Editing Position (geometry) (section)

===''n'' dimensions===

[[Linear algebra]] allows for the abstraction of an ''n''-dimensional position vector. A position vector can be expressed as a linear combination of [[basis (linear algebra)|basis]] vectors:<ref>{{cite book |title=Mathematical methods for physics and engineering |url=https://archive.org/details/mathematicalmeth00rile |url-access=registration |first1=K. F. |last1=Riley |first2=M. P.|last2=Hobson |first3=S. J. |last3=Bence | publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref><ref>{{cite book |last1=Lipschutz |first1=S. |url=https://archive.org/details/linearalgebra0000lips_a2h3 |title=Linear Algebra |last2=Lipson |first2=M. |publisher=McGraw Hill |year=2009 |isbn=978-0-07-154352-1 |url-access=registration}}</ref>

:<math>\mathbf{r} = \sum_{i=1}^n x_i \mathbf{e}_i = x_1 \mathbf{e}_1 + x_2 \mathbf{e}_2 + \dotsb + x_n \mathbf{e}_n. </math>

The [[set (mathematics)|set]] of all position vectors forms [[position space]] (a [[vector space]] whose elements are the position vectors), since positions can be added ([[vector addition]]) and scaled in length ([[scalar multiplication]]) to obtain another position vector in the space. The notion of "space" is intuitive, since each ''x<sub>i</sub>'' (''i'' = 1, 2, …, ''n'') can have any value, the collection of values defines a point in space.

The ''[[dimension (linear algebra)|dimension]]'' of the position space is ''n'' (also denoted dim(''R'') = ''n''). The ''[[coordinates]]'' of the vector '''r''' with respect to the basis vectors '''e'''<sub>''i''</sub> are ''x''<sub>''i''</sub>. The vector of coordinates forms the [[coordinate vector]] or ''n''-[[tuple]] (''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x<sub>n</sub>'').

Each coordinate ''x<sub>i</sub>'' may be parameterized a number of [[parameter]]s ''t''. One parameter ''x<sub>i</sub>''(''t'') would describe a curved 1D path, two parameters ''x<sub>i</sub>''(''t''<sub>1</sub>, ''t''<sub>2</sub>) describes a curved 2D surface, three ''x<sub>i</sub>''(''t''<sub>1</sub>, ''t''<sub>2</sub>, ''t''<sub>3</sub>) describes a curved 3D volume of space, and so on.

The [[linear span]] of a basis set ''B'' = {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>,  …, '''e'''<sub>''n''</sub>} equals the position space ''R'', denoted span(''B'') = ''R''.