Editing Cartesian coordinate system (section)

===Three dimensions===
{{Further|Three-dimensional space}}
[[File:Coord system CA 0.svg|thumb|A three dimensional Cartesian coordinate system, with origin ''O'' and axis lines ''X'', ''Y'' and ''Z'', oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates {{math|1=''x'' = 2}}, {{math|1=''y'' = 3}}, and {{math|1=''z'' = 4}}, or {{math|(2, 3, 4)}}.]]

A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the ''axes'') that go through a common point (the ''origin''), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point ''P'' of space, one considers a plane through ''P'' perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of ''P'' are those three numbers, in the chosen order. The reverse construction determines the point ''P'' given its three coordinates.

Alternatively, each coordinate of a point ''P'' can be taken as the distance from ''P'' to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.

Each pair of axes defines a ''coordinate plane''. These planes divide space into eight ''[[octant (solid geometry)|octants]]''. The octants are:

<math display=block>
\begin{align}
(+x,+y,+z) && (-x,+y,+z) && (+x,-y,+z) && (+x,+y,-z) \\
(+x,-y,-z) && (-x,+y,-z) && (-x,-y,+z) && (-x,-y,-z)
\end{align}
</math>

The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in {{math|(3, −2.5, 1)}} or {{math|(''t'', ''u'' + ''v'', ''π''/2)}}. Thus, the origin has coordinates {{math|(0, 0, 0)}}, and the unit points on the three axes are {{math|(1, 0, 0)}}, {{math|(0, 1, 0)}}, and {{math|(0, 0, 1)}}.

Standard names for the coordinates in the three axes are ''abscissa'', ''ordinate'' and ''applicate''.<ref>{{Cite web |title=Cartesian coordinates |url=https://planetmath.org/cartesiancoordinates |access-date=2024-08-25 |website=planetmath.org}}</ref> The coordinates are often denoted by the letters ''x'', ''y'', and ''z''. The axes may then be referred to as the ''x''-axis, ''y''-axis, and ''z''-axis, respectively. Then the coordinate planes can be referred to as the ''xy''-plane, ''yz''-plane, and ''xz''-plane.

In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called ''height'' or ''altitude''. The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point {{math|(0, 0, 1)}}; a convention that is commonly called ''the [[right-hand rule]]''.

[[File:Cartesian coordinate surfaces.png|thumb| The [[Coordinate system#Coordinate surface|coordinate surfaces]] of the Cartesian coordinates {{math|(''x'', ''y'', ''z'')}}. The ''z''-axis is vertical and the ''x''-axis is highlighted in green. Thus, the red plane shows the points with {{math|1=''x'' = 1}}, the blue plane shows the points with {{math|1=''z'' = 1}}, and the yellow plane shows the points with {{math|1=''y'' = −1}}. The three surfaces intersect at the point ''P'' (shown as a black sphere) with the Cartesian coordinates {{math|(1, −1, 1}}).]]