Definition 1. Non-empty set ${\mathbf{E}}^{n}\left(R\right)$ 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 $\left|XY\right|$ - 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 $\left({x}_{1}\mathrm{,...,}{x}_{n}\right)$ and point Y n-tuple $\left({y}_{1}\mathrm{,...,}{y}_{n}\right)$, then
$$\left|XY\right|=\underset{i=1}{\overset{n}{\sum \sqrt{{({x}_{i}-{y}_{i})}^{2}}}}$$ |
Distance of points $\left|XY\right|$ in the Euclidean space satisfies some important properties.
The one-to-one mapping as defined in the Definition 1. is called the Cartesian coordinate system in the n-dimensional Euclidean space ${\mathbf{E}}^{n}\left(R\right)$. Origin of the coordinate system is the point attached the n-tuple consisting from zeros only.
Line is one-dimensional Euclidean space ${\mathbf{E}}^{1}\left(R\right)$, plane is two-dimensional Euclidean space ${\mathbf{E}}^{2}\left(R\right)$ and ${\mathbf{E}}^{3}\left(R\right)\phantom{\rule{2mm}{2mm}}$is three-dimensional Euclidean space.
In the Euclidean plane ${\mathbf{E}}^{2}\left(R\right)$, the Cartesian coordinate system $\left(O\mathrm{,}\phantom{\rule{2mm}{2mm}}x\mathrm{,}\phantom{\rule{2mm}{2mm}}y\right)$ can be defined, while:
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 $M=\left({x}_{M}\mathrm{,}\phantom{\rule{2mm}{2mm}}{y}_{M}\right)$ called Cartesian coordinates of the point in plane.
The two coordinates ${x}_{M}\mathrm{,}\phantom{\rule{2mm}{2mm}}{y}_{M}$ determine distances of he point M from the coordinate axes y and x respectively.
In the Euclidean space ${\mathbf{E}}^{3}\left(R\right)$, the Cartesian coordinate system $\left(O\mathrm{,}\phantom{\rule{2mm}{2mm}}x\mathrm{,}\phantom{\rule{2mm}{2mm}}y\mathrm{,}\phantom{\rule{2mm}{2mm}}z\right)$ can be defined, while:
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 ${x}_{M}\mathrm{,}\phantom{\rule{2mm}{2mm}}{y}_{M}\mathrm{,}\phantom{\rule{2mm}{2mm}}{z}_{M}$ determine distances of he point M from the coordinate planes $\mu \mathrm{,}\phantom{\rule{2mm}{2mm}}\nu \phantom{\rule{2mm}{2mm}}\mathrm{and}\phantom{\rule{2mm}{2mm}}\pi $ respectively.