# Definition:Vector (Physics)

*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:

This can be informally interpreted as "something that points in some direction".

This can be rendered on the page like so:

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

- $O = \underbrace {\tuple {0, 0, \ldots, 0} }_n$

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 $\set {x_1, x_2, \ldots, x_n}$ 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:

\(\displaystyle \bsx\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \vec x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \hat x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \underline x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) | |||||||||||

\(\displaystyle \tilde x\) | \(=\) | \(\displaystyle \tuple {x_1, x_2, \ldots, x_n}\) |

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 $\bsx$ 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

- 1966: Isaac Asimov:
*Understanding Physics*... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Forces and Vectors - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Vectors and Scalars - 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE: Example $1$ - 1995: John B. Fraleigh and Raymond A. Beauregard:
*Linear Algebra*(3rd ed.)

- Weisstein, Eric W. "Vector." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Vector.html - For a video presentation of the contents of this page, visit the Khan Academy.