Editing Euclidean vector (section)

=== {{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]]).