Definition:Vector Quantity

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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 vector space, often the real vector space $\R^n$.

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

A magnitude
A direction.


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:

Vector.png

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.


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

  • Results about vectors can be found here.


Sources