Editing Cartesian coordinate system (section)

==Representing a vector in the standard basis==
A point in space in a Cartesian coordinate system may also be represented by a position [[Euclidean vector|vector]], which can be thought of as an arrow pointing from the origin of the coordinate system to the point.<ref>{{harvnb|Brannan|Esplen|Gray|1998|loc=Appendix 2, pp. 377–382}}</ref> If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as <math>\mathbf{r}</math>. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:

<math display=block> \mathbf{r} = x \mathbf{i} + y \mathbf{j},</math>

where <math>\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}</math> and <math>\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}</math> are [[unit vectors]] in the direction of the ''x''-axis and ''y''-axis respectively, generally referred to as the ''[[standard basis]]'' (in some application areas these may also be referred to as [[versor]]s). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates <math>(x,y,z)</math> can be written as:<ref>{{harvnb|Griffiths|1999}}</ref>
<math display=block> \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k},</math>

where <math>\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},</math> <math>\mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix},</math> and <math>\mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.</math>

There is no ''natural'' interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use [[complex number]]s to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates {{nowrap|(''x'', ''y'')}} with the complex number {{nowrap|1=''z'' = ''x'' + ''iy''}}. Here, ''i'' is the [[imaginary unit]] and is identified with the point with coordinates {{nowrap|(0, 1)}}, so it is ''not'' the unit vector in the direction of the ''x''-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three-dimensional cartesian space a similar identification can be made with a subset of the [[quaternion]]s.