Definition:Vector (Physics)

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This page is about vectors in physics and applied mathematics. For other uses, see Definition:Vector.

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:

A magnitude
A direction.

This can be rendered on the page like so:

Vector.png

In a Euclidean $n$-space $\R^n$, it is implied that the arrow issues from the origin:

$O = \underbrace{\left({0, 0, \ldots, 0}\right)}_n$


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 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.


Component

A vector $\mathbf a$ can be represented with its initial point at the origin of a Cartesian coordinate system.

Let $\mathbf i, \mathbf j, \mathbf k$ be the unit vectors in the positive direction of the $x$-axis, $y$-axis and $z$-axis respectively.

Then:

$\mathbf a = a_1 \mathbf i + a_2 \mathbf j + a_3 \mathbf k$

where:

$a_1 \mathbf i, a_2 \mathbf j, a_3 \mathbf k$ are the component vectors of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions
$a_1, a_2, a_3$ are the components of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions.


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


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


Vector 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:

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


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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

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.

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.


Also see

Sources