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Euclidean space: concept, properties, signs

Back in 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 in aggregate form what has long been known by the term "Euclidean space".

The Euclidean space, whose definition is based on the scalar multiplication of vectors, is a particular case of a linear (affine) space that satisfies a number of requirements. First, the scalar product of the vectors is absolutely symmetric, that is, the vector with the coordinates (x; y) is quantitatively identical to the vector with the coordinates (y; x), however, it is opposite in direction.

Secondly, if the scalar product of the vector is produced with itself, the result of this action will be positive. The only exception is the case when the initial and final coordinate of this vector is zero: in this case, and its product with itself will be equal to zero.

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

In the event that all these four conditions are met, we can say with confidence that we have Euclidean space before us.

Euclidean space from the practical point of view can be characterized by the following concrete examples:

  1. The simplest case is the presence of a set of vectors with a scalar product defined by the basic laws of geometry.
  2. Euclidean space is obtained also in the case when 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 Euclidean space is the so-called zero space, which is obtained if the scalar length of both vectors is zero.

Euclidean space has a number of specific properties. First, the scalar multiplier can be taken out of parentheses both from the first and from the second co-factor of the scalar product, the result of this will not undergo any changes. Second, 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 the vectors, distributivity also occurs in the case of subtraction of vectors. Finally, thirdly, with the scalar multiplication of the vector by zero, the result will also be zero.

Thus, the Euclidean space is the most important geometric concept used in solving problems with relative positions of vectors relative to one another, for the characterization of which a notion is used such as a scalar product.

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