solve problems
• Use $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$ as a geometric expression of the scalar product of non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ in two dimensions and three dimensions, where $\theta$ is the angle between the vectors and $0°\le \theta \le 180°$
• Verify the equivalence of $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$ with the algebraic definition of the scalar product, $\mathbf{a}\cdot \mathbf{b}=x_1{x}_2+y_1{y}_2$ for two dimensions and $\mathbf{a}\cdot \mathbf{b}=x_1{x}_2+y_1{y}_2+z_1{z}_2$ for three dimensions
• Derive and use the property $\mathbf{a}\cdot \mathbf{a}=|\mathbf{a}{|}^2$ to establish the scalar product definition of the magnitude of a vector $(|\mathbf{a}|=\sqrt{\mathbf{a}\cdot \mathbf{a}})$ in two dimensions and three dimensions
• Calculate the angle between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$, in both two dimensions and three dimensions, using the scalar product by deriving and applying the relationship $\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}=(\hat{\mathbf{a}}\cdot \hat{\mathbf{b}})$
• Establish $\mathbf{a}\cdot \mathbf{b}=0$ as a condition for two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ to be perpendicular to each other and use it to determine if two vectors are perpendicular
• Establish $\mathbf{a}\cdot \mathbf{b}=\pm |\mathbf{a}||\mathbf{b}|$ as another way to determine if two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel
• Define the projection of a vector $\mathbf{a}$ onto a vector $\mathbf{b}$, denoted by ${\mathrm{proj}}_{\mathbf{b}}\mathbf{a}$, to be the vector component of $\mathbf{a}$ in the direction of vector $\mathbf{b}$
• Examine the proof of the formula ${\mathrm{proj}}_{\mathbf{b}}\mathbf{a}=(\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{b}{|}^2})\mathbf{b}=(\mathbf{a}\cdot \hat{\mathbf{b}})\hat{\mathbf{b}}$ and use the formula to solve problems
• Determine that the component of a vector $\mathbf{a}$ perpendicular to another vector $\mathbf{b}$ is $\mathbf{a}-{\mathrm{proj}}_{\mathbf{b}}\mathbf{a}$

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 $\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}$, or $\mathbf{r}=x\mathbf{i}+y\mathbf{j}$ 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 $\mathbf{v}=\dot{x}\mathbf{i}+\dot{y}\mathbf{j}$ and acceleration vector $\mathbf{a}=\ddot{x}\mathbf{i}+\ddot{y}\mathbf{j}$ 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