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Definition of a Cartesian Coordinate System

Through any space point O we draw three mutually perpendicular straight lines. Their accepted names are: axis Ox (axis of the abscissas), axis Oy (axis of the ordinates), axis Oz. Point O is called the origin of the coordinates. On these straight lines one gives unit vectors i (in the direction of axis Ox), j (in the direction of axis Oy), k (in the direction of axis Oz).

Let M be a space point. The point gives a vector OM; M1, M2, M3 are correspondingly the orthogonal projections of the point M on the axes Ox, Oy, Oz. Then

OM = OM1 + OM2 + OM3 = xi + yj + zk

The numbers x, y, z are called the coordinates of the point M or the vector OM. This coordinate system is called the Cartesian coordinate system or the rectangular coordinate system. Each the three numbers (x;y;z) give the only point M.

It follows that a Cartesian coordinate system states one-to-one correspondence between a space point set and an ordered three real number set.

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