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

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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, but they are also essential in economics, engineering, and many other fields.

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 a vector are called its components. For example, the vector $\vec{v} = (v_1, v_2)$ in $\mathbb{R}^2$ has two components.
Example: The vector $\vec{a} = (2, 3)$ represents a movement 2 units to the right and 3 units up from the origin.

Types of Numbers and Spaces

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

Real Numbers ($\mathbb{R}$): The set of all numbers that can be represented on the number line, including integers, fractions, and irrational numbers.
Vector Spaces ($\mathbb{R}^n$): The set of all ordered n-tuples of real numbers. For example, $\mathbb{R}^2$ is the set of all ordered pairs, and $\mathbb{R}^3$ is the set of all ordered triples.
Example: $\mathbb{R}^2$ is the plane, and $\mathbb{R}^3$ is three-dimensional space.

Graphical Representation of Vectors

Vectors are often represented graphically as arrows in the coordinate plane or space.

Origin: The starting point of a vector is usually the origin (0,0) in $\mathbb{R}^2$ or (0,0,0) in $\mathbb{R}^3$.
Direction and Magnitude: The arrow points from the origin to the point defined by the vector's components.
Example: The vector $\vec{v} = (1,2)$ starts at the origin and ends at the point (1,2).

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 all 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 through the origin. Parallel vectors are scalar multiples of each other.

Definition: Two vectors are collinear if they are scalar multiples of each other.
Example: $\vec{a} = (1,2)$ and $\vec{b} = (2,4)$ are collinear because $\vec{b} = 2\vec{a}$.

Distance Between Vectors

The distance between two vectors is the length of the line segment connecting 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 $\mathbb{R}^3$:

$ d(\vec{u}, \vec{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + (u_3 - v_3)^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 or magnitude of a vector is the distance from the origin to the point defined by the vector.

Definition: The magnitude or norm of $\vec{v} = (v_1, v_2)$ is:

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

In $\mathbb{R}^n$:

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

Unit Vector: A vector whose magnitude is 1.
Example: $\vec{v} = (3, 4)$ has $\|\vec{v}\| = \sqrt{3^2 + 4^2} = 5$.

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: $2\vec{a} = 2(1,2) = (2,4)$; $-1\vec{a} = (-1, -2)$.

Unit Vectors in a Given Direction

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

Formula: If $\vec{v} \neq \vec{0}$, then the unit vector in the direction of $\vec{v}$ is:

$ \vec{u} = \frac{\vec{v}}{\|\vec{v}\|} $

Example: For $\vec{v} = (3, 4)$, $\vec{u} = \frac{1}{5}(3, 4) = (0.6, 0.8)$.

Collinearity and Scalar Multiples

If two vectors are collinear, one is a scalar multiple of the other. This is useful for determining if vectors are parallel.

Theorem: Vectors $\vec{u}$ and $\vec{v}$ are collinear if and only if there is a scalar $k$ such that $\vec{u} = k\vec{v}$.
Example: If $\vec{u} = (-2,7)$ and $\vec{v} = (4, -14)$, then $\vec{v} = -2\vec{u}$, so they are collinear.

Summary Table: Key Vector Properties

Property	Definition	Formula	Example
Zero Vector	All components are zero	$\vec{0} = (0, 0, ..., 0)$	$\vec{0} = (0,0)$
Magnitude	Length of vector	$\\|\vec{v}\\| = \sqrt{v_1^2 + v_2^2}$	$\\| (3,4) \\| = 5$
Distance	Between two vectors	$d(\vec{u}, \vec{v}) = \sqrt{(u_1-v_1)^2 + (u_2-v_2)^2}$	$d((1,2),(2,4)) = \sqrt{5}$
Unit Vector	Magnitude is 1	$\vec{u} = \frac{\vec{v}}{\\|\vec{v}\\|}$	$\frac{1}{5}(3,4) = (0.6,0.8)$
Scalar Multiplication	Multiply by real number	$c\vec{v} = (cv_1, cv_2)$	$2(1,2) = (2,4)$
Collinearity	Vectors are scalar multiples	$\vec{u} = k\vec{v}$	$(-2,7)$ and $(4,-14)$

Additional info:

These notes are from 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 economics for modeling and optimization.