Euclidean n-Space

# Euclidean n-Space

So far we have looked strictly at $\mathbb{R}$ - the set of real numbers. We will now extend our reach to higher dimensions and looked at Euclidean $n$-space.

 Definition: For each positive integer $n$, the Euclidean $n$-Space denoted $\mathbb{R}^n$ is the set of all points $\mathbf{x} = (x_1, x_2, ..., x_n)$ such that $x_1, x_2, ..., x_n \in \mathbb{R}$. The $k^{\mathrm{th}}$ coordinate of the point $\mathbf{x}$ is the real number $x_k$.

In the case where $n = 1$ we have that Euclidean 1-space is simply the real line $\mathbb{R}$. When $n = 2$ we are looking at points $(x, y)$ in the plane, and when $n = 3$ we are looking in at points $(x, y, z)$ in three-dimensional space.

The graphic below illustrates how we can visualize Euclidean $n$-space for $n = 1, 2, 3$: Of course when $n \geq 4$ it is practically impossibly to visualize Euclidean $n$-space and so, we will usually talk merely about the points (or vectors) which make up the space. Like with the cases above, the point $\mathbf{x} = (x_1, x_2, ..., x_n)$ for $x_1, x_2, ..., x_n \in \mathbb{R}$ symbolically imply the existence of $n$ mutually perpendicular axes that intersect at a point called the origin we denote by:

(1)
\begin{align} \quad \mathbf{0} = (0, 0, ..., 0) \end{align}

The point $\mathbf{x}$ is described to be located in respect to $\mathbf{0}$, i.e., the point $\mathbf{x}$ is located $x_1$ along the first axis, $x_2$ along the the second axis, …, $x_n$ along the $n^{\mathrm{th}}$ axis. Sometimes we instead prefer to visualize $\mathbf{x}$ as a vector (arrow) the starts at the origin and whose arrowhead ends at the point $(x_1, x_2, ..., x_n)$