study of mathematics leads to a constant increase in the enrichment and diversity of the modeling of objects and phenomena of the environment.Thus, the expansion of the concept of number allows to present a quantitative description of the objects of the environment, with new classes of geometrical figures obtained to describe the variety of their forms.But the development of science and mathematics itself requests require the introduction and study of new and emerging modeling tools.In particular, a large number of physical quantities can not be characterized only by the numbers, because it's important and the direction of their actions.And thanks to that characterize directed segments and areas, numerical values, then, on this basis, and obtain a new notion of mathematics - the concept of vector.
perform basic mathematical operations on them, too, defined by physical considerations, and this eventually led to the founding of vector algebra, which now carries a huge role in the formation of physical theories.At the same time, in mathematics, a kind of algebra and its generalizations have become a very convenient language and means of receipt and identification of new results.
What is a vector?
vector is called the set of all directed line segments of equal length and given direction.Each of the segments of this set is called a vector image.
It is clear that the vector is denoted by its image.All directed segments that represent a vector , have the same length and direction, which are called, respectively, in length (module, the absolute value) and the direction vector.Its length is designated IaI .Two vectors are said to be equal if they have the same direction and the same length.
directed segment, which is the beginning point A and end - point B, is uniquely characterized by an ordered pair of points (A, B).Consider also a plurality of pairs (A, A), (B; C) ....This set represents a vector, which is called zero and is denoted 0 .The image of the zero vector is any point.Module zero vector is assumed to be zero.The notion of the direction of the zero vector is not defined.
For any non-zero vector is determined, given the opposite, that is, one that has the same length, but in the opposite direction.Vectors that have the same or opposite directions, called collinear.
Possible applications of vectors associated with the introduction of actions on creation of vectors and vector algebra, which has many properties in common with the usual "number" algebra (although, of course, there are also significant differences).
Addition of two vectors (collinear) is carried out according to the rule of the triangle (place the origin of the vector b the end of the vector a , then the vector a + b connects the beginning of the vector a the end of the vector b ) or parallelogram (putstart vectors a and b at one point, then vector a + b , with the start at the same point, is the diagonal of a parallelogram, which is built on the vectors a and b ).Addition of vectors (a few) can be performed by using the rule of the polygon.If terms are collinear, the corresponding geometric design cut.
operations with vectors are specified coordinates are reduced to operations with numbers: addition of vectors - addition of the corresponding coordinates, for example, if a = (x1, y1) and b = (x2, y2), then a +b = (x1 + x2; y1 + y2).
rule of vector addition has all the algebraic properties, which are inherent to addition of numbers:
- From permutation sum is not changed:
a + b = b + a
Addition of vectors with this property should be the rule of the parallelogram.Indeed, what a difference in what order to summarize the vectors a and b, if the diagonal of a parallelogram is still the same? - associative:
(a + b) + c = a + (b + c). - Adding to the vector of the zero vector does not change anything:
a +0 = a
It is quite obvious if we imagine such addition in terms of the rules of the triangle. - Each vector a has the opposite vector, referred to - a;vector addition, positive and negative, will be equal to zero: a + (- a) = 0.
