Cartesian Coordinate System: Algebraic Equations

Cartesian Coordinate System

-A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Cartesian coordinate system is a way of locating objects in either two- or three-dimensional space by specifying their X(horizontal) position, Y (vertical) position and Z (through) position. It is used in graphics and in positioning text on documents.

-Algebraic equations involving the coordinates of the points lying on the shape. For example, a circle o f radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

A system
For instance, in Figure 1 two lines intersect each other at right angles in the point 0, the origin. One axis is the line O-X, the other O-Y and any point in the plane can be denoted by two numbers giving it's perpendicular distances from O-X and from O-Y.

A general point P can reached by traveling a distance x along a line O-X, and then a distance y along a line parallel to O-Y. O-X is called the x-axis, O-Y the y-axis, and the point P is said to have Cartesian coordinates (x, y). In the coordinate system shown, as is indicated in the diagram, the x-coordinate is positive for points above the right of the y-axis and negative for points to the left of this axis. The y-coordinate is positive for points above the x- axis and negative points below it. The coordinates of the origin are (0,
They intersect in a point O, again called origin. The lines are x-, y-, and z-axis. As in two-dimensional case, the axes consist of two half-lines: a positive and a negative part of the axis. The frame is right handed, if we rotate the positive x-axis to the positive y-axis the rotation direction (by the corkscrew rule) is the direction of the positive z-axis. In older literature ans\d some special application one may find a left-handed Cartesian set of axes, in which the x- and the y-axis are interchanged.
Generalization

One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not