The study of mathematics leads to constant enrichment and an increase in the variety of modeling tools for objects and environmental phenomena. Thus, the expansion of the concept of number allows us to present a quantitative characteristic of environmental objects, and with the help of new classes of geometric figures it turns out to describe the diversity of their forms. But the development of the 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 cannot be characterized only by numbers, because the direction of their action is also important. And due to the fact that directed segments characterize both directions and numerical values, on this basis a new concept of mathematics is obtained - the concept of a vector.
The implementation of basic mathematical operations on them was also determined for physical reasons, and this ultimately led to the founding of 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 generalization became a very convenient language, as well as a means of obtaining and determining new results.
What is a vector?
A vector is a set of all directed segments having the same length and a given direction. Each of the segments of this population is called a vector image.
It is clear that the vector is indicated by its image. All directional segments that depict vector a have the same length and direction, which are called the length (absolute value, absolute value) and direction of the vector, respectively. Its length is indicated 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 is point B, is uniquely characterized by an ordered pair of points (A; B). Consider also the many pairs (A; A), (B; B) .... This set denotes a vector that is called zero and denoted by 0 . The image of the zero vector is any point. The absolute value of the zero vector is considered equal to zero. The concept of the direction of the zero vector is not defined.
For any nonzero vector, a vector is determined that is opposite to the given one, 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, which has many common properties with the usual "numerical" algebra (although, of course, there are significant differences).
The addition of two vectors (noncollinear) is carried out according to the triangle rule (place the beginning of vector b at the end of vector a , then vector a + b connects the beginning of vector a with the end of vector b ) or a parallelogram (put the beginnings of vectors a and b at one point, then vector a + b , having a beginning at the same point, is the diagonal of a parallelogram, which is built on the vectors a and b ). The addition of vectors (several) can be done 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 is the addition of the corresponding coordinates, for example, if a = (x1; y1) and b = (x2; y2), then a + b = (x1 + x2 ; y1 + y2).
The rule of addition of vectors has all the algebraic properties that are inherent in addition of numbers:
- From the rearrangement of the terms, the sum does not change:
a + b = b + a
The addition of vectors using this property follows from the parallelogram rule. Indeed, what difference does it make in what order to sum the vectors a and b if the diagonal of the parallelogram is still the same? - Associativity property:
(a + b) + c = a + (b + c). - Adding a vector of zero vector does not change anything:
a +0 = a
This is quite obvious if you imagine such an addition from the point of view of the rule of a triangle. - Each vector a has an opposite vector, denoted by - a; addition of vectors, positive and negative, will be equal to zero: a + (- a) = 0.