Editing Euclidean vector (section)

==Representations==
{{further|Vector representation}}
[[Image:vector from A to B.svg|class=skin-invert-image|right|200px|Vector arrow pointing from ''A'' to ''B'']]

Vectors are usually denoted in [[lowercase]] boldface, as in <math>\mathbf{u}</math>''', <math>\mathbf{v}</math>''' and <math>\mathbf{w}</math>, or in lowercase italic boldface, as in '''''a'''''. ([[Uppercase]] letters are typically used to represent [[matrix (mathematics)|matrices]].) Other conventions include <math>\vec{a}</math> or <u>''a''</u>, especially in handwriting. Alternatively, some use a [[tilde]] (~) or a wavy underline drawn beneath the symbol, e.g. <math>\underset{^\sim}a</math>, which is a convention for indicating boldface type. If the vector represents a directed [[distance]] or [[displacement (vector)|displacement]] from a point ''A'' to a point ''B'' (see figure), it can also be denoted as <math>\stackrel{\longrightarrow}{AB}</math> or <u>''AB''</u>. In [[German language|German]] literature, it was especially common to represent vectors with small [[fraktur]] letters such as <math>\mathfrak{a}</math>.

Vectors are usually shown in graphs or other diagrams as arrows (directed [[line segment]]s), as illustrated in the figure. Here, the point ''A'' is called the ''origin'', ''tail'', ''base'', or ''initial point'', and the point ''B'' is called the ''head'', ''tip'', ''endpoint'', ''terminal point'' or ''final point''. The length of the arrow is proportional to the vector's [[magnitude (mathematics)|magnitude]], while the direction in which the arrow points indicates the vector's direction.

[[Image:Notation for vectors in or out of a plane.svg|class=skin-invert-image|right|200px]]
On a two-dimensional diagram, a vector [[perpendicular]] to the [[plane (mathematics)|plane]] of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an [[arrow (weapon)|arrow]] head on and viewing the flights of an arrow from the back.

[[Image:Position vector.svg|class=skin-invert-image|thumb|right|A vector in the Cartesian plane, showing the position of a point ''A'' with coordinates (2, 3).]]
[[Image:3D Vector.svg|class=skin-invert-image|300px|right]]
In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an ''n''-dimensional Euclidean space can be represented as [[coordinate vector]]s in a [[Cartesian coordinate system]]. The endpoint of a vector can be identified with an ordered list of ''n'' real numbers (''n''-[[tuple]]). These numbers are the [[Cartesian coordinate|coordinates]] of the endpoint of the vector, with respect to a given [[Cartesian coordinate system]], and are typically called the ''[[scalar component]]s'' (or ''scalar projections'') of the vector on the axes of the coordinate system.

As an example in two dimensions (see figure), the vector from the origin ''O'' = (0, 0) to the point ''A'' = (2, 3) is simply written as
<math display=block>\mathbf{a} = (2,3).</math>

The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation <math>\overrightarrow{OA}</math> is usually deemed not necessary (and is indeed rarely used).

In ''three dimensional'' Euclidean space (or {{math|'''R'''<sup>3</sup>}}), vectors are identified with triples of scalar components:
<math display=block>\mathbf{a} = (a_1, a_2, a_3).</math>
also written,
<math display=block>\mathbf{a} = (a_x, a_y, a_z).</math>

This can be generalised to ''n-dimensional'' Euclidean space (or {{math|'''R'''<sup>''n''</sup>}}).
<math display=block>\mathbf{a} = (a_1, a_2, a_3, \cdots, a_{n-1}, a_n).</math>

These numbers are often arranged into a [[column vector]] or [[row vector]], particularly when dealing with [[matrix (mathematics)|matrices]], as follows:
<math display=block>\mathbf{a} =
  \begin{bmatrix}
    a_1\\
    a_2\\
    a_3\\
  \end{bmatrix} =
  [ a_1\ a_2\ a_3 ]^{\operatorname{T}}.
</math>

Another way to represent a vector in ''n''-dimensions is to introduce the [[standard basis]] vectors. For instance, in three dimensions, there are three of them:
<math display=block>{\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\mathbf e}_3 = (0,0,1).</math>
These have the intuitive interpretation as vectors of unit length pointing up the ''x''-, ''y''-, and ''z''-axis of a [[Cartesian coordinate system]], respectively. In terms of these, any vector '''a''' in {{math|'''R'''<sup>3</sup>}} can be expressed in the form:
<math display=block>\mathbf{a} = (a_1,a_2,a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1), \ </math>

or
<math display=block>\mathbf{a} = \mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3 = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3,</math>

where '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub> are called the '''[[vector component]]s''' (or '''vector projections''') of '''a''' on the basis vectors or, equivalently, on the corresponding Cartesian axes ''x'', ''y'', and ''z'' (see figure), while ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> are the respective [[scalar component]]s (or scalar projections).

In introductory physics textbooks, the standard basis vectors are often denoted <math>\mathbf{i},\mathbf{j},\mathbf{k}</math> instead (or <math>\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}</math>, in which the [[hat symbol]] <math>\mathbf{\hat{}}</math> typically denotes [[unit vector]]s). In this case, the scalar and vector components are denoted respectively ''a<sub>x</sub>'', ''a<sub>y</sub>'', ''a<sub>z</sub>'', and '''a'''<sub>''x''</sub>, '''a'''<sub>''y''</sub>, '''a'''<sub>''z''</sub> (note the difference in boldface). Thus,

<math display=block>\mathbf{a} = \mathbf{a}_x + \mathbf{a}_y + \mathbf{a}_z = a_x{\mathbf i} + a_y{\mathbf j} + a_z{\mathbf k}.</math>

The notation '''e'''<sub>''i''</sub> is compatible with the [[index notation]] and the [[summation convention]] commonly used in higher level mathematics, physics, and engineering.

=== {{anchor|Vector component|Decomposition}} Decomposition or resolution===
{{Further|Basis (linear algebra)}}
As explained [[#Representations|above]], a vector is often described by a set of vector components that [[#Addition and subtraction|add up]] to form the given vector. Typically, these components are the [[Vector projection|projections]] of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be ''decomposed'' or ''resolved with respect to'' that set.

[[Image:Surface normal tangent.svg|class=skin-invert-image|right|thumb|Illustration of tangential and normal components of a vector to a surface.]]

The decomposition or resolution<ref>[[Josiah Willard Gibbs|Gibbs, J.W.]] (1901). ''Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs'', by E.B. Wilson, Chares Scribner's Sons, New York, p. 15: "Any vector {{math|'''r'''}} coplanar with two non-collinear vectors {{math|'''a'''}} and {{math|'''b'''}} may be resolved into two components parallel to {{math|'''a'''}} and {{math|'''b'''}} respectively. This resolution may be accomplished by constructing the parallelogram ..."</ref> of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.

Moreover, the use of Cartesian unit vectors such as <math>\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}</math> as a [[Basis (linear algebra)|basis]] in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a [[cylindrical coordinate system]] (<math>\boldsymbol{\hat{\rho}}, \boldsymbol{\hat{\phi}}, \mathbf{\hat{z}}</math>) or [[spherical coordinate system]] (<math>\mathbf{\hat{r}}, \boldsymbol{\hat{\theta}}, \boldsymbol{\hat{\phi}}</math>). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively.

The choice of a basis does not affect the properties of a vector or its behaviour under transformations.

A vector can also be broken up with respect to "non-fixed" basis vectors that change their [[orientation (geometry)|orientation]] as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively ''normal'', and ''tangent'' to a surface (see figure). Moreover, the ''radial'' and ''[[tangential component]]s'' of a vector relate to the ''[[radius]] of [[rotation]]'' of an object. The former is [[Parallel (geometry)|parallel]] to the radius and the latter is [[Perpendicular|orthogonal]] to it.<ref>{{Cite web |url=http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.angacc.html |title=U. Guelph Physics Dept., "Torque and Angular Acceleration" |access-date=2007-01-05 |archive-date=2007-01-22 |archive-url=https://web.archive.org/web/20070122155954/http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.angacc.html |url-status=dead }}</ref>

In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a ''global'' coordinate system, or [[inertial reference frame]]).