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.