Definition:Vector (Euclidean Space)

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This page is about vectors in Euclidean space. For other uses, see Definition:Vector.


Definition

A vector is defined as an element of a vector space.

We have that $\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space.

Hence a vector in $\R^n$ is defined as any element of $\R^n$.


$\R^1$

As $\R$ forms a vector space, every real number is a vector.

Note that as $\R$ is a vector space over itself, every real number is also a scalar.

Hence a vector in $\R^n$ is sometimes imprecisely used to mean "a vector in $\R^n$, $n > 1$".


Geometric Interpretation

From the Cantor-Dedekind Hypothesis, the real number line $\R$ can be represented by an infinite straight line.

By the same token, a vector in $\R$ can be represented by a directed line segment.

Formally, a vector $\sequence {x_1}, x_1 \in \R$ is accurately represented by the set of all directed line segments having:

  • Direction dependent on whether $x_1 < 0$ or $x_1 > 0$

By convention, if only one axis is under consideration, the line is placed horizontally, such that segments oriented towards the right are positive, to the left negative.

Note that in such a context the zero vector can be interpreted as a directed line segment beginning and terminating at the same point.


$\R^2$

We have that $\R^2$ is a vector space.

Hence any ordered $2$-tuple of $2$ real numbers is a vector.


Geometric Interpretation

From the definition of the real number plane, we can represent the vector space $\R^2$ by points on the plane.

That is, every pair of coordinates $\tuple {x_1, x_2}$ can be uniquely defined by a point in the plane.

An arrow with base at the origin and terminal point $\tuple {x_1, x_2}$ is defined to have the length equal to the magnitude of the vector, and direction defined by the relative location of $\tuple {x_1, x_2}$ with the origin as the point of reference.

Each vector is then represented by the set of all directed line segments with:

  • Direction equal to the direction of $\overrightarrow {\tuple {0, 0} \tuple {x_1, x_2} }$


Vector Notation

Several conventions are found in the literature for annotating a general vector in a style that distinguishes it from a scalar, 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 $\bsx$ 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.

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.

Because of this method of rendition, some sources refer to vectors as arrows.


Comment

The reader should be aware that a vector in $\R^n$ is and only is an ordered $n$-tuple of $n$ real numbers. The geometric interpretations given above are only representations of vectors.

Further, the geometric interpretation of a vector is accurately described as the set of all line segments equivalent to a given directed line segment, rather than any particular line segment.


Sources