quality:corruption, stylistic:tone
Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as r {\displaystyle \mathbf {r} }. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:

r = x i + y j, {\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j},}

where i = (1 0) {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}} and j = (0 1) {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}} 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 versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates (x, y, z) {\displaystyle (x,y,z)} can be written as: r = x i + y j + z k, {\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k},}

where i = (1 0 0), {\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},} j = (0 1 0), {\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},} and k = (0 0 1). {\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}

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 numbers to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (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 quaternions.