# Definition:Vector (Euclidean Space)

*This page is about vectors in Euclidean space. For other uses, see Definition:Vector.*

## Contents

## Definition

A vector is defined as an element of a vector space.

We have that $\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space.

Hence a **vector in $\R^n$** is defined as any element of $\R^n$.

### $\R^1$

As $\R$ forms a vector space, every real number is a vector.

Note that as $\R$ is a vector space over itself, every real number is also a scalar.

Hence **a vector in $\R^n$** is sometimes imprecisely used to mean "a vector in $\R^n$, $n > 1$".

### Geometric Interpretation

From the Cantor-Dedekind Hypothesis, the real number line $\R$ can be represented by an infinite straight line.

By the same token, a vector in $\R$ can be represented by a directed line segment.

Formally, a vector $\sequence {x_1}, x_1 \in \R$ is accurately represented by **the set of all directed line segments** having:

- Magnitude $\size {x_1}$

- Direction dependent on whether $x_1 < 0$ or $x_1 > 0$

By convention, if only one axis is under consideration, the line is placed horizontally, such that segments oriented towards the right are positive, to the left negative.

Note that in such a context the zero vector can be interpreted as a directed line segment beginning and terminating at the same point.

## $\R^2$

We have that $\R^2$ is a vector space.

Hence any ordered $2$-tuple of $2$ real numbers is a **vector**.

### Geometric Interpretation

From the definition of the real number plane, we can represent the vector space $\R^2$ by points on the plane.

That is, every pair of coordinates $\tuple {x_1, x_2}$ can be uniquely defined by a point in the plane.

An arrow with base at the origin and terminal point $\tuple {x_1, x_2}$ is defined to have the length equal to the magnitude of the vector, and direction defined by the relative location of $\tuple {x_1, x_2}$ with the origin as the point of reference.

Each vector is then represented by the set of all directed line segments with:

- Magnitude $\sqrt {x_1^2 + x_2^2}$

- Direction equal to the direction of $\overrightarrow {\tuple {0, 0} \tuple {x_1, x_2} }$

## Vector Notation

Several conventions are found in the literature for annotating a general vector in a style that distinguishes it from a scalar, as follows.

Let $\set {x_1, x_2, \ldots, x_n}$ be a collection of scalars which form the components of an $n$-dimensional vector.

The vector $\tuple {x_1, x_2, \ldots, x_n}$ can be annotated as:

\(\displaystyle \bsx\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \vec x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \hat x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \underline x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \tilde x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) |

To emphasize the arrow interpretation of a vector, we can write:

- $\bsv = \sqbrk {x_1, x_2, \ldots, x_n}$

or:

- $\bsv = \sequence {x_1, x_2, \ldots, x_n}$

In printed material the boldface $\bsx$ is common. This is the style encouraged and endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.

However, for handwritten material (where boldface is difficult to render) it is usual to use the **underline** version $\underline x$.

Also found in handwritten work are the **tilde** version $\tilde x$ and **arrow** version $\vec x$, but as these are more intricate than the simple underline (and therefore more time-consuming and tedious to write), they will only usually be found in fair copy.

It is also noted that the **tilde** over $\tilde x$ does not render well in MathJax under all browsers, and differs little visually from an overline.

The **hat** version $\hat x$ usually has a more specialized meaning, namely to symbolize a unit vector.

In computer-rendered materials, the **arrow** version $\vec x$ is popular, as it is descriptive and relatively unambiguous, and in $\LaTeX$ it is straightforward.

However, it does not render well in all browsers, and is therefore (reluctantly) not recommended for use on this website.

Because of this method of rendition, some sources refer to vectors as **arrows**.

### Comment

The reader should be aware that a vector in $\R^n$ is and only is an ordered $n$-tuple of $n$ real numbers. The geometric interpretations given above are only *representations* of vectors.

Further, the geometric interpretation of a vector is accurately described as **the set of all line segments equivalent to a given directed line segment**, rather than any particular line segment.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**vector**