# Definition:Vector Quantity

*This page is about Vector Quantity. For other uses, see Vector.*

## Contents

## Definition

A **vector quantity** is a 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 vector space, often the real vector space $\R^n$.

The usual intellectual frame of reference is to interpret a **vector quantity** as having:

### Arrow Representation

A **vector quantity** $\mathbf v$ is often represented diagramatically in the form of an **arrow** such that:

- its length is proportional to the magnitude of $\mathbf v$
- its direction corresponds to the direction of $\mathbf v$.

The head of the arrow then indicates the positive sense of the direction of $\mathbf v$.

It can be rendered on the page like so:

It is important to note that a vector quantity, when represented in this form, is not in general fixed in space.

All that is being indicated using such a notation is its magnitude and direction, and not, in general, a point at which it acts.

### Component

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.

A vector quantity $\mathbf a$ 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**.

## Also known as

A **vector quantity** in this context is frequently referred to just as a **vector**.

Some sources use the term **free vector**, so as to distinguish it from what is referred to in $\mathsf{Pr} \infty \mathsf{fWiki}$ as a **directed line segment**.

Further to the **arrow representation** of a **vector quantity**, some sources refer to such quantities as **arrows**.

## 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:

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

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

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

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

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

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** $\bsx$ 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.

## Examples

### Displacement

The **(physical) displacement** of a body is a measure of its position relative to a given point of reference in a particular frame of reference.

**Displacement** is a vector quantity, so it specifies a magnitude and direction from the point of reference.

### Velocity

The **velocity** $\mathbf v$ of a body $M$ is defined as the first derivative of the displacement $\mathbf s$ of $M$ from a given point of reference with respect to time $t$:

- $\mathbf v = \dfrac {\d \mathbf s} {\d t}$

### Acceleration

The **acceleration** $\mathbf a$ of a body $M$ is defined as the first derivative of the velocity $\mathbf v$ of $M$ relative to a given point of reference with respect to time $t$:

- $\mathbf a = \dfrac {\d \mathbf v} {\d t}$

## Also see

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Definitions: $1$. - 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: 1. Scalar and Vector Quantities - 1957: D.E. Rutherford:
*Vector Methods*(9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 1$. - 1960: M.B. Glauert:
*Principles of Dynamics*... (previous) ... (next): Chapter $1$: Vector Algebra: $1.1$ Definition of a Vector - 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 - 1970: George Arfken:
*Mathematical Methods for Physicists*(2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach - 1972: M.A. Akivis and V.V. Goldberg:
*An Introduction to Linear Algebra & Tensors*(translated by Richard A. Silverman) ... (next): Chapter $1$: Linear Spaces: $1$. Basic Concepts - 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 $1$ - 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.1$ Geometric and Algebraic Definitions of a Vector - 1995: John B. Fraleigh and Raymond A. Beauregard:
*Linear Algebra*(3rd ed.) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**vector** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**vector** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**vector**

- 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.

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