Number Plane.Cartesian Coordinate System in the Plane and Space

A pair of numbers is usually described as two numbers, which are consider in a specific order (an ordered pair).The set of all pairs of real numbers is called number plane.

As for the set of all real numbers (or number scale) is a geometric model such as the direct-axis and the set of all pairs of real numbers (number plane) is a geometric model such as the coordinate plane.

Coordinate plane xOy is defined by two mutually perpendicular lines with a common coordinate origin O and the same scale. The point O is called the coordinate origin.

The horizontal line is called the abscissa axis or axis Ox and the vertical line is called ordinate axis or axis Oy. These axes form cartesian coordinate system in the plane.

Every point of plane xOy corresponds to the pair of numbers coordinates of this point relative to given coordinate system.

Let′s consider a orthogonal projection of a point M on the axis Ox and Oy, corresponding points are denoted as M_x and M_y. The point M_x has coordinate (abscissa) x, the point M_y has coordinate (ordinate) y. These two numbers, that are written in the order, is called coordinates of point M and we write M(x;y).

Coordinate axes divide coordinate plate on four quadrants, that are numbered with roman digits.

The points, which are lie on the axis O_x have ordinate y that equal zero; the points, which are lie on the axis Oy - abscissa  x that equal zero.

Analogically, we can bring cartesian coordinate system in the space. For this we take three mutually perpendicular lines with a common origin O and the same scale. Let′s draw the plate through each pair of these lines. The plate, that passes through the lines Ox and Oy is called the plate xOy and two others - the plate xOz and yOz. The point O is called coordinate origin; lines Ox, Oy and Oz are called coordinate axes; the plates xOy,xOz and yOz are called coordinate plates. Thus, the axis   Ox is called abscissa axis, the axis   Oy is called ordinate axis and the axis Oz is applicate axis.

Let′s take an arbitrary point M and will draw a plane through it, that are parallel to the plane yOz, then the built plane will pass the axis Ox at the point Mx.

The coordinate x of point M has number that equals to absolute value the length of segment (it will be positive if lies on the positive semi-axis and it will be negatine if lies on the negative semi-axis). Analogically, we can determine the coordinates y and z of point M. The point M with the coordinates x, y, z we will write as M (x;y;z), and x is called abscissa, y - ordinate, z is calelld applicate.

So, every point M in the space corresponds to the three numbers, that are taken in the same order-the coordinates of point M in the space.