📐Analytic Geometry and Calculus Unit 14 – Vectors in 2D and 3D

Key Concepts and Definitions

• Vectors quantities have both magnitude and direction, in contrast to scalar quantities which only have magnitude
• Magnitude the length or size of a vector, denoted as $|v|$ for a vector $v$
• Direction orientation of a vector in space, often specified using angles or unit vectors
• Components the scalar values that make up a vector when it is expressed in a particular coordinate system (rectangular, polar, etc.)
  • In 2D, a vector has two components $(x, y)$
  • In 3D, a vector has three components $(x, y, z)$
• Unit vectors vectors with a magnitude of 1 that point in a specific direction
  • In 2D: $\hat{i}$ (points along the positive x-axis) and $\hat{j}$ (points along the positive y-axis)
  • In 3D: $\hat{i}$, $\hat{j}$, and $\hat{k}$ (points along the positive z-axis)
• Zero vector a vector with a magnitude of 0, denoted as $\vec{0}$, has no specific direction

Vector Basics in 2D

• Vectors in 2D can be represented using an ordered pair $(x, y)$, where $x$ and $y$ are the vector's components
• Graphically, a 2D vector is represented as an arrow with its tail at the origin and its head at the point $(x, y)$
• The magnitude of a 2D vector $\vec{v} = (x, y)$ is calculated using the Pythagorean theorem: $|\vec{v}| = \sqrt{x^2 + y^2}$
• The direction of a 2D vector can be described using the angle $\theta$ it makes with the positive x-axis
  • This angle can be found using the arctangent function: $\theta = \tan^{-1}(\frac{y}{x})$
• Unit vectors in 2D:
  • $\hat{i} = (1, 0)$ points along the positive x-axis
  • $\hat{j} = (0, 1)$ points along the positive y-axis
• Any 2D vector can be expressed as a linear combination of unit vectors: $\vec{v} = x\hat{i} + y\hat{j}$

Extending to 3D Vectors

• In 3D, vectors are represented using an ordered triplet $(x, y, z)$, where $x$, $y$, and $z$ are the vector's components
• Graphically, a 3D vector is represented as an arrow with its tail at the origin and its head at the point $(x, y, z)$
• The magnitude of a 3D vector $\vec{v} = (x, y, z)$ is calculated using an extended Pythagorean theorem: $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$
• The direction of a 3D vector can be described using two angles:
  • The angle $\theta$ it makes with the positive x-axis in the xy-plane
  • The angle $\phi$ it makes with the positive z-axis
• Unit vectors in 3D:
  • $\hat{i} = (1, 0, 0)$ points along the positive x-axis
  • $\hat{j} = (0, 1, 0)$ points along the positive y-axis
  • $\hat{k} = (0, 0, 1)$ points along the positive z-axis
• Any 3D vector can be expressed as a linear combination of unit vectors: $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$

Vector Operations

• Vector addition adding two or more vectors component-wise
  • For 2D vectors: $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$
  • For 3D vectors: $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2)$
• Scalar multiplication multiplying a vector by a scalar (real number) to change its magnitude
  • For a scalar $c$ and a vector $\vec{v} = (x, y)$: $c\vec{v} = (cx, cy)$
  • For a scalar $c$ and a vector $\vec{v} = (x, y, z)$: $c\vec{v} = (cx, cy, cz)$
• Dot product (scalar product) a multiplication operation that results in a scalar value
  • For 2D vectors: $(x_1, y_1) \cdot (x_2, y_2) = x_1x_2 + y_1y_2$
  • For 3D vectors: $(x_1, y_1, z_1) \cdot (x_2, y_2, z_2) = x_1x_2 + y_1y_2 + z_1z_2$
• Cross product (vector product) a multiplication operation that results in a vector perpendicular to the two input vectors (only defined for 3D vectors)
  • For 3D vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$: $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$
• Vector subtraction can be performed by adding the negative of the vector being subtracted
  • For 2D vectors: $(x_1, y_1) - (x_2, y_2) = (x_1 - x_2, y_1 - y_2)$
  • For 3D vectors: $(x_1, y_1, z_1) - (x_2, y_2, z_2) = (x_1 - x_2, y_1 - y_2, z_1 - z_2)$

Geometric Applications

• Vectors can be used to represent displacements, velocities, and forces in physics and engineering
• The dot product of two vectors $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between the vectors
  • This can be used to find the angle between two vectors or to determine if vectors are orthogonal (perpendicular)
• The cross product of two vectors $\vec{a} \times \vec{b}$ results in a vector perpendicular to both $\vec{a}$ and $\vec{b}$, with a magnitude equal to the area of the parallelogram formed by the two vectors
• Vectors can be used to find the distance between points in 2D and 3D space
  • The distance between points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  • The distance between points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
• Vectors can be used to find the midpoint of a line segment connecting two points
  • For 2D points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the midpoint is $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
  • For 3D points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the midpoint is $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})$

Algebraic Representations

• Vectors can be represented using matrix notation, with each component as an element in a column matrix
  • A 2D vector $(x, y)$ can be written as $\begin{bmatrix} x \\ y \end{bmatrix}$
  • A 3D vector $(x, y, z)$ can be written as $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$
• Vector operations can be performed using matrix operations
  • Addition and subtraction of vectors correspond to element-wise addition and subtraction of their matrix representations
  • Scalar multiplication of a vector corresponds to multiplying each element of its matrix representation by the scalar
• The dot product of two vectors can be calculated using matrix multiplication
  • For 2D vectors $\vec{a} = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$: $\vec{a} \cdot \vec{b} = \begin{bmatrix} a_1 & a_2 \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = a_1b_1 + a_2b_2$
  • For 3D vectors $\vec{a} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$: $\vec{a} \cdot \vec{b} = \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = a_1b_1 + a_2b_2 + a_3b_3$
• The cross product of two 3D vectors can be calculated using the determinant of a 3x3 matrix
  • For 3D vectors $\vec{a} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Problem-Solving Strategies

• When solving problems involving vectors, first identify the given information and the desired outcome
• Sketch the problem situation, including any relevant vectors, to visualize the relationships between the given information
• Break down complex problems into smaller, more manageable steps
  • Solve for intermediate values or components before tackling the main problem
• Use the properties of vector operations to simplify expressions and equations
  • Distributive property: $a(\vec{u} + \vec{v}) = a\vec{u} + a\vec{v}$
  • Associative property: $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
  • Commutative property: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$
• Apply the appropriate vector operation based on the problem context
  • Use vector addition and subtraction for problems involving displacements, velocities, or forces
  • Use the dot product for problems involving angles, projections, or work
  • Use the cross product for problems involving torque, angular momentum, or finding perpendicular vectors
• Double-check your solution by verifying that it makes sense in the context of the problem and that the units are consistent

Real-World Applications

• Vectors are used extensively in physics to represent quantities such as displacement, velocity, acceleration, and force
  • Newton's second law of motion: $\vec{F} = m\vec{a}$, where $\vec{F}$ is the net force, $m$ is the mass, and $\vec{a}$ is the acceleration
• In engineering, vectors are used to analyze and design structures, machines, and systems
  • Statics: Analyzing forces and moments acting on a system in equilibrium
  • Dynamics: Studying the motion and forces of objects and systems
• Computer graphics and game development rely heavily on vector mathematics for rendering 2D and 3D objects, simulating physics, and controlling character movements
• Navigation systems, such as GPS, use vectors to represent positions, velocities, and directions on Earth's surface
• Vectors are used in fluid dynamics to describe the flow of liquids and gases, with applications in aerodynamics, hydraulics, and meteorology
• In electromagnetics, vectors are used to represent electric and magnetic fields, as well as the propagation of electromagnetic waves
• Quantum mechanics employs vector spaces and linear algebra to describe the states and evolution of quantum systems
• Machine learning and data analysis often involve vector representations of data points and use vector operations for tasks such as clustering, classification, and dimensionality reduction