structural:scholar_edition, linguistic:translation
Cartesian coordinates

Every Euclidean vector space has an orthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a basis (e 1, …, e n) {\displaystyle (e_{1},\dots,e_{n})} of unit vectors (‖ e i ‖ = 1 {\displaystyle \|e_{i}\|=1}) that are pairwise orthogonal (e i ⋅ e j = 0 {\displaystyle e_{i}\cdot e_{j}=0} for i ≠ j). More precisely, given any basis (b 1, …, b n), {\displaystyle (b_{1},\dots,b_{n}),} the Gram–Schmidt process computes an orthonormal basis such that, for every i, the linear spans of (e 1, …, e i) {\displaystyle (e_{1},\dots,e_{i})} and (b 1, …, b i) {\displaystyle (b_{1},\dots,b_{i})} are equal.

Given a Euclidean space E, a Cartesian frame is a set of data consisting of an orthonormal basis of E →, {\displaystyle {\overrightarrow {E}},} and a point of E, called the origin and often denoted O. A Cartesian frame (O, e 1, …, e n) {\displaystyle (O,e_{1},\dots,e_{n})} allows defining Cartesian coordinates for both E and E → {\displaystyle {\overrightarrow {E}}} in the following way.

The Cartesian coordinates of a vector v of E → {\displaystyle {\overrightarrow {E}}} are the coefficients of v on the orthonormal basis e 1, …, e n. {\displaystyle e_{1},\dots,e_{n}.} For example, the Cartesian coordinates of a vector v {\displaystyle v} on an orthonormal basis (e 1, e 2, e 3) {\displaystyle (e_{1},e_{2},e_{3})} (that may be named as (x, y, z) {\displaystyle (x,y,z)} as a convention) in a 3-dimensional Euclidean space is (α 1, α 2, α 3) {\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})} if v = α 1 e 1 + α 2 e 2 + α 3 e 3 {\displaystyle v=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}. As the basis is orthonormal, the i -th coefficient α i {\displaystyle \alpha _{i}} is equal to the dot product v ⋅ e i. {\displaystyle v\cdot e_{i}.}

The Cartesian coordinates of a point P of E are the Cartesian coordinates of the vector O P →. ({\displaystyle {\overrightarrow {OP}}.{\vphantom {\frac {(}{}}}}