Definition:Vector/Real Euclidean Space

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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, forms a real Euclidean space.

Hence a vector in $\R^n$ is defined as an element of the real Euclidean space $\R^n$.


This article is complete as far as it goes, but it could do with expansion.
In particular: While it is possible to identify a vector by a tuple in $\R^n$ we need to explain that the vector is not the point.
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This article, or a section of it, needs explaining.
In particular: Give some consideration as to whether we merge to, or just reference, Definition:Vector Quantity, which is the conventional "vector" in applied maths and physics in ordinary space. See talk -- this is already being thought about.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
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$\R^2$: Plane Vector

Consider the real Euclidean space $\R^2$.

A vector in $\R^2$ can be referred to as a plane vector.


$\R^3$: Space Vector

Consider the real Euclidean space $\R^3$.

A vector in $\R^3$ can be referred to as a space vector.


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.


Geometric Interpretation


This page or section has statements made on it that ought to be extracted and proved in a Theorem page.
In particular: see if something coherent can be made from this. If so, then it needs to be a result in its own right.
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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 $\overrightarrow {\tuple {0, 0} \tuple {x_1, x_2} }$


This article is complete as far as it goes, but it could do with expansion.
In particular: This could be the destination for the elaboration of a vector as the equivalence class of all line segments of given length and slope, as defined in one of the sources that are still to be processed. Work in progress.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
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When this work has been completed, you may remove this instance of {{Expand}} from the code.
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Comment


This page has been identified as a candidate for refactoring of medium complexity.
In particular: Move this into an interpretation of a vector as an equivalence class of line segments
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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.


Also see

  • Results about vectors can be found here.


Sources

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