Euclidean space: concept, properties, signs

Back in school, all students become familiar with the concept“Euclidean geometry”, the main provisions of which are focused around several axioms based on such geometric elements as point, plane, straight line, movement. All of them together form what has long been known by the term "Euclidean space."

Евклидово пространство, определение которого based on the position of scalar multiplication of vectors, is a special case of linear (affine) space that satisfies a variety 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 opposite in direction.

Во-вторых, в том случае, если производится scalar product of a vector with itself, 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 zero: in this case and its product with itself, the same will be zero.

Third, there is distributivity.scalar product, that is, the possibility of decomposition of one of its coordinates into the sum of two values, which does 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 amount.

In the event that all these four conditions are fulfilled, 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:

1. The simplest case is the presence of a set of vectors with a scalar product determined by the basic laws of geometry.
2. Euclidean space will turn out in the event thatif by vectors we mean a certain finite set of real numbers with a given formula describing their scalar sum or product.
3. A special case of the Euclidean space is the so-called zero space, which is obtained if the scalar length of both vectors is zero.

Euclidean space has a variety ofspecific properties. First, the scalar factor can be bracketed from both the first and the second factors of the scalar product, the result will not change. Secondly, along with the distributivity of the first element of the scalar product, the distributivity of the second element also acts. In addition, in addition to the scalar sum of vectors, distributivity occurs in the case of subtracting vectors. Finally, thirdly, with scalar multiplication of the vector by zero, the result will also be equal to zero.

Таким образом, евклидово пространство – это the most important geometric concept used in solving problems with the mutual arrangement of vectors relative to each other, to characterize which such a concept as the scalar product is used.