Definition:Vector Quantity

Definition
A vector is a mathematical entity which needs more than one component to specify it.

Formally, a vector is an element of a vector space, often the real vector space $\R^n$.

The usual intellectual frame of reference is to interpret a vector as having: which can be rendered on the page like this:
 * A magnitude
 * A direction


 * Vector.png

In an Euclidean n-space $\R^n$, it is implied that the arrow issues from the coordinate $O = \underbrace{\left({0,0,\cdots,0}\right)}_{n}$.

Alternatively, and frequently more usefully, a vector can also expressed in terms of coordinates. In the above diagram, this would be the "head" of the vector.

It is important to note that there is no mathematical difference between interpreting a vector in $n$-space as "just the tip of the arrow" or "an arrow issuing from $O$ ending at the tip of the arrow". It is only a manner of connotation: both an arrow and a point have the same defining property of an ordered $n$-tuple.

In the contexts of physics and applied mathematics, it is a real-world physical quantity that needs for its model a mathematical object which contains more than one (usually numeric) component.

In this context it is frequently referred to as a vector quantity.

An example is a velocity.

The number of components in a vector is determined by the number of dimensions in the coordinate system of its frame of reference.

In more than three dimensions, the concepts of magnitude and direction are usually abandoned in favour of an ordered tuple of scalars.

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

The vector $\left({x_1, x_2, \ldots, x_n}\right)$ can be annotated as:


 * $\mathbf x = \left({x_1, x_2, \ldots, x_n}\right)$
 * $\vec x = \left({x_1, x_2, \ldots, x_n}\right)$
 * $\hat x = \left({x_1, x_2, \ldots, x_n}\right)$
 * $\underline x = \left({x_1, x_2, \ldots, x_n}\right)$
 * $\tilde x = \left({x_1, x_2, \ldots, x_n}\right)$

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


 * $\mathbf{v} = \left [{x_1, x_2, \ldots, x_n}\right]$

or:


 * $\mathbf{v} = \left \langle{x_1, x_2, \ldots, x_n}\right \rangle$

In printed material the boldface $\mathbf x$ is common. This is the style encouraged and endorsed by this website.

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, they will only usually be found in fair copy.

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.

A vector with $n$ components is sometimes called an $n$-vector.

Vector Space
In the context of abstract algebra, a vector is an element of a module or a vector space.

Also see

 * Module
 * Vector Space
 * Scalar Quantity
 * Scalar
 * Scalar Field