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 $\R^n$.

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:

$\ds \bsx$

$=$