At school, all students are introduced to the concept of "Euclidean geometry", the main provisions of which are focused around a few axioms based on geometric elements such as points, planes, straight line movement.All of them together form what is already known by the term "Euclidean space".
Euclidean space, the definition of which is based on the position of the scalar multiplication of vectors is a special case of a linear (affine) space, which satisfies a number of requirements.Firstly, scalar product perfectly symmetrical, i.e. the vector with coordinates (x; y) in terms of quantity is identical to the vector coordinates (y; x), but opposite in direction.
Secondly, in the event that produced the scalar product of the vector with itself, the result of this action will be positive.The only exception would be the case when the initial and final coordinates of this vector is equal to zero: in this case, and his work with himself the same will be zero.
Third, there is a scalar product is distributive, ie the possibility of expanding one of its coordinates on the sum of the two values, which does not entail any change in the final result of the scalar multiplication of vectors.Finally, in the fourth, with the multiplication of vectors by the same real number of their scalar product is also increased by the same factor.
In that case, if all four of these conditions, we can safely say that this is 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 plurality of vectors determined from the basic laws of geometry of the inner product.
- Euclidean space and in turn if the vectors for we understand some finite set of real numbers with a given formula which describes the scalar sum or product.
- particular case of Euclidean space is necessary to recognize the so-called zero space, which is obtained when the scalar length of both vectors is zero.
Euclidean space has a number of specific properties.First, the scalar factor can be taken out of the brackets from both the first and the second factor of the scalar product, the result of this will not undergo any changes.Secondly, along with the distributed first element scalar product works and Distributivity second element.In addition to the scalar sum of vectors Distributivity occurs in case of subtraction of vectors.Finally, in the third, when the scalar multiplication of vectors to zero, the result will be zero.
Thus Euclidean space - is the most important geometric concept used in solving problems with the mutual arrangement of the vectors relative to each other, which is used to characterize such a thing as a scalar product.
