Vectors: Definitions, Properties, and Applications (Math 1229A/B Unit 1)

Vectors

Introduction to Vectors

Vectors are fundamental mathematical objects used to represent quantities that have both magnitude and direction. In mathematics, vectors are often introduced in the context of geometry and physics, and they play a crucial role in many areas of science and engineering.

• Definition: A vector is an ordered list of numbers, which can be represented as a directed line segment in space.

• Notation: Vectors are typically denoted by boldface letters (e.g., v) or with an arrow above the letter (e.g., v).

• Components: The numbers in the list are called components of the vector.

• Example: The vector $\vec{v} = (2, 3)$ in $\mathbb{R}^2$ has components 2 and 3.

Types of Numbers and Spaces

Vectors can exist in different spaces depending on the type and number of components.

• Real Numbers ($\mathbb{R}$): Includes all rational and irrational numbers.

• Vector Spaces: $\mathbb{R}^2$ (plane), $\mathbb{R}^3$ (space), $\mathbb{R}^n$ (n-dimensional space).

• Example: $\vec{v} = (1, 2, 3)$ is a vector in $\mathbb{R}^3$.

Geometric Representation of Vectors

Vectors are often represented as arrows in space, starting at one point and ending at another.

• Directed Line Segment: The arrow points from the initial point to the terminal point.

• Position Vector: A vector whose initial point is the origin.

• Example: The vector $\vec{v} = (2, 1)$ starts at the origin and ends at the point (2, 1).

Zero Vector

The zero vector is a special vector whose components are all zero.

• Definition: The zero vector in $\mathbb{R}^n$ is $\vec{0} = (0, 0, ..., 0)$.

• Properties: The zero vector has no direction and zero magnitude.

Equality of Vectors

Two vectors are equal if and only if their corresponding components are equal.

• Definition: $\vec{u} = (u_1, u_2, ..., u_n)$ and $\vec{v} = (v_1, v_2, ..., v_n)$ are equal if $u_i = v_i$ for all $i$.

• Example: $\vec{a} = (2, 1)$ and $\vec{b} = (2, 1)$ are equal vectors.

Collinearity and Parallelism

Vectors are collinear if they lie on the same line, and parallel if they are scalar multiples of each other.

• Definition: Two vectors are collinear if they are parallel or anti-parallel.

• Example: $\vec{a} = (1, 2)$ and $\vec{b} = (2, 4)$ are collinear since $\vec{b} = 2\vec{a}$.

Distance Between Vectors

The distance between two vectors is the length of the line segment joining their endpoints.

• Definition: The distance between $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^2$ is:

$d(\vec{u}, \vec{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2}$

• In $\mathbb{R}^n$:

$d(\vec{u}, \vec{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + \cdots + (u_n - v_n)^2}$

• Example: For $\vec{u} = (1, 2)$ and $\vec{v} = (2, 4)$, $d(\vec{u}, \vec{v}) = \sqrt{(1-2)^2 + (2-4)^2} = \sqrt{1 + 4} = \sqrt{5}$.

Length (Magnitude) of a Vector

The length, magnitude, or norm of a vector is a measure of its size.

• Definition: The length of $\vec{v} = (v_1, v_2, ..., v_n)$ is:

$\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$

• Unit Vector: A vector whose length is 1.

• Example: $\vec{v} = (3, 4)$, $\|\vec{v}\| = \sqrt{3^2 + 4^2} = 5$.

Scalar Multiplication

Vectors can be multiplied by scalars (real numbers), which changes their magnitude but not their direction (unless the scalar is negative).

• Definition: For $c \in \mathbb{R}$ and $\vec{v} = (v_1, v_2, ..., v_n)$, $c\vec{v} = (cv_1, cv_2, ..., cv_n)$.

• Properties: If $c > 0$, the direction is preserved; if $c < 0$, the direction is reversed.

• Example: $2\vec{v} = (2v_1, 2v_2, ..., 2v_n)$.

Finding Unit Vectors

To find a unit vector in the direction of a given vector, divide the vector by its magnitude.

• Formula: $\vec{u} = \frac{\vec{v}}{\|\vec{v}\|}$

• Example: For $\vec{v} = (3, 4)$, $\vec{u} = \frac{(3, 4)}{5} = (0.6, 0.8)$

Summary Table: Key Vector Properties

Property	Definition	Formula	Example
Zero Vector	All components are zero	$\vec{0} = (0, 0, ..., 0)$	$\vec{0}$ in $\mathbb{R}^3$ is (0, 0, 0)
Length (Norm)	Magnitude of vector	$\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}$	$\| (3, 4) \| = 5$
Distance	Between two vectors	$d(\vec{u}, \vec{v}) = \sqrt{(u_1 - v_1)^2 + ... + (u_n - v_n)^2}$	$d((1,2), (2,4)) = \sqrt{5}$
Unit Vector	Length is 1	$\vec{u} = \frac{\vec{v}}{\|\vec{v}\|}$	$\frac{(3,4)}{5} = (0.6, 0.8)$
Scalar Multiplication	Multiply by real number	$c\vec{v} = (cv_1, cv_2, ..., cv_n)$	$2(1,2) = (2,4)$

Additional info:

• These notes are for a mathematics course (Math 1229A/B) and focus on vectors, which are not directly part of a standard Microeconomics curriculum. However, vectors are sometimes used in advanced microeconomics (e.g., optimization, utility functions), but the content here is purely mathematical and not economics-specific.