Vectors and Euclidean Space: Foundations and Operations

Euclidean Space and Vectors

Introduction to Euclidean Space

Euclidean space, denoted as $\mathbb{R}^n$, is a fundamental concept in mathematics, especially in algebra and geometry. It consists of all ordered n-tuples of real numbers and provides the setting for studying vectors and their operations.

Vectors in $\mathbb{R}^2$

Definition and Representation

• Vector in $\mathbb{R}^2$: A vector $\vec{x}$ in $\mathbb{R}^2$ is an ordered pair of real numbers, written as $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, where $x_1, x_2 \in \mathbb{R}$.

• Graphical Interpretation: The vector $\vec{x}$ is represented as an arrow from the origin $(0,0)$ to the point $(x_1, x_2)$ in the plane.

Example: The vector $\vec{x} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$ starts at the origin and ends at the point (2, 3).

Set Definition of $\mathbb{R}^2$

• $\mathbb{R}^2 = \left\{ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} : x_1, x_2 \in \mathbb{R} \right\}$

Vectors in $\mathbb{R}^3$

Definition and Representation

• Vector in $\mathbb{R}^3$: A vector $\vec{x}$ in $\mathbb{R}^3$ is an ordered triple of real numbers, written as $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$, where $x_1, x_2, x_3 \in \mathbb{R}$.

• Graphical Interpretation: The vector $\vec{x}$ is represented as an arrow from the origin $(0,0,0)$ to the point $(x_1, x_2, x_3)$ in three-dimensional space.

Example: The vector $\vec{x} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ starts at the origin and ends at the point (1, 2, 3).

Vectors in $\mathbb{R}^n$

General Definition

• Vector in $\mathbb{R}^n$: A vector $\vec{x}$ in $\mathbb{R}^n$ is an ordered n-tuple of real numbers, written as $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, where $x_1, x_2, ..., x_n \in \mathbb{R}$.

• Components: The entries $x_1, x_2, ..., x_n$ are called the components of the vector.

Set Definition: $\mathbb{R}^n = \left\{ \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} : x_1, x_2, ..., x_n \in \mathbb{R} \right\}$

Transpose of a Vector

Definition

• The transpose of a column vector $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ is the row vector $\vec{x}^T = [x_1, x_2, ..., x_n]$.

Application: The transpose operation is important in matrix algebra and when defining dot products.

Standard (Elementary) Vectors

Standard Vectors in $\mathbb{R}^2$

• The standard vectors (or elementary vectors) in $\mathbb{R}^2$ are:

  • $\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

  • $\vec{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

• These vectors form the basis for $\mathbb{R}^2$ and are often denoted as $\vec{i}$ and $\vec{j}$, respectively.

Standard Vectors in $\mathbb{R}^3$

• The standard vectors in $\mathbb{R}^3$ are:

  • $\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ (denoted $\vec{i}$)

  • $\vec{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ (denoted $\vec{j}$)

  • $\vec{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ (denoted $\vec{k}$)

Standard Vectors in $\mathbb{R}^n$

• In $\mathbb{R}^n$, the standard vectors are $\vec{e}_1, \vec{e}_2, ..., \vec{e}_n$, where $\vec{e}_i$ has a 1 in the $i$-th position and 0 elsewhere.

Directed Vectors

Definition of a Directed Vector

• Given two points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ in $\mathbb{R}^2$, the directed vector from $P$ to $Q$ is:

  • $\vec{d} = \overrightarrow{PQ} = Q - P = \begin{bmatrix} x_2 - x_1 \\ y_2 - y_1 \end{bmatrix}$

Example: If $P = (1,2)$ and $Q = (3,-1)$, then $\overrightarrow{PQ} = \begin{bmatrix} 3-1 \\ -1-2 \end{bmatrix} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}$.

Operations on Vectors

Addition and Subtraction

• Addition: For vectors $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and $\vec{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ in $\mathbb{R}^2$:

  • $\vec{x} + \vec{y} = \begin{bmatrix} x_1 + y_1 \\ x_2 + y_2 \end{bmatrix}$

• Subtraction: $\vec{x} - \vec{y} = \begin{bmatrix} x_1 - y_1 \\ x_2 - y_2 \end{bmatrix}$

Example: $\begin{bmatrix} 2 \\ 3 \end{bmatrix} + \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 1 \\ 6 \end{bmatrix}$

Graphical Representation

• Vector Addition: Place the tail of the second vector at the head of the first; the resultant vector goes from the tail of the first to the head of the second.

• Vector Subtraction: The vector $\vec{x} - \vec{y}$ is found by adding $\vec{x}$ and $-\vec{y}$ (the negative of $\vec{y}$).

Summary Table: Standard Vectors in $\mathbb{R}^n$

Space	Standard Vectors	Component Form
$\mathbb{R}^2$	$\vec{e}_1, \vec{e}_2$	$\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, $\vec{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
$\mathbb{R}^3$	$\vec{e}_1, \vec{e}_2, \vec{e}_3$	$\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\vec{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\vec{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$
$\mathbb{R}^n$	$\vec{e}_1, \vec{e}_2, ..., \vec{e}_n$	$\vec{e}_i$ has 1 in the $i$-th position, 0 elsewhere

Key Takeaways

• Vectors in $\mathbb{R}^n$ are ordered n-tuples of real numbers.

• Standard vectors form the basis for vector spaces.

• Vector operations (addition, subtraction, transpose) are foundational for further study in algebra and geometry.