Even at school, all students get acquainted with the concept of "Euclidean geometry", the main provisions of which are focused around several axioms based on such geometric elements as a point, plane, line, movement. All of them together form what has long been known under the term "Euclidean space."
Euclidean space, the definition of which is based on the statement on scalar multiplication of vectors, is a special case of a linear (affine) space that satisfies a number of requirements. First, the scalar product of vectors is absolutely symmetrical, that is, a vector with coordinates (x; y) is quantitatively identical to a vector with coordinates (y; x), but it is opposite in direction.
Secondly, if the scalar product of the vector with itself is produced, then the result of this action will be positive. The only exception will be the case when the initial and final coordinate of this vector is equal to zero: in this case, the product of it with itself will be equal to zero.
Thirdly, the scalar product is distributive, that is, it is possible to decompose one of its coordinates into the sum of two values, which will not entail any changes in the final result of scalar multiplication of vectors. Finally, fourthly, when the vectors are multiplied by the same real number, their scalar product will also increase by the same amount.
In the event that all these four conditions are satisfied, we can say with confidence that we have Euclidean space.
Euclidean space from a practical point of view can be characterized by the following specific examples:
- The simplest case is the presence of many vectors with a scalar product defined by the basic laws of geometry.
- Euclidean space will turn out in the case if by vectors we mean a certain finite set of real numbers with a given formula that describes their scalar sum or product.
- A special case of Euclidean space should be recognized as the so-called zero space, which is obtained if the scalar length of both vectors is equal to zero.
Euclidean space has a number of specific properties. Firstly, the scalar factor can be taken out of brackets from both the first and second factors of the scalar product, the result of this will not undergo any changes. Secondly, along with the distributivity of the first element of a scalar product, the distributivity of the second element also acts. In addition, in addition to the scalar sum of vectors, distributivity also takes place in the case of subtraction of vectors. Finally, thirdly, with scalar multiplication of the vector by zero, the result will also be zero.
Thus, Euclidean space is the most important geometric concept used in solving problems with the mutual arrangement of vectors relative to each other, for the characterization of which such a concept as a scalar product is used.