# 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:

A magnitude
A direction.

## 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

• Results about vectors can be found here.