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 point, plane, line, motion. All of them together form what has long been known under the term "Euclidean space".
Euclidean space, whose definition is based on the concept of 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, the vector with coordinates (x;y) is quantitatively identical to the vector with coordinates (y;x), but opposite in direction.
Secondly, if the scalar product of a vector with itself is performed, then the result of this action will be positive. The only exception will be the case when the initial and final coordinates of this vector are equal to zero: in this case, its product with itself will also 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 vectors are multiplied by the same real number, their scalar product will also increase by the same factor.
If all these four conditions are met, we can say with confidence that we have a 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 a set of vectors with a scalar product defined according to the basic laws of geometry.
- Euclidean space will also be obtained if by vectors we mean a certain finite set of real numbers with a given formula describing their scalar sum or product.
- A special case of Euclidean space is 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 both from the first and the second factor of the scalar product, the result from this will not change in any way. Second, along with the distributivity of the first element of the scalarproduct, 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 vector subtraction. Finally, thirdly, when a vector is scalarly multiplied by zero, the result will also be zero.
Thus, the Euclidean space is the most important geometric concept used in solving problems with the mutual arrangement of vectors relative to each other, which is characterized by such a concept as the scalar product.
