# Euclidean Space

There are many types of space that we will refer to when we discuss vectors.

## 1-Dimensional Real Space

We will first look at 1-dimensional real space or $\mathbb{R}$, which is essentially just a number line:

A vector in 1-space has only one component, for example $\vec{u} = (4)$. Often times vectors in $\mathbb{R}$ are uninteresting.

## 2-Dimensional Real Space

The next space we will look at is 2-dimensional real space or $\mathbb{R}^2$. This is the space we are likely most familiar with and is typically represented with two coordinate axes. The first coordinate that represents horizontal location is called the **abscissa**, while the second coordinate that represents the vertical location is called the **ordinate**.

A vector in 2-space is often denoted $\vec{u} = (u_1, u_2)$.

## 3-Dimensional Real Space

Of course we must mention 3-dimensional real space or $\mathbb{R}^3$. This space will be hard to illustrate since we will try to represent 3-space on a 2-dimensional surface (such as your computer screen). Often times we will be looking at an $xyz$-coordinate system. The orientation of the axes isn't particularly important though some orientations are easier to manage than others.

A vector in 3-space has three components and is denoted $\vec{u} = (u_1, u_2, u_3)$.

## Higher Dimensional Real Space

When we start looking at higher dimensional space, we typically use the umbrella term "**Euclidean n-space**". For example, 4-dimensional real space can also be called Euclidean 4-space. We will generally not illustrate vectors that are in greater dimensional space because our imaging is restricted to projecting these vectors to 2-space.