Cartesian coordinate system

Definition 1. Non-empty set En(R) is called the n-dimensional Euclidean space defined over the set of real numbers iff

1. any two points X,Y from the space can be attached a unique real number |XY| - their distance

2. there exists at least one one-to-one mapping of this set to the set of all ordered n-tuples of real numbers such, that if the point X is attached n-tuple (x1,...,xn) and point Y n-tuple (y1,...,yn), then

| X Y | = ( x i y i ) 2 i = 1 n

Distance of points |XY| in the Euclidean space satisfies some important properties.

  1. For any two points in the space En(R) the relation |XY|0 is true and |XY|=0,ifX=Y.
  2. For any two points in the space En(R) the equality |XY|=|YX| holds.
  3. For any three points in the space En(R) the triangular inequality |XY|+|YZ||XZ| holds.

The one-to-one mapping as defined in the Definition 1. is called the Cartesian coordinate system in the n-dimensional Euclidean space En(R). Origin of the coordinate system is the point attached the n-tuple consisting from zeros only.

Line is one-dimensional Euclidean space E1(R), plane is two-dimensional Euclidean space E2(R) and E3(R)is three-dimensional Euclidean space.

In the Euclidean plane E2(R), the Cartesian coordinate system (O,x,y) can be defined, while:

  1. O be an arbitrary fixed point called the origin of the coordinate system,
  2. x and y called coordinate axes be two perpendicular lines sharing the common point O.

obr1
Figure 1: Cartesian coordinates of point in plane

Choosing the positive orientation on the half-lines with the start point O and the measurement unit for the length on the coordinate axes x and y, any point M in the plane can be attached a unique ordered pair of real numbers M=(xM,yM) called Cartesian coordinates of the point in plane.

The two coordinates xM,yM determine distances of he point M from the coordinate axes y and x respectively.

In the Euclidean space E3(R), the Cartesian coordinate system (O,x,y,z) can be defined, while:

  1. O be an arbitrary fixed point called the origin of the coordinate system,
  2. x, y and z called coordinate axes be three perpendicular lines sharing the common point O, and forming 3 coordinate planes π=xy,ν=xz,μ=yz.

obr4
Figure 2: Cartesian coordinate system in space

Choosing the positive orientation on the half-lines with the start point O and the measurement unit for the length on the coordinate axes x, y and z, any point M in the space can be attached a unique ordered triple of real numbers

M = ( x M , y M , y M )

called Cartesian coordinates of the point in space. The three coordinates xM,yM,zM determine distances of he point M from the coordinate planes μ,νandπ respectively.