Introduction to Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates which are signed distances from the point to two fixed perpendicular directed lines, which are measured in the same unit of length. Each reference line is called coordinate axis or just axis of the system and the point where they meet is its origin, generally at ordered pair (0,0). The coordinates can also defined as the positions of the perpendicular projections of the point onto the two axes expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to 3 mutually perpendicular planes (or equivalently by its perpendicular projection onto three mutually perpendicular lines). Generally, one can specify a point in a space of any dimension n by use of n Cartesian coordinates the signed distances from n mutually perpendicular hyperplane.
Cartesian formulas for the plane
Distance between two points
The Euclidean distance between two points of the plane with Cartesian coordinates (x1, y1) and (x2,y2) is:
This is the Cartesian version of Pythagoras' theorem. In 3-dimensional space, the distance between points (x1, y1, z1) and (x2, y2, z2) is
which can be obtained by 2 consecutive applications of Pythagoras' theorem
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