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.