To the question: "What is the vector?" - usually answer: "This is a mathematical value that has a size and direction". Seeking to dispel such a common misconception and reveal the essence of the vector quantity, the well-known mechanic A. Minakov (1893-1954) gave an illustrative example in his lectures. The flows of cars are characterized by the magnitude and direction, therefore, according to the definition given here - they are vectors. Now imagine a crossroads of streets with one-way traffic. One passes 300, and another 400 cars per hour. We fold the vectors according to the rule of the parallelogram and get: every hour a 3002 + 4002 = 500 cars crashes into the building at the intersection corner, and only 200 pass the intersection without an accident.

To the question: what is the vector? - usually answer: this is a mathematical value that has a size and direction

Nonsense? Of course. Why? Yes, because addition by the rule of a parallelogram is an element of the definition of a vector. A vector is a mathematical quantity that has a size, is characterized by a direction, and adds up with a similar magnitude by the parallelogram rule. The last definition is the most important.

Therefore, as Minakov said figuratively, if two mathematical quantities come to you and say: "We are vectors!", They need to say: "Get well!" And if they are built according to the rule of the parallelogram, they are vectors, and if they do not, they will not.

A vector can be denoted by a small Latin letter with an arrow (sometimes a dash) above it. Another common way of writing is to select a character in bold type, for example, a.

The vector in geometry is naturally associated with the transfer (parallel transfer), which obviously clarifies the origin of its name (Latin vector, bearing). Indeed, every directed segment uniquely determines some parallel transfer of a plane or space: for example, a vector naturally defines a transfer for which the point A passes to the point B, also back, a parallel transfer at which B is transferred to A.

Vectors are widely used in geometry and applied sciences, where they are used to represent quantities having a direction (forces, velocities, etc.). Application of them simplifies a number of operations - for example, determination of angles between straight lines or segments, calculation of areas of figures.

When working with vectors, you often enter Cartesian coordinate system and in it determine the coordinates of the vector, expanding it along the basis vectors. The expansion in terms of the basis can be geometrically represented by the projections of the vector onto the coordinate axes. If the coordinates of the origin and the end are known, the coordinates of the vector itself are obtained by subtracting from the coordinates of the end of the vector of coordinates of its origin.