# Definition:Vector Quantity

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

## Definition

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 $\R^n$.

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.

Let $\mathbf a$ be a vector quantity embedded in an $n$-dimensional Cartesian coordinate system $C_n$.

Let $\mathbf a$ be represented with its initial point at the origin of $C_n$.

Let $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ be the unit vectors in the positive direction of the coordinate axes of $C_n$.

Then:

- $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_3 \mathbf e_n$

where:

- $a_1 \mathbf e_1, a_2 \mathbf e_2, \ldots, a_3 \mathbf e_n$ are the
**component vectors**of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ - $a_1, a_2, \ldots, a_3$ are the
**components**of $\mathbf a$ in the directions of $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$.

The number of **components** in $\mathbf a$ is determined by the number of dimensions in the Cartesian coordinate system of its frame of reference.

### Equivalence Class

A vector quantity is an instance of an equivalence class of directed line segments whose lengths and directions are equal.

The canonical instance is the one whose initial point is located at the origin.

## 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:

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

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

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

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

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

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.

## 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.

### 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

- Definition:Scalar Quantity
- Definition:Scalar (Vector Space)
- Definition:Scalar Field (Linear Algebra)

- Results about
**vectors**can be found**here**.

## Sources

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Definitions: $1$. Scalar and vector quantities - 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 - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules - 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 - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space - 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):**vector** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**vector** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**vector**

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