# Definition:Vector

## Definition

### Vector (Module)

Let $\struct {G, +_G, \circ}_R$ be a module, where:

- $\struct {G, +_G}$ is an abelian group

- $\struct {R, +_R, \times_R}$ is the scalar ring of $\struct {G, +_G, \circ}_R$.

The elements of the abelian group $\struct {G, +_G}$ are called **vectors**.

### Vector (Linear Algebra)

Let $V = \struct {G, +_G, \circ}_K$ be a vector space over $K$, where:

- $\struct {G, +_G}$ is an abelian group

- $\struct {K, +_K, \times_K}$ is the scalar field of $V$.

The elements of the abelian group $\struct {G, +_G}$ are called **vectors**.

### Vector (Real Euclidean Space)

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, forms a real Euclidean space.

Hence a **vector in $\R^n$** is defined as an element of the real Euclidean space $\R^n$.

### Vector (Affine Geometry)

Let $\EE$ be an affine space.

Let $V$ be the tangent space of $\EE$.

An element $v$ of $V$ is called a **vector**.

### Vector Quantity

A **vector quantity** is a real-world concept that needs for its model a mathematical object with more than one **component** to specify it.

Formally, a **vector quantity** is an element of a normed vector space, often the real vector space $\R^3$.

The usual intellectual frame of reference is to interpret a **vector quantity** as having:

## Vector Notation

Several conventions are found in the literature for annotating a general vector quantity in a style that distinguishes it from a scalar quantity, 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:

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

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

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

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

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

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** $\bsx$ or $\mathbf x$ 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**: $\overline x$.

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.

## Also see

- Definition:Directed Line Segment: a vector whose vector space is a Euclidean space.

- Results about
**vectors**can be found**here**.