# Definition:Vector Notation

## Definition

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.

## Historical Note

During the course of development of vector analysis, various notations were introduced, as follows:

Symbol | Used by |
---|---|

$\mathbf a$ | Josiah Willard Gibbs and Edwin Bidwell Wilson Oliver Heaviside Cesare Burali-Forti and Roberto Marcolongo |

$\mathscr A$ | Max Abraham |

$\mathfrak A$ | Vladimir Sergeyevitch Ignatowski |

$\mathbf A$ | Hendrik Antoon Lorentz |

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $2$. Graphical Representation of Vectors - 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 - 1964: D.E. Rutherford:
*Classical Mechanics*(3rd ed.) ... (previous) ... (next): Introduction - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 22$: Notation for Vectors - 1970: George Arfken:
*Mathematical Methods for Physicists*(2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach