The polar coordinates of P

Suppose that Cartesian coordinates are taken with the same origin and the same unit of length, with positive x-axis along the directed line Ox. Then the Cartesian coordinates (x,y) of a point P can be found from (r,θ), by x =  rcosθ, y = rsinθ. Conversely, the polar coordinates can be found from the Cartesian coordinates by , and θ is such that

(Note that θ = tan−1(y/x) but this is not, of itself, sufficient to uniquely determine θ.) In certain circumstances, authors may allow r to be negative, in which case the polar coordinates (r,θ) give the same point as (–r,θ + π).

In mechanics, it is useful, when a point P has polar coordinates (r,θ), to define unit vectors er and e_θ by

where i and j are unit vectors in the directions of the positive x- and y-axes. Then er is a unit vector along OP in the direction of increasing r, and e_θ is a unit vector perpendicular to this in the direction of increasing θ. These vectors satisfy er∙e_θ =0 and er×e_θ =k, where k =  i×j. See radial and transverse components.