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.