Vector. Adding vectors

The study of mathematics leads to a constant enrichment and an increase in the variety of means for modeling objects and phenomena of the environment. Thus, the expansion of the concept of number allows us to present a quantitative characteristic of the objects of the environment, with the help of new classes of geometric figures it is possible to describe the diversity of their forms. But the development of natural sciences and the demands of mathematics itself require the introduction and study of new and new modeling tools. In particular, a large number of physical quantities can not be characterized only by numbers, because the direction of their action is also important. And due to the fact that the directed segments characterize the directions, numerical values, then on this basis a new concept of mathematics - the concept of a vector - has turned out.

The fulfillment of the basic mathematical actions on them was also determined by physical considerations, and this eventually led to the foundation of the vector algebra, which now plays a huge role in the formation of physical theories. At the same time, in mathematics, this kind of algebra and its generalizations have become very convenient language, as well as a means of obtaining and determining new results.

What is a vector?

A vector is the set of all directed segments having the same length and a given direction. Each of the segments of this set is called the image of the vector.

It is clear that the vector is denoted by its image. All the directed segments that represent the vector a have the same length and direction, which are called, respectively, the length (modulus, absolute value) and direction of the vector. Its length is denoted by IaI . Two vectors are called equal if they have the same direction and the same length.

A directed segment, the beginning of which is point A, and the end point B, is uniquely characterized by an ordered pair of points (A; B). We also consider the set of pairs (A; A), (B; B) .... This set designates a vector, which is called zero and is denoted by 0 . The image of the zero vector is any point. The zero-vector module is considered to be zero. The concept of the direction of the zero vector is not defined.

For any nonzero vector, a vector is defined that is opposite to a given vector, that is, one that has the same length but the opposite direction. Vectors having the same or opposite directions are called collinear.

The possibilities of using vectors are associated with the introduction of actions on vectors and the creation of a vector algebra that has many common properties with the usual "numerical" algebra (although, of course, there are also significant differences).

The addition of two vectors (noncollinear) is performed according to the rule of the triangle (we place the beginning of the vector b at the end of the vector a , then the vector a + b connects the origin of the vector a to the end of the vector b ) or the parallelogram (we place the vectors a and b in the same point, then the vector a + B , having a beginning at the same point, is the diagonal of the parallelogram, which is constructed on the vectors a and b ). Adding vectors (several) can be performed using the polygon rule. If the terms are collinear, then the corresponding geometric constructions are reduced.

Operations with vectors that are given by coordinates are reduced to operations with numbers: addition of vectors - addition of corresponding coordinates, for example, if a = (x1; y1), and b = (x2; y2), then a + b = (x1 + x2 ; Y1 + y2).

The rule for adding vectors has all the algebraic properties that are inherent in the addition of numbers:

1. From the permutation of the terms, the sum does not change:
   A + b = b + a
   The addition of vectors by means of this property follows from the parallelogram rule. Indeed, what difference does it make in which order to sum the vectors a and b, if the diagonal of the parallelogram is still the same?
2. Associativity property:
   (A + b) + c = a + (b + c).
3. The addition to the vector of the zero vector does not change anything:
   A +0 = a
   This is quite obvious if we imagine such an addition from the point of view of the rule of a triangle.
4. Each vector a has an opposite vector, denoted by - a; The addition of vectors, positive and negative, will be zero: a + (- a) = 0.