Definition:Vector Quantity

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

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:

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.


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

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