What does vector mean? Vectors for dummies. Actions with vectors. Vector coordinates. The simplest problems with vectors. Dot product of vectors

Standard definition: “A vector is a directed segment.” This is usually the extent of a graduate’s knowledge about vectors. Who needs any “directional segments”?

But really, what are vectors and what are they for?
Weather forecast. “Wind northwest, speed 18 meters per second.” Agree, both the direction of the wind (where it blows from) and the magnitude (that is, the absolute value) of its speed matter.

Quantities that have no direction are called scalar. Mass, work, electric charge are not directed anywhere. They are characterized only by a numerical value - “how many kilograms” or “how many joules”.

Physical quantities that have not only an absolute value, but also a direction, are called vector quantities.

Speed, force, acceleration - vectors. For them, “how much” is important and “where” is important. For example, the acceleration of gravity is directed towards the surface of the Earth, and its value is 9.8 m/s 2. Impulse, electric field strength, magnetic field induction are also vector quantities.

You remember that physical quantities are denoted by letters, Latin or Greek. The arrow above the letter indicates that the quantity is vector:

Here's another example.
A car moves from A to B. The end result is its movement from point A to point B, that is, movement by a vector .

Now it’s clear why a vector is a directed segment. Please note that the end of the vector is where the arrow is. Vector length is called the length of this segment. Indicated by: or

Until now, we have worked with scalar quantities, according to the rules of arithmetic and elementary algebra. Vectors are a new concept. This is another class of mathematical objects. They have their own rules.

Once upon a time we didn’t even know anything about numbers. My acquaintance with them began in elementary school. It turned out that numbers can be compared with each other, added, subtracted, multiplied and divided. We learned that there is a number one and a number zero.
Now we are introduced to vectors.

The concepts of “more” and “less” for vectors do not exist - after all, their directions can be different. Only vector lengths can be compared.

But there is a concept of equality for vectors.
Equal vectors that have the same length and the same direction are called. This means that the vector can be transferred parallel to itself to any point in the plane.
Single is a vector whose length is 1. Zero is a vector whose length is zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in a rectangular coordinate system - the same one in which we draw function graphs. Each point in the coordinate system corresponds to two numbers - its x and y coordinates, abscissa and ordinate.
The vector is also specified by two coordinates:

Here the coordinates of the vector are written in parentheses - in x and y.
They are found simply: the coordinate of the end of the vector minus the coordinate of its beginning.

If the vector coordinates are given, its length is found by the formula

Vector addition

There are two ways to add vectors.

1 . Parallelogram rule. To add the vectors and , we place the origins of both at the same point. We build up to a parallelogram and from the same point we draw a diagonal of the parallelogram. This will be the sum of the vectors and .

Remember the fable about the swan, crayfish and pike? They tried very hard, but they never moved the cart. After all, the vector sum of the forces they applied to the cart was equal to zero.

2. The second way to add vectors is the triangle rule. Let's take the same vectors and . We will add the beginning of the second to the end of the first vector. Now let's connect the beginning of the first and the end of the second. This is the sum of the vectors and .

Using the same rule, you can add several vectors. We arrange them one after another, and then connect the beginning of the first to the end of the last.

Imagine that you are going from point A to point B, from B to C, from C to D, then to E and to F. The end result of these actions is movement from A to F.

When adding vectors and we get:

Vector subtraction

The vector is directed opposite to the vector. The lengths of the vectors and are equal.

Now it’s clear what vector subtraction is. The vector difference and is the sum of the vector and the vector .

Multiplying a vector by a number

When a vector is multiplied by the number k, a vector is obtained whose length is k times different from the length . It is codirectional with the vector if k is greater than zero, and opposite if k is less than zero.

Dot product of vectors

Vectors can be multiplied not only by numbers, but also by each other.

The scalar product of vectors is the product of the lengths of the vectors and the cosine of the angle between them.

Please note that we multiplied two vectors, and the result was a scalar, that is, a number. For example, in physics, mechanical work is equal to the scalar product of two vectors - force and displacement:

If the vectors are perpendicular, their scalar product is zero.
And this is how the scalar product is expressed through the coordinates of the vectors and:

From the formula for the scalar product you can find the angle between the vectors:

This formula is especially convenient in stereometry. For example, in Problem 14 of the Profile Unified State Exam in Mathematics, you need to find the angle between intersecting lines or between a straight line and a plane. Problem 14 is often solved several times faster using the vector method than using the classical method.

In the school mathematics curriculum, only the scalar product of vectors is taught.
It turns out that, in addition to the scalar product, there is also a vector product, when the result of multiplying two vectors is a vector. Anyone who takes the Unified State Exam in physics knows what the Lorentz force and the Ampere force are. The formulas for finding these forces include vector products.

Vectors are a very useful mathematical tool. You will see this in your first year.

VECTOR
In physics and mathematics, a vector is a quantity that is characterized by its numerical value and direction. In physics, there are many important quantities that are vectors, for example, force, position, speed, acceleration, torque, momentum, electric and magnetic field strength. They can be contrasted with other quantities such as mass, volume, pressure, temperature and density, which can be described by an ordinary number and are called "scalars". Vector notation is used when working with quantities that cannot be completely specified using ordinary numbers. For example, we want to describe the position of an object relative to some point. We can tell how many kilometers an object is from a point, but we cannot fully determine its location until we know the direction in which it is located. Thus, the location of an object is characterized by a numerical value (distance in kilometers) and direction. Graphically, vectors are depicted as directed straight segments of a certain length, as in Fig. 1. For example, in order to graphically represent a force of five kilograms, you need to draw a straight line segment five units long in the direction of the force. The arrow indicates that the force is acting from A to B; if the force were acting from B to A, then we would write or For convenience, vectors are usually denoted in bold capital letters (A, B, C, and so on); vectors A and -A have equal numerical values, but opposite in direction. The numerical value of the vector A is called the modulus or length and is denoted A or |A|. This quantity is, of course, a scalar. A vector whose beginning and end coincide is called zero and is denoted by O.

Two vectors are called equal (or free) if their magnitudes and directions coincide. In mechanics and physics, however, this definition must be used with caution, since two equal forces applied to different points of the body will generally lead to different results. In this regard, vectors are divided into “connected” or “sliding”, as follows: Connected vectors have fixed points of application. For example, a radius vector indicates the position of a point relative to some fixed origin. Connected vectors are considered equal if they not only have the same modules and directions, but they also have a common point of application. Sliding vectors are vectors that are equal to each other and located on the same straight line.
Vector addition. The idea of ​​vector addition comes from the idea that we can find a single vector that has the same effect as two other vectors combined. If, in order to get to a certain point, we first need to walk A kilometers in one direction and then B kilometers in the other direction, then we could reach our final point by walking C kilometers in the third direction (Fig. 2). In this sense it can be said that

A + B = C.
The vector C is called the "resulting vector" of A and B, and is given by the construction shown in the figure; a parallelogram is constructed on vectors A and B as sides, and C is a diagonal connecting the beginning of A and the end of B. From Fig. 2 it is clear that the addition of vectors is “commutative”, i.e. A + B = B + A. In a similar way, you can add several vectors, sequentially connecting them in a “continuous chain”, as shown in Fig. 3 for three vectors D, E and F. From Fig. 3 it is also clear that

(D + E) + F = D + (E + F), i.e. addition of vectors is associative. Any number of vectors can be summed, and the vectors do not necessarily have to lie in the same plane. Subtraction of vectors is represented as addition with a negative vector. For example, A - B = A + (-B), where, as defined earlier, -B is a vector equal to B in magnitude, but opposite in direction. This addition rule can now be used as a real criterion for checking whether some quantity is a vector or not. Movements are usually subject to the conditions of this rule; the same can be said about speeds; the forces add up in the same way as could be seen from the “triangle of forces”. However, some quantities that have both numerical values ​​and directions do not obey this rule, and therefore cannot be considered as vectors. An example is finite rotations.
Multiplying a vector by a scalar. The product mA or Am, where m (m # 0) is a scalar and A is a nonzero vector, is defined as another vector that is m times longer than A and has the same direction as A if m is positive, and the opposite direction if m negative, as shown in Fig. 4, where m is 2 and -1/2, respectively. In addition, 1A = A, i.e. When multiplied by 1, the vector does not change. The quantity -1A is a vector equal to A in length but opposite in direction, usually written as -A. If A is a zero vector and/or m = 0, then mA is a zero vector. Multiplication is distributive, i.e.

We can add any number of vectors, and the order of the terms does not affect the result. The converse is also true: any vector can be decomposed into two or more “components”, i.e. into two or more vectors, which, when added, give the original vector as the result. For example, in Fig. 2, A and B are components of C. Many mathematical operations with vectors are simplified if the vector is decomposed into three components along three mutually perpendicular directions. Let us choose a right-handed Cartesian coordinate system with axes Ox, Oy and Oz as shown in Fig. 5. By right-handed coordinate system we mean that the x, y and z axes are positioned as the thumb, index and middle fingers of the right hand might be positioned respectively. From one right-handed coordinate system it is always possible to obtain another right-handed coordinate system by appropriate rotation. In Fig. 5, the decomposition of vector A into three components is shown and they add up to vector A, since

Hence,

One could also first add and get and then add to it. The projections of vector A onto the three coordinate axes designated Ax, Ay and Az are called the “scalar components” of vector A:

where a, b and g are the angles between A and the three coordinate axes. Now we introduce three vectors of unit length i, j and k (unit vectors) having the same direction as the corresponding axes x, y and z. Then, if Ax is multiplied by i, then the resulting product is a vector equal to and

Two vectors are equal if and only if their corresponding scalar components are equal. Thus, A = B if and only if Ax = Bx, Ay = By, Az = Bz. Two vectors can be added by adding their components:

In addition, according to the Pythagorean theorem:

Linear functions. The expression aA + bB, where a and b are scalars, is called a linear function of the vectors A and B. It is a vector in the same plane as A and B; if A and B are not parallel, then when a and b change, the vector aA + bB will move across the entire plane (Fig. 6). If A, B and C do not all lie in the same plane, then the vector aA + bB + cC (a, b and c changing) moves throughout space. Suppose A, B and C are the unit vectors of i, j and k. Vector ai lies on the x axis; the vector ai + bj can move throughout the xy plane; the vector ai + bj + ck can move throughout space.

One could choose four mutually perpendicular vectors i, j, k and l and define the four-dimensional vector as the quantity A = Axi + Ayj + Azk + Awl
with length

and one could continue to five, six, or any number of dimensions. Although it is impossible to visually represent such a vector, no mathematical difficulties arise here. Such a record is often useful; for example, the state of a moving particle is described by a six-dimensional vector P (x, y, z, px, py, pz), the components of which are its position in space (x, y, z) and momentum (px, py, pz). Such a space is called "phase space"; if we consider two particles, then the phase space is 12-dimensional, if there are three, then 18-dimensional, and so on. The number of dimensions can be increased unlimitedly; Moreover, the quantities we will be dealing with behave in much the same way as those we will consider in the rest of this article, namely, three-dimensional vectors.
Multiplying two vectors. The rule for adding vectors was derived by studying the behavior of quantities represented by vectors. There is no apparent reason why two vectors could not be multiplied in some way, but this multiplication will only make sense if it can be shown to be mathematically valid; in addition, it is desirable that the work have a certain physical meaning. There are two ways to multiply vectors that meet these conditions. The result of one of them is a scalar, such a product is called the “dot product” or “inner product” of two vectors and is written AÇB or (A, B). The result of another multiplication is a vector called the "cross product" or "outer product" and is written A*B or []. Dot products have a physical meaning for one, two, or three dimensions, whereas cross products are defined only for three dimensions.
Dot products. If, under the influence of some force F, the point to which it is applied moves a distance r, then the work done is equal to the product of r and the component of F in the direction of r. This component is equal to F cos bF, rc, where bF, rc is the angle between F and r, i.e. Work done = Fr cos bF, rs. This is an example of the physical justification of the scalar product defined for any two vectors A, B by means of the formula
A*B = AB cos bA, Bc.
Since all quantities on the right side of the equation are scalars, then A*B = B*A; therefore, scalar multiplication is commutative. Scalar multiplication also has the distributive property: A*(B + C) = A*B + A*C. If vectors A and B are perpendicular, then cos bA, Bc is zero, and therefore A*B = 0, even if neither A nor B are zero. This is why we cannot divide by a vector. Suppose we divided both sides of the equation A*B = A*C by A. This would give B = C, and if division could be done, then this equality would be the only possible result. However, if we rewrite the equation A*B = A*C as A*(B - C) = 0 and remember that (B - C) is a vector, then it is clear that (B - C) is not necessarily zero and, therefore, B must not be equal to C. These conflicting results show that vector division is not possible. The scalar product provides another way to write the numerical value (modulus) of a vector: A*A = AA*cos 0° = A2;
That's why

The scalar product can be written in another way. To do this, remember that: A = Ax i + Ayj + Azk. notice, that

Then,

Since the last equation contains x, y, and z as subscripts, the equation would seem to depend on the particular coordinate system chosen. However, this is not the case, as can be seen from the definition, which does not depend on the chosen coordinate axes.
Vector works. A vector or outer product of vectors is a vector whose modulus is equal to the product of their moduli by the sine of the angle perpendicular to the original vectors and together with them constitutes a right-hand triple. This product is most easily introduced by considering the relationship between velocity and angular velocity. The first is a vector; we will now show that the latter can also be interpreted as a vector. The angular velocity of a rotating body is determined as follows: select any point on the body and draw a perpendicular from this point to the axis of rotation. Then the angular velocity of the body is the number of radians by which this line rotates per unit time. If angular velocity is a vector, it must have a numerical value and a direction. The numerical value is expressed in radians per second, the direction can be chosen along the axis of rotation, it can be determined by directing the vector in the direction in which the right-hand propeller would move when rotating with the body. Consider the rotation of a body around a fixed axis. If we install this axis inside a ring, which in turn is attached to an axis inserted inside another ring, we can rotate the body inside the first ring with angular velocity w1 and then cause the inner ring (and body) to rotate with angular velocity w2. Figure 7 explains the point; circular arrows indicate the direction of rotation. This body is a solid sphere with center O and radius r.

Rice. 7. A SPHERE WITH CENTER O rotates with angular velocity w1 inside the ring BC, which, in turn, rotates inside the ring DE with angular velocity w2. The sphere rotates with an angular velocity equal to the sum of the angular velocities and all points on the straight line POP" are in a state of instantaneous rest.

Let's give this body a motion that is the sum of two different angular velocities. This movement is quite difficult to visualize, but it is quite obvious that the body no longer rotates about a fixed axis. However, we can still say that it rotates. To show this, let us choose a certain point P on the surface of the body, which at the moment of time we are considering is located on a great circle connecting the points at which two axes intersect the surface of the sphere. Let us drop perpendiculars from P to the axis. These perpendiculars will become the radii PJ and PK of the circles PQRS and PTUW respectively. Let's draw a straight line POPў passing through the center of the sphere. Now point P, at the moment in time under consideration, simultaneously moves along circles that touch at point P. Over a short time interval Dt, P moves a distance

This distance is zero if

In this case, the point P is in a state of instantaneous rest, and similarly all points on the straight line POP. The rest of the sphere will be in motion (the circles along which other points move do not touch, but intersect). POPў is therefore instantaneous axis of rotation of the sphere, just as a wheel rolling along the road at each moment of time rotates about its lowest point. What is the angular velocity of the sphere? For simplicity, let us choose point A at which the axis w1 intersects the surface. At the moment of time that we are considering , it moves in time Dt by a distance

In a circle of radius r sin w1. By definition, angular velocity

From this formula and relation (1) we obtain

In other words, if you write down a numerical value and choose the direction of the angular velocity as described above, then these quantities add up as vectors and can be considered as such. Now you can enter the cross product; Consider a body rotating with angular velocity w. Let us choose any point P on the body and any origin O, which is located on the axis of rotation. Let r be a vector directed from O to P. Point P moves in a circle with speed V = w r sin (w, r). The velocity vector V is tangent to the circle and points in the direction shown in Fig. 8.

This equation gives the dependence of the speed V of a point on the combination of two vectors w and r. We use this relationship to determine a new type of product and write: V = w * r. Since the result of such multiplication is a vector, this product is called a vector product. For any two vectors A and B, if A * B = C, then C = AB sin bA, Bc, and the direction of vector C is such that it is perpendicular to the plane passing through A and B and points in the direction coinciding with the direction of motion of the right-handed screw if it is parallel to C and rotates from A to B. In other words, we can say that A, B and C, arranged in this order, form a right-handed set of coordinate axes. The cross product is anticommutative; the vector B * A has the same modulus as A * B, but is directed in the opposite direction: A * B = -B * A. This product is distributive, but not associative; it can be proven that

Let's see how the vector product is written in terms of components and unit vectors. First of all, for any vector A, A * A = AA sin 0 = 0.
Therefore, in case of unit vectors, i * i = j * j = k * k = 0 and i * j = k, j * k = i, k * i = j. Then,

This equality can also be written as a determinant:

If A * B = 0, then either A or B is equal to 0, or A and B are collinear. Thus, as with the dot product, division by a vector is not possible. The value A * B is equal to the area of ​​a parallelogram with sides A and B. This is easy to see, since B sin bA, Bс is its height and A is its base. There are many other physical quantities that are cross products. One of the most important cross products appears in the theory of electromagnetism and is called the Pointing vector P. This vector is given by: P = E * H, where E and H are the electric and magnetic field vectors, respectively. Vector P can be thought of as a given energy flow in watts per square meter at any point. Let's give a few more examples: the moment of force F (torque) relative to the origin of coordinates acting on a point whose radius vector r is defined as r * F; a particle located at point r, with mass m and velocity V, has angular momentum mr * V relative to the origin; the force acting on a particle carrying an electric charge q through a magnetic field B with a speed V is qV * B.
Triple works. From three vectors we can form the following triple products: vector (A*B) * C; vector (A * B) * C; scalar (A * B)*C. The first type is the product of a vector C and a scalar A*B; We have already talked about such works. The second type is called the double cross product; the vector A * B is perpendicular to the plane where A and B lie, and therefore (A * B) * C is a vector lying in the plane of A and B and perpendicular to C. Therefore, in general, (A * B) * C is not equals A * (B * C). By writing A, B and C in terms of their coordinates (components) along the x, y and z axes and multiplying, we can show that A * (B * C) = B * (A*C) - C * (A*B). The third type of product, which arises in lattice calculations in solid state physics, is numerically equal to the volume of a parallelepiped with edges A, B, C. Since (A * B)*C = A*(B * C), the signs of scalar and vector multiplications can be swap places, and the piece is often written as (A B C). This product is equal to the determinant

Note that (A B C) = 0 if all three vectors lie in the same plane or if A = 0 or (and) B = 0 or (and) C = 0.
VECTOR DIFFERENTIATION
Suppose that the vector U is a function of one scalar variable t. For example, U could be the radius vector drawn from the origin to the moving point, and t could be the time. Let t change by a small amount Dt, which will lead to a change in U by the amount DU. This is shown in Fig. 9. The ratio DU/Dt is a vector directed in the same direction as DU. We can define the derivative of U with respect to t as

provided that such a limit exists. On the other hand, we can represent U as the sum of components along three axes and write

If U is the radius vector r, then dr/dt is the speed of the point expressed as a function of time. Differentiating with respect to time again, we get acceleration. Let's assume that the point moves along the curve shown in Fig. 10. Let s be the distance traveled by a point along a curve. During a small time interval Dt, the point will travel a distance Ds along the curve; the position of the radius vector will change to Dr. Therefore Dr/Ds is a vector directed like Dr. Further

Vector Dr - change in radius vector.

is a unit vector tangent to the curve. This can be seen from the fact that as point Q approaches point P, PQ approaches the tangent and Dr approaches Ds. The formulas for differentiating a product are similar to the formulas for differentiating the product of scalar functions; however, since the cross product is anticommutative, the order of multiplication must be preserved. That's why,

Thus, we see that if a vector is a function of one scalar variable, then we can represent the derivative in much the same way as in the case of a scalar function.
Vector and scalar fields. Gradient. In physics, you often have to deal with vector or scalar quantities that vary from point to point in a given region. Such areas are called "fields". For example, the scalar could be temperature or pressure; the vector can be the speed of a moving fluid or the electrostatic field of a system of charges. If we have chosen a certain coordinate system, then any point P (x, y, z) in a given area corresponds to a certain radius vector r (= xi + yj + zk) and also the value of the vector quantity U (r) or scalar f (r) associated with it. Let us assume that U and f are uniquely defined in the domain; those. Each point corresponds to one and only one U or f value, although different points can, of course, have different values. Let's say we want to describe the rate at which U and f change as we move through this area. Simple partial derivatives, such as dU/dx and df/dy, do not suit us, because they depend on the specifically chosen coordinate axes. However, it is possible to introduce a vector differential operator independent of the choice of coordinate axes; this operator is called a "gradient". Let us deal with a scalar field f. First, as an example, consider a contour map of a region of the country. In this case, f is the height above sea level; contour lines connect points with the same f value. When moving along any of these lines, f does not change; if you move perpendicular to these lines, then the rate of change of f will be maximum. We can associate with each point a vector indicating the magnitude and direction of the maximum change in speed f; such a map and some of these vectors are shown in Fig. 11. If we do this for each point in the field, we get a vector field associated with a scalar field f. This is the field of a vector called the "gradient" f, which is written as grad f or Cf (the symbol C is also called "nabla").

In the case of three dimensions, contour lines become surfaces. A small shift Dr (= iDx + jDy + kDz) leads to a change in f, which is written as

where the dots indicate terms of higher orders. This expression can be written as a scalar product

Let us divide the right and left sides of this equality by Ds, and let Ds tend to zero; Then

where dr/ds is the unit vector in the selected direction. The expression in parentheses is a vector depending on the selected point. Thus df/ds has a maximum value when dr/ds points in the same direction, the expression in parentheses being the gradient. Thus,

- a vector equal in magnitude and coinciding in direction with the maximum rate of change f relative to the coordinates. The gradient f is often written as

This means that operator C exists on its own. In many cases it behaves like a vector and is in fact a "vector differential operator" - one of the most important differential operators in physics. Despite the fact that C contains unit vectors i, j and k, its physical meaning does not depend on the chosen coordinate system. What is the relationship between Cf and f? First of all, suppose that f determines the potential at any point. For any small displacement Dr, the value of f will change by

If q is a quantity (for example, mass, charge) moved by Dr, then the work done when moving q by Dr is

Since Dr is displacement, then qСf is force; -Cf is the tension (force per unit quantity) associated with f. For example, let U be the electrostatic potential; then E is the electric field strength, given by the formula E = -CU. Let us assume that U is created by a point electric charge of q coulombs placed at the origin. The value of U at point P (x, y, z) with radius vector r is given by

Where e0 is the dielectric constant of free space. That's why

whence it follows that E acts in the direction r and its magnitude is equal to q/(4pe0r3). Knowing the scalar field, we can determine the vector field associated with it. The opposite is also possible. From the point of view of mathematical processing, scalar fields are easier to operate than vector ones, since they are specified by a single coordinate function, while a vector field requires three functions corresponding to the vector components in three directions. So the question arises: given a vector field, can we write down the associated scalar field?
Divergence and rotor. We saw the result of C acting on a scalar function. What happens when C is applied to a vector? There are two possibilities: let U(x, y, z) be a vector; then we can form the cross product and scalar product as follows:

The first of these expressions is a scalar called the divergence of U (denoted divU); the second is a vector called the rotor U (denoted rotU). These differential functions, divergence and curl, are widely used in mathematical physics. Imagine that U is some vector and that it and its first derivatives are continuous in some region. Let P be a point in this region surrounded by a small closed surface S bounding the volume DV. Let n be a unit vector perpendicular to this surface at every point (n changes direction as it moves around the surface, but always has unit length); let n point outward. Let's show that

Here S indicates that these integrals are taken over the entire surface, da is an element of the surface S. For simplicity, we will choose the shape S that is convenient for us in the form of a small parallelepiped (as shown in Fig. 12) with sides Dx, Dy and Dz; point P is the center of the parallelepiped. Let us calculate the integral from equation (4) first over one face of the parallelepiped. For the front face n = i (the unit vector is parallel to the x-axis); Da = DyDz. The contribution to the integral from the front face is equal to

On the opposite side n = -i; this face contributes to the integral

Using Taylor's theorem, we obtain the total contribution from the two faces

Note that DxDyDz = DV. In a similar way, you can calculate the contribution from the other two pairs of faces. The total integral is equal to

and if we set DV(r) 0, then the higher order terms disappear. According to formula (2), the expression in brackets is divU, which proves equality (4). Equality (5) can be proven in the same way. Let's use Fig. again. 12; then the contribution from the front face to the integral will be equal to

And, using Taylor’s theorem, we find that the total contribution to the integral from the two faces has the form

those. these are two terms from the expression for rotU in equation (3). The other four terms are obtained after taking into account the contributions from the other four faces. What do these ratios actually mean? Let's consider equality (4). Let's assume that U is the speed (of a fluid, for example). Then nНU da = Un da, where Un is the normal component of the vector U to the surface. Therefore, Un da ​​is the volume of liquid flowing through da per unit time, and is the volume of liquid flowing through S per unit time. Hence,

The rate of expansion of a unit volume around point P. This is where divergence gets its name; it shows the rate at which the fluid expands out of (i.e. diverges from) P. To explain the physical meaning of the rotor U, consider another surface integral over a small cylindrical volume of height h surrounding the point P; plane-parallel surfaces can be oriented in any direction we choose. Let k be the unit vector perpendicular to each surface, and let the area of ​​each surface be DA; then the total volume DV = hDA (Fig. 13). Let us now consider the integral

The integrand is the previously mentioned triple scalar product. This product will be zero on flat surfaces where k and n are parallel. On a curved surface

Where ds is the element of the curve as shown in Fig. 13. Comparing these equalities with relation (5), we obtain that

We still assume that U is the speed. In this case, what will be the average angular velocity of the fluid around k? It's obvious that

if DA is not 0. This expression is maximal when k and rotU point in the same direction; this means that rotU is a vector equal to twice the angular velocity of the fluid at point P. If the fluid is rotating relative to P, then rotU #0, and the U vectors will rotate around P. This is where the name rotor comes from. The divergence theorem (Ostrogradsky-Gauss theorem) is a generalization of formula (4) for finite volumes. It states that for some volume V bounded by a closed surface S,

And it is valid for all continuous vector functions U that have continuous first derivatives everywhere in V and on S. We will not give a proof of this theorem here, but its validity can be understood intuitively by imagining the volume V divided into cells. The flux U through a surface common to two cells vanishes, and only cells located on the boundary S will contribute to the surface integral. Stokes' theorem is a generalization of equation (6) for finite surfaces. She claims that

The content of the article

VECTOR. In physics and mathematics, a vector is a quantity that is characterized by its numerical value and direction. In physics, there are many important quantities that are vectors, for example, force, position, speed, acceleration, torque, momentum, electric and magnetic field strength. They can be contrasted with other quantities such as mass, volume, pressure, temperature and density, which can be described by an ordinary number, and are called “scalars”.

Vector notation is used when working with quantities that cannot be completely specified using ordinary numbers. For example, we want to describe the position of an object relative to some point. We can tell how many kilometers an object is from a point, but we cannot fully determine its location until we know the direction in which it is located. Thus, the location of an object is characterized by a numerical value (distance in kilometers) and direction.

Graphically, vectors are depicted as directed straight segments of a certain length, as in Fig. 1. For example, in order to graphically represent a force of five kilograms, you need to draw a straight line segment five units long in the direction of the force. The arrow indicates that the force acts from A To B; if the force acted from B To A, then we would write or . For convenience, vectors are usually denoted in bold capital letters ( A, B, C and so on); vectors A And - A have equal numerical values, but opposite in direction. Numerical value of the vector A called module or length and is designated A or | A|. This quantity is, of course, a scalar. A vector whose beginning and end coincide is called zero and is denoted O.

The two vectors are called equal(or free), if their modules and directions coincide. In mechanics and physics, however, this definition must be used with caution, since two equal forces applied to different points of the body will generally lead to different results. In this regard, vectors are divided into “connected” or “sliding”, as follows:

Related Vectors have fixed application points. For example, a radius vector indicates the position of a point relative to some fixed origin. Connected vectors are considered equal if they not only have the same modules and directions, but they also have a common point of application.

Sliding vectors Vectors that are equal to each other and located on the same straight line are called.

Vector addition.

The idea of ​​vector addition comes from the idea that we can find a single vector that has the same effect as two other vectors combined. If in order to get to a certain point, we need to go first A kilometers in one direction and then B kilometers in the other direction, then we could reach our final point by walking C kilometers in the third direction (Fig. 2). In this sense it can be said that

A + B = C.

Vector C is called the "resulting vector" A And B, it is given by the construction shown in the figure; on vectors A And B how a parallelogram is built on the sides, and C– diagonal connecting the beginning A and the end IN. From Fig. 2 it is clear that the addition of vectors is “commutative”, i.e.

A + B = B + A.

In a similar way, you can add several vectors, sequentially connecting them in a “continuous chain”, as shown in Fig. 3 for three vectors D, E And F. From Fig. 3 it is also clear that

(D + E) + F = D+ (E+ F),

those. addition of vectors is associative. Any number of vectors can be summed, and the vectors do not necessarily have to lie in the same plane. Subtraction of vectors is represented as addition with a negative vector. For example,

A – B = A + (–B),

where, as defined earlier, – B– vector equal to IN modulus, but opposite in direction.

This addition rule can now be used as a real criterion for checking whether some quantity is a vector or not. Movements are usually subject to the conditions of this rule; the same can be said about speeds; the forces add up in the same way as could be seen from the “triangle of forces”. However, some quantities that have both numerical values ​​and directions do not obey this rule, and therefore cannot be considered as vectors. An example is finite rotations.

Multiplying a vector by a scalar.

Work mA or Am, Where m (m No. 0) is a scalar, and A– non-zero vector, defined as another vector that is in m times longer A and has the same direction as A, if number m positive, and the opposite if m negative, as shown in Fig. 4, where m equal to 2 and –1/2 respectively. In addition, 1 A = A, i.e. When multiplied by 1, the vector does not change. Value –1 A– vector equal to A in length but opposite in direction, usually written as – A. If A– zero vector and/or m= 0, then mA– zero vector. Multiplication is distributive, i.e.

We can add any number of vectors, and the order of the terms does not affect the result. The converse is also true: any vector can be decomposed into two or more “components”, i.e. into two or more vectors, which, when added, give the original vector as the result. For example, in Fig. 2, A And B- Components C.

Many mathematical operations with vectors are simplified if the vector is decomposed into three components along three mutually perpendicular directions. Let us choose a right-handed Cartesian coordinate system with axes Ox, Oy And Oz as shown in fig. 5. By right-handed coordinate system we mean that the axes x, y And z positioned in the same way as the thumb, index and middle fingers of the right hand could be positioned respectively. From one right-handed coordinate system it is always possible to obtain another right-handed coordinate system by appropriate rotation. In Fig. 5, vector decomposition shown A into three components and . They add up to a vector A, because

You could also first add and get , and then add to .

Vector projections A on three coordinate axes designated A x, A y And A z are called the "scalar components" of the vector A:

Where a, b And g– angles between A and three coordinate axes. Now we introduce three vectors of unit length i, j And k(orts) having the same direction as the corresponding axes x, y And z. Then if A x multiply by i, then the resulting product is a vector equal to , and

Two vectors are equal if and only if their corresponding scalar components are equal. Thus, A= B then and only when A x = B x, A y = B y, A z = B z.

Two vectors can be added by adding their components:

In addition, according to the Pythagorean theorem:

Linear functions.

Expression aA + bB, Where a And b- scalars, called linear function vectors A And B. This is a vector located in the same plane as A And B; If A And B are not parallel, then when changing a And b vector aA + bB will move throughout the plane (Fig. 6). If A, B And C not all lie in the same plane, then the vector aA + bB + cC (a, b And c change) moves throughout the space. Let's pretend that A, B And C– unit vectors i, j And k. Vector ai lies on the axis x; vector ai + bj can move across the entire plane xy; vector ai + bj+ ck can move throughout the space.

One could choose four mutually perpendicular vectors i, j, k And l and define a four-dimensional vector as the quantity

A =A xi+ A yj+ Azk +Awl

and one could continue to five, six, or any number of dimensions. Although it is impossible to visually represent such a vector, no mathematical difficulties arise here. Such a record is often useful; for example, the state of a moving particle is described by a six-dimensional vector P(x, y, z, p x, p y, p z), the components of which are its position in space ( x, y, z) and impulse ( p x, p y, p z). Such a space is called "phase space"; if we consider two particles, then the phase space is 12-dimensional, if there are three, then 18-dimensional, and so on. The number of dimensions can be increased unlimitedly; Moreover, the quantities we will be dealing with behave in much the same way as those we will consider in the rest of this article, namely, three-dimensional vectors.

Multiplying two vectors.

The rule for adding vectors was derived by studying the behavior of quantities represented by vectors. There is no apparent reason why two vectors could not be multiplied in some way, but this multiplication will only make sense if it can be shown to be mathematically valid; in addition, it is desirable that the work have a certain physical meaning.

There are two ways to multiply vectors that meet these conditions. The result of one of them is a scalar, such a product is called the “dot product” or “inner product” of two vectors and is written A H B or ( A, B). The result of another multiplication is a vector called the "cross product" or "outer product" and is written Aґ B or [ A, B]. Dot products have a physical meaning for one, two, or three dimensions, whereas cross products are defined only for three dimensions.

Dot products.

If under the influence of some force F the point to which it is applied moves a distance r, then the work done is equal to the product r and components F in the direction r. This component is equal to F cos b F, r s , where b F, r c – angle between F And r, i.e.

Work done = Fr cos b F, r With .

This is an example of the physical justification of the scalar product defined for any two vectors A, B through the formula

AC B =AB cos b A, B With .

Since all quantities on the right side of the equation are scalars, then

A H B = B H A;

therefore, scalar multiplication is commutative.

Scalar multiplication also has the distributive property:

A H ( B + WITH) = A H B + A H WITH.

If the vectors A And B are perpendicular, then cos b A, B c is equal to zero, and therefore, A H B= 0, even if neither A,nor B are not equal to zero. This is why we cannot divide by a vector. Let's say we divide both sides of the equation A H B= A H C on A. This would give B= C, and if division could be performed, then this equality would be the only possible result. However, if we rewrite the equation A H B= A H C as A H ( B – C) = 0 and remember that ( B – C) is a vector, then it is clear that ( B – C) is not necessarily zero and therefore B should not be equal C. These conflicting results indicate that vector division is not possible.

The scalar product gives another way to write the numerical value (modulus) of a vector:

A H A =A.A.H cos 0° = A 2 ;

The scalar product can be written in another way. To do this, remember that:

A =A xi+ A yj+ Azk.

Since the last equation contains x, y And z as subscripts, the equation would seem to depend on the particular coordinate system chosen. However, this is not the case, as can be seen from a definition that is independent of the chosen coordinate axes.

Vector works.

A vector or outer product of vectors is a vector whose modulus is equal to the product of their moduli by the sine of the angle perpendicular to the original vectors and together with them constitutes a right-hand triple. This product is most easily introduced by considering the relationship between velocity and angular velocity. The first is a vector; we will now show that the latter can also be interpreted as a vector.

The angular velocity of a rotating body is determined as follows: select any point on the body and draw a perpendicular from this point to the axis of rotation. Then the angular velocity of the body is the number of radians by which this line rotates per unit time.

If angular velocity is a vector, it must have a numerical value and direction. The numerical value is expressed in radians per second, the direction can be chosen along the axis of rotation, it can be determined by directing the vector in the direction in which the right-hand propeller would move when rotating with the body.

Consider the rotation of a body around a fixed axis. If we install this axis inside a ring, which in turn is fixed to an axis inserted inside another ring, we can rotate the body inside the first ring with angular velocity w 1 and then cause the inner ring (and body) to rotate at angular velocity w 2.Figure 7 explains the essence of the matter; circular arrows indicate the direction of rotation. This body is a solid sphere with a center ABOUT and radius r.

Let's give this body a motion that is the sum of two different angular velocities. This movement is quite difficult to visualize, but it is quite obvious that the body no longer rotates about a fixed axis. However, we can still say that it rotates. To show this, let's choose some point P on the surface of the body, which at the moment of time we are considering is on a great circle connecting the points at which two axes intersect the surface of the sphere. Let's drop the perpendiculars from P on the axis. These perpendiculars will become radii P.J. And PK circles PQRS And PTUW respectively. Let's make a direct POPў passing through the center of the sphere. Now the point P, at the moment in time under consideration simultaneously moves along circles that touch at the point P. For a short time interval D t, P moves a distance

This distance is zero if

In this case, the point P is in a state of instantaneous rest, and in the same way all points on a straight line POPў. The rest of the sphere will be in motion (the circles along which other points move do not touch, but intersect). POPў is, therefore, the instantaneous axis of rotation of the sphere, just as a wheel rolling along the road at each moment of time rotates relative to its lowest point.

What is the angular velocity of the sphere? For simplicity, let’s choose a point A, in which the axis w 1 intersects the surface. At the moment of time that we are considering, it moves in time D t to a distance

in a circle of radius r sin w 1. By definition, angular velocity

From this formula and relation (1) we obtain

In other words, if you write down a numerical value and choose the direction of the angular velocity as described above, then these quantities add up as vectors and can be considered as such.

Now you can enter the cross product; consider a body rotating with angular velocity w. Let's choose any point P on the body and any origin ABOUT, which is located on the axis of rotation. Let r– vector directed from ABOUT To P. Dot P moves in a circle at speed

V = w r sin( w, r).

Speed ​​vector V is tangent to the circle and points in the direction shown in Fig. 8.

This equation gives the speed dependence V points from a combination of two vectors w And r. We use this relationship to determine a new type of product and write:

V= wґ r.

Since the result of such multiplication is a vector, this product is called a vector product. For any two vectors A And B, If

Aґ B= C,

C = AB sin b A, B With ,

and vector direction C such that it is perpendicular to the plane passing through A And B and points in the direction coinciding with the direction of movement of the clockwise rotating propeller, if it is parallel C and rotates from A To B. In other words, we can say that A, B And C, arranged in this order, form the right set of coordinate axes. The cross product is anticommutative; vector B ґ A has the same module as A ґ B, but directed in the opposite direction:

A ґ B = –B ґ A.

This work is distributive, but not associative; it can be proven that

Let's see how the vector product is written in terms of components and unit vectors. First of all, for any vector A,

A ґ A = A.A. sin 0 = 0.

Therefore, in the case of unit vectors,

iґ i=jґ j=kґ k=0

i ґ j=k, jґ k =i, kґ i=j.

This equality can also be written as a determinant:

If A ґ B = 0 , then either A or B equals 0 , or A And B collinear. Thus, as with the dot product, division by a vector is not possible. Magnitude A ґ B equal to the area of ​​a parallelogram with sides A And B. It's easy to see because B sin b A, B c – its height and A– foundation.

There are many other physical quantities that are cross products. One of the most important vector products appears in the theory of electromagnetism and is called the Pointing vector P. This vector is given as follows:

P = E ґ H,

Where E And H are the vectors of electric and magnetic fields, respectively. Vector P can be thought of as a given flow of energy in watts per square meter at any point. Let's give a few more examples: moment of force F(torque) relative to the origin acting on a point whose radius vector r, is defined as r ґ F; particle located at a point r, mass m and speed V, has angular momentum mr ґ V relative to the origin; force acting on a particle carrying an electric charge q through a magnetic field B with speed V, There is qV ґ B.

Triple works.

From three vectors we can form the following triple products: vector ( A H B) ґ C; vector ( Aґ B)ґ C; scalar ( Aґ B)H C.

The first type is the product of a vector C and scalar A H B; We have already talked about such works. The second type is called the double cross product; vector Aґ B perpendicular to the plane where they lie A And B, and therefore ( Aґ B)ґ C– vector lying in the plane A And B and perpendicular C. Therefore, in the general case, ( Aґ B)ґ C № Aґ (Bґ C). Having written down A, B And C through their coordinates (components) along the axes x, y And z and multiplying, we can show that Aґ (Bґ C) = Bґ (A H C) – Сґ ( A H B). The third type of product, which arises in lattice calculations in solid state physics, is numerically equal to the volume of a parallelepiped with edges A, B, C. Because ( Aґ B)H C = A H ( Bґ C), the scalar and vector multiplication signs can be swapped, and the product is often written as ( A B C). This product is equal to the determinant

Notice, that ( A B C) = 0 if all three vectors lie in the same plane or if A = 0 or/and IN = 0 or/and WITH = 0 .

VECTOR DIFFERENTIATION

Let's assume that the vector U is a function of one scalar variable t. For example, U can be a radius vector drawn from the origin to the moving point, and t- time. Let t will change by a small amount D t, which will lead to a change U by the value D U. This is shown in Fig. 9. Ratio D U/D t– vector directed in the same direction as D U. We can define the derivative U By t, How

provided that such a limit exists. On the other hand, one can imagine U as the sum of the components along three axes and write

If U– radius vector r, That dr/dt– point speed expressed as a function of time. Differentiating with respect to time again, we get acceleration. Let's assume that the point moves along the curve shown in Fig. 10. Let s– the distance traveled by a point along a curve. Over a short period of time D t the point will travel a distance D s along the curve; the position of the radius vector will change to D r. Therefore D r/D s– vector directed as D r. Further

is a unit vector tangent to the curve. This can be seen from the fact that as the point approaches Q to the point P, PQ approaches tangent and D r approaches D s.

The formulas for differentiating a product are similar to the formulas for differentiating the product of scalar functions; however, since the cross product is anticommutative, the order of multiplication must be preserved. That's why,

Thus, we see that if a vector is a function of one scalar variable, then we can represent the derivative in much the same way as in the case of a scalar function.

Vector and scalar fields.

Gradient.

In physics, you often have to deal with vector or scalar quantities that vary from point to point in a given region. Such areas are called "fields". For example, the scalar could be temperature or pressure; the vector can be the speed of a moving fluid or the electrostatic field of a system of charges. If we have chosen a certain coordinate system, then at any point P (x, y, z) in a given area corresponds to a certain radius vector r (= xi + yj + zk) and also the value of the vector quantity U(r) or scalar f(r) associated with it. Let's pretend that U And f are uniquely defined in the area; those. each point corresponds to one and only one value U or f, although different points can of course have different meanings. Let's say we want to describe the speed at which U And f change as you move around this area.

Simple partial derivatives such as ¶ U/¶x And ¶f/¶y, we are not satisfied with them because they depend on the specifically selected coordinate axes. However, it is possible to introduce a vector differential operator independent of the choice of coordinate axes; this operator is called a "gradient".

Let us deal with a scalar field f. First, as an example, consider a contour map of a region of the country. In this case f- height above sea level; contour lines connect points with the same value f. When moving along any of these lines f does not change; if you move perpendicular to these lines, then the rate of change f will be maximum. We can associate a vector with each point indicating the magnitude and direction of the maximum change in speed f; such a map and some of these vectors are shown in Fig. 11. If we do this for each point in the field, we get a vector field associated with a scalar field f. This is a vector field called a "gradient" f, which is written as grad f or with f (the symbol C is also called “nabla”).

In the case of three dimensions, contour lines become surfaces. Small offset D r (= i D x + j D y + k D z) leads to a change f, which is written as

where the dots indicate terms of higher orders. This expression can be written as a scalar product

Let us divide the right and left sides of this equality by D s, and let D s tends to zero; Then

Where dr/ds - unit vector in the selected direction. The expression in parentheses is a vector depending on the selected point. Thus, df/ds has maximum value when dr/ds points in the same direction, the expression in parentheses is the gradient. Thus,

– a vector equal in magnitude and coinciding in direction with the maximum rate of change f relative to coordinates. Gradient f often written as

This means that operator C exists on its own. In many cases it behaves like a vector, and is in fact a "vector differential operator" - one of the most important differential operators in physics. Despite the fact that C contains unit vectors i,j And k, its physical meaning does not depend on the chosen coordinate system.

What is the relationship between C f And f? First of all, let's assume that f determines the potential at any point. For any small displacement D r magnitude f will change to

If q– quantity (for example mass, charge) moved to D r, then the work done while moving q

whence it follows that E acts in the direction r and its value is equal q/(4pe 0r 3).

Knowing the scalar field, we can determine the vector field associated with it. The opposite is also possible. From the point of view of mathematical processing, scalar fields are easier to operate than vector ones, since they are specified by a single coordinate function, while a vector field requires three functions corresponding to the vector components in three directions. So the question arises: given a vector field, can we write down the associated scalar field?

Divergence and rotor.

We saw the result of C acting on a scalar function. What happens when C is applied to a vector? There are two possibilities: let U n a da D

if D A® 0. This expression is maximum when k and rot U point in the same direction; this means rot U– vector equal to twice the angular velocity of the fluid at a point P. If the fluid rotates relative to P, then rot U No. 0, and vectors U will revolve around P. This is where the name rotor came from.

Divergence theorem (Ostrogradsky–Gauss theorem)

The divergence theorem (Ostrogradsky–Gauss theorem) is a generalization of formula (4) for finite volumes. She claims that for a certain volume V, limited by a closed surface S,

and is valid for all continuous vector functions U, having continuous first derivatives everywhere in V and on S. We will not give a proof of this theorem here, but its validity can be understood intuitively by imagining the volume V divided into cells. Flow U through a surface common to two cells vanishes, and only cells located on the boundary S will contribute to the surface integral.

Stokes' theorem

is a generalization of equation (6) for finite surfaces. She claims that

Where C– closed curve and S– any surface bounded by this curve. U and its first derivatives must be continuous everywhere on S And C.

What is Vector? The meaning of the word “Vector” in popular dictionaries and encyclopedias, examples of the use of the term in everyday life.

Vector of Constructive Tension – Philosophical Dictionary

A necessary element of constructive tension that determines the orientation, direction of reproduction, personal culture, personality, its activities, communities at all stages of the social whole; brigades, enterprises, departments, etc. reproduction of subcultures corresponding to communities. V.K.S. is a necessary element of any dual opposition as an indicator of value orientation, built into any reproductive activity of the subject. Thus, not only is there a division of reality into good and evil, but also the need for the subject to strive for good and avoid evil. Dual opposition carries within itself positive and negative, direct and reverse V.K. By mastering the corresponding (sub)cultures, the individual thereby acquires a certain orientation in the fight against disorganization. Each of the cells of society is characterized by a certain specific orientation that is opposed to entropy and disorganization. In this regard, the most important problem in any society is the degree of coincidence of vectors at different levels of society, the degree of coincidence of V.K.N. individuals and organizations, teams and enterprises, etc. Any community can work normally if its inherent V.K. coincides with, does not significantly diverge from V.K.N. its members recreating its people. Otherwise, a sociocultural contradiction arises, giving rise to disorganization, which threatens both the growth of innovations above the level of novelty acceptable in a given subculture, and a decrease in social energy below the lower threshold.

Vector M. – Explanatory Dictionary by Efremova

1. A straight segment, characterized by a numerical value and a certain direction.

Expected Return Vector – Economic dictionary

a vector of numbers corresponding to expected returns for a given set of securities.

Vector of Ranks – Sociological Dictionary

– vector statistics constructed from a random vector of observations X = (X1, ... ,Xn) (see Vector), the components of which are obtained as follows. If all Xt are different, then the components of V.r. are natural numbers from 1 to n: in place of each Xi there is a number expressing the number of such components of the vector Xi, the value of which is less than the value of Xi. In other words, in the place of the largest Xi there is the number n, in the place of the next largest (in descending order) - (n-1), etc. In the place of the smallest there is 1. If certain X. are equal to each other , then V.r. is constructed as follows: the largest X is assigned rank n, the next largest is assigned rank (n-1), etc. until, after assigning rank (n-k), equal Xi are encountered. Let these be Xkl,...,Xkl. We assign a rank to each of them. The next largest Xkl 1 we assign a rank n-(to l 1), if it is not equal to any other component of X, and the rank of Yu.N. Tolstov

State Vector –

same as wave function.

Vectorcardiography – Psychological Dictionary

(vectorcardiography) - see Electrocardiography.

Vectorcardiography – Psychological Encyclopedia

Vectorcardiography – Medical dictionary

see Electrocardiography.

Vectormeter – Big Encyclopedic Dictionary

(from vector and...meter) - a device for measuring currents, voltages and phases of alternating current.

Vectormeter M. – Explanatory Dictionary by Efremova

1. An electrical instrument for measuring voltage or strength and phase of alternating current.

Vector Diagram – Big Encyclopedic Dictionary

graphical representation of the values ​​of physical quantities that change according to a harmonic law, and the relationships between them in the form of vectors. Used in calculations in electrical engineering, acoustics, optics, etc.

Vector Psychology – Sociological Dictionary

See FIELD THEORY.

Vector Psychology – Psychological Dictionary

See the discussion of Lewin's theory in the article vector(1).

Vector Psychology – Psychological Encyclopedia

Vector Calculus – Big Encyclopedic Dictionary

branch of mathematics in which operations with supervectors are studied. includes vector algebra and vector analysis. The rules of vector algebra reflect the properties of actions by supervector quantities. For example, the sum of vectors a and b is a vector going from the beginning of vector a to the end of vector b, provided that the beginning of vector b is applied to the end of vector a; this rule is related to the rule of addition of forces or velocities (see Parallelogram of forces). In vector calculus, two types of vector multiplication are established (see Dot product, Vector product). If i, j, k are three mutually perpendicular unit vectors in space, then any vector a can be uniquely represented in the form a=a1i+a2j+a3k. The numbers a1, a2, a3 are called components (coordinates) of vector a. Vector analysis is based on the operations of differentiation and integration of vector functions.

Vector Field – Big Encyclopedic Dictionary

a region at each point P of which a vector a(P) is specified. Many physical phenomena and processes lead to the concept of a vector field (for example, the velocity vectors of particles of a moving fluid at each moment of time form a vector field).

Vector Product – Big Encyclopedic Dictionary

vector a to vector b - vector p = VECTOR SPACE - a mathematical concept that generalizes the concept of the set of all vectors of a 3-dimensional space to the case of an arbitrary number of dimensions.

Vector Approach to Psychotherapy – Psychological Dictionary

(vector approach to psychotherapy) V. p. p. postulates that the entire variety of therapies is essentially distributed along 6 main lines. vectors, or modalities, indicating the direction of growth. Choosing one of many therapeutic methods, mainly On these vectors, the eclectically oriented therapist can achieve highly effective balanced therapeutic integration, as well as gain freedom to express their personal preferences and talents. Below is the classification. therapy methods based on these vectors. 1. Rational vector, characterized by insight, expansion of awareness and learning: a) psychoan; b) rational-emotive therapy; c) transactional analysis; d) behavioral therapy. 2. Neuromuscular vector, characterized by muscle tension, muscle relaxation and movement, accompanied by changes in breathing and release of emotions: a) Reichian therapy; b) bioenergy; c) rolfing; d) Alexander's method; e) Feldenkrais method; e) dance therapy. 3. Interpersonal vector, characterized by relationships between people: a) groups of meetings; b) psychodrama; c) joint family therapy; d) Gestalt therapy. 4. Vector of fantasy, characterized by intrapersonal experience when external stimulation is turned off: a) hypnotherapy; b) psychosynthesis; c) guided daydreams. 5. Transpersonal vector, characterized by the transcendence of the closed state of consciousness of the individual: a) spiritual healing; b) parapsychological phenomena; c) Jungian psychology; d) meditation. 6. Biochemical vector, characterized by chemical changes in the body that have internal or external origin: a) orthomolecular therapy; b) carbogen; c) dietary procedures and exercises; d) psychedelic and psycholytic drug therapy; e) sedatives, stimulants and tranquilizers. See also Innovative psychotherapies, Methods of psychotherapy by P. Bindrim

Finally, I got my hands on this vast and long-awaited topic. analytical geometry. First, a little about this section of higher mathematics... Surely you now remember a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “analytical solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical same method involves solving problems mainly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent; often it is enough to carefully apply the necessary formulas - and the answer is ready! No, of course, we won’t be able to do this without drawings at all, and besides, for a better understanding of the material, I will try to cite them beyond necessity.

The newly opened course of lessons on geometry does not pretend to be theoretically complete; it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need more complete help on any subsection, I recommend the following quite accessible literature:

1) A thing that, no joke, several generations are familiar with: School textbook on geometry, authors - L.S. Atanasyan and Company. This school locker room hanger has already gone through 20 (!) reprints, which, of course, is not the limit.

2) Geometry in 2 volumes. Authors L.S. Atanasyan, Bazylev V.T.. This is literature for high school, you will need first volume. Rarely encountered tasks may fall out of my sight, and the tutorial will be of invaluable help.

Both books can be downloaded for free online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download examples in higher mathematics.

Among the tools, I again propose my own development - software package in analytical geometry, which will greatly simplify life and save a lot of time.

It is assumed that the reader is familiar with basic geometric concepts and figures: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello to repeaters)

And now we will consider sequentially: the concept of a vector, actions with vectors, vector coordinates. I recommend reading further the most important article Dot product of vectors, and also Vector and mixed product of vectors. A local task - Division of a segment in this respect - will also not be superfluous. Based on the above information, you can master equation of a line in a plane With simplest examples of solutions, which will allow learn to solve geometry problems. The following articles are also useful: Equation of a plane in space, Equations of a line in space, Basic problems on a straight line and a plane, other sections of analytical geometry. Naturally, standard tasks will be considered along the way.

Vector concept. Free vector

First, let's repeat the school definition of a vector. Vector called directed a segment for which its beginning and end are indicated:

In this case, the beginning of the segment is the point, the end of the segment is the point. The vector itself is denoted by . Direction is essential, if you move the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must agree, entering the doors of an institute or leaving the doors of an institute are completely different things.

It is convenient to consider individual points of a plane or space as the so-called zero vector. For such a vector, the end and beginning coincide.

!!! Note: Here and further, you can assume that the vectors lie in the same plane or you can assume that they are located in space - the essence of the material presented is valid for both the plane and space.

Designations: Many immediately noticed the stick without an arrow in the designation and said, there’s also an arrow at the top! True, you can write it with an arrow: , but it is also possible the entry that I will use in the future. Why? Apparently, this habit developed for practical reasons; my shooters at school and university turned out to be too different-sized and shaggy. In educational literature, sometimes they don’t bother with cuneiform writing at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was stylistics, and now about ways to write vectors:

1) Vectors can be written in two capital Latin letters:
and so on. In this case, the first letter Necessarily denotes the beginning point of the vector, and the second letter denotes the end point of the vector.

2) Vectors are also written in small Latin letters:
In particular, our vector can be redesignated for brevity by a small Latin letter.

Length or module a non-zero vector is called the length of the segment. The length of the zero vector is zero. Logical.

The length of the vector is indicated by the modulus sign: ,

We will learn how to find the length of a vector (or we will repeat it, depending on who) a little later.

This was basic information about vectors, familiar to all schoolchildren. In analytical geometry, the so-called free vector.

To put it simply - the vector can be plotted from any point:

We are accustomed to calling such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, they are the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” this or that vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a vector of arbitrary length and direction - it can be “cloned” an infinite number of times and at any point in space, in fact, it exists EVERYWHERE. There is such a student saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is mathematically correct - the vector can be attached there too. But don’t rush to rejoice, it’s the students themselves who often suffer =)

So, free vector- This a bunch of identical directed segments. The school definition of a vector, given at the beginning of the paragraph: “A directed segment is called a vector...” implies specific a directed segment taken from a given set, which is tied to a specific point in the plane or space.

It should be noted that from the point of view of physics, the concept of a free vector is generally incorrect, and the point of application of the vector matters. Indeed, a direct blow of the same force on the nose or forehead, enough to develop my stupid example, entails different consequences. However, unfree vectors are also found in the course of vyshmat (don’t go there :)).

Actions with vectors. Collinearity of vectors

A school geometry course covers a number of actions and rules with vectors: addition according to the triangle rule, addition according to the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, let us repeat two rules that are especially relevant for solving problems of analytical geometry.

The rule for adding vectors using the triangle rule

Consider two arbitrary non-zero vectors and :

You need to find the sum of these vectors. Due to the fact that all vectors are considered free, we will set aside the vector from end vector:

The sum of vectors is the vector. For a better understanding of the rule, it is advisable to put a physical meaning into it: let some body travel along the vector , and then along the vector . Then the sum of vectors is the vector of the resulting path with the beginning at the departure point and the end at the arrival point. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way very lean along a zigzag, or maybe on autopilot - along the resulting vector of the sum.

By the way, if the vector is postponed from started vector, then we get the equivalent parallelogram rule addition of vectors.

First, about collinearity of vectors. The two vectors are called collinear, if they lie on the same line or on parallel lines. Roughly speaking, we are talking about parallel vectors. But in relation to them, the adjective “collinear” is always used.

Imagine two collinear vectors. If the arrows of these vectors are directed in the same direction, then such vectors are called co-directed. If the arrows point in different directions, then the vectors will be opposite directions.

Designations: collinearity of vectors is written with the usual parallelism symbol: , while detailing is possible: (vectors are co-directed) or (vectors are oppositely directed).

The work a non-zero vector on a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .

The rule for multiplying a vector by a number is easier to understand with the help of a picture:

Let's look at it in more detail:

1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.

2) Length. If the multiplier is contained within or , then the length of the vector decreases. So, the length of the vector is half the length of the vector. If the modulus of the multiplier is greater than one, then the length of the vector increases in time.

3) Please note that all vectors are collinear, while one vector is expressed through another, for example, . The reverse is also true: if one vector can be expressed through another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to the original) vector.

4) The vectors are co-directed. Vectors and are also co-directed. Any vector of the first group is oppositely directed with respect to any vector of the second group.

Which vectors are equal?

Two vectors are equal if they are in the same direction and have the same length. Note that codirectionality implies collinearity of vectors. The definition would be inaccurate (redundant) if we said: “Two vectors are equal if they are collinear, codirectional, and have the same length.”

From the point of view of the concept of a free vector, equal vectors are the same vector, as discussed in the previous paragraph.

Vector coordinates on the plane and in space

The first point is to consider vectors on the plane. Let us depict a Cartesian rectangular coordinate system and plot it from the origin of coordinates single vectors and :

Vectors and orthogonal. Orthogonal = Perpendicular. I recommend that you slowly get used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity And orthogonality.

Designation: The orthogonality of vectors is written with the usual perpendicularity symbol, for example: .

The vectors under consideration are called coordinate vectors or orts. These vectors form basis on surface. What a basis is, I think, is intuitively clear to many; more detailed information can be found in the article Linear (non) dependence of vectors. Basis of vectors In simple words, the basis and origin of coordinates define the entire system - this is a kind of foundation on which a full and rich geometric life boils.

Sometimes the constructed basis is called orthonormal basis of the plane: “ortho” - because the coordinate vectors are orthogonal, the adjective “normalized” means unit, i.e. the lengths of the basis vectors are equal to one.

Designation: the basis is usually written in parentheses, inside which in strict sequence basis vectors are listed, for example: . Coordinate vectors it is forbidden rearrange.

Any plane vector the only way expressed as:
, Where - numbers which are called vector coordinates in this basis. And the expression itself called vector decompositionby basis .

Dinner served:

Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing a vector into a basis, the ones just discussed are used:
1) the rule for multiplying a vector by a number: and ;
2) addition of vectors according to the triangle rule: .

Now mentally plot the vector from any other point on the plane. It is quite obvious that his decay will “follow him relentlessly.” Here it is, the freedom of the vector - the vector “carries everything with itself.” This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be plotted from the origin; one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change! True, you don’t need to do this, since the teacher will also show originality and draw you a “credit” in an unexpected place.

Vectors illustrate exactly the rule for multiplying a vector by a number, the vector is codirectional with the base vector, the vector is directed opposite to the base vector. For these vectors, one of the coordinates is equal to zero; you can meticulously write it like this:


And the basis vectors, by the way, are like this: (in fact, they are expressed through themselves).

And finally: , . By the way, what is vector subtraction, and why didn’t I talk about the subtraction rule? Somewhere in linear algebra, I don’t remember where, I noted that subtraction is a special case of addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , . Rearrange the terms and see in the drawing how well the good old addition of vectors according to the triangle rule works in these situations.

The considered decomposition of the form sometimes called vector decomposition in the ort system(i.e. in a system of unit vectors). But this is not the only way to write a vector; the following option is common:

Or with an equal sign:

The basis vectors themselves are written as follows: and

That is, the coordinates of the vector are indicated in parentheses. In practical problems, all three notation options are used.

I doubted whether to speak, but I’ll say it anyway: vector coordinates cannot be rearranged. Strictly in first place we write down the coordinate that corresponds to the unit vector, strictly in second place we write down the coordinate that corresponds to the unit vector. Indeed, and are two different vectors.

We figured out the coordinates on the plane. Now let's look at vectors in three-dimensional space, almost everything is the same here! It will just add one more coordinate. It’s hard to make three-dimensional drawings, so I’ll limit myself to one vector, which for simplicity I’ll set aside from the origin:

Any 3D space vector the only way expand over an orthonormal basis:
, where are the coordinates of the vector (number) in this basis.

Example from the picture: . Let's see how the vector rules work here. First, multiplying the vector by a number: (red arrow), (green arrow) and (raspberry arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector begins at the initial point of departure (beginning of the vector) and ends at the final point of arrival (end of the vector).

All vectors of three-dimensional space, naturally, are also free; try to mentally set aside the vector from any other point, and you will understand that its decomposition “will remain with it.”

Similar to the flat case, in addition to writing versions with brackets are widely used: either .

If one (or two) coordinate vectors are missing in the expansion, then zeros are put in their place. Examples:
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write .

The basis vectors are written as follows:

This, perhaps, is all the minimum theoretical knowledge necessary to solve problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that teapots re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time to better assimilate the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I note that the materials on the site are not enough to pass the theoretical test or colloquium on geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of presentation, but a plus to your understanding of the subject. To receive detailed theoretical information, please bow to Professor Atanasyan.

And we move on to the practical part:

The simplest problems of analytical geometry.
Actions with vectors in coordinates

It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, you don’t even have to remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on your shirt; many things are familiar to you from school.

The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas... you will see for yourself.

How to find a vector from two points?

If two points of the plane and are given, then the vector has the following coordinates:

If two points in space and are given, then the vector has the following coordinates:

That is, from the coordinates of the end of the vector you need to subtract the corresponding coordinates beginning of the vector.

Exercise: For the same points, write down the formulas for finding the coordinates of the vector. Formulas at the end of the lesson.

Example 1

Given two points of the plane and . Find vector coordinates

Solution: according to the corresponding formula:

Alternatively, the following entry could be used:

Aesthetes will decide this:

Personally, I'm used to the first version of the recording.

Answer:

According to the condition, it was not necessary to construct a drawing (which is typical for problems of analytical geometry), but in order to clarify some points for dummies, I will not be lazy:

You definitely need to understand difference between point coordinates and vector coordinates:

Point coordinates– these are ordinary coordinates in a rectangular coordinate system. I think everyone knows how to plot points on a coordinate plane from the 5th-6th grade. Each point has a strict place on the plane, and they cannot be moved anywhere.

The coordinates of the vector– this is its expansion according to the basis, in this case. Any vector is free, so if necessary, we can easily move it away from some other point in the plane. It is interesting that for vectors you don’t have to build axes or a rectangular coordinate system at all; you only need a basis, in this case an orthonormal basis of the plane.

The records of coordinates of points and coordinates of vectors seem to be similar: , and meaning of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, also applies to space.

Ladies and gentlemen, let's fill our hands:

Example 2

a) Points and are given. Find vectors and .
b) Points are given And . Find vectors and .
c) Points and are given. Find vectors and .
d) Points are given. Find vectors .

Perhaps that's enough. These are examples for you to decide on your own, try not to neglect them, it will pay off ;-). There is no need to make drawings. Solutions and answers at the end of the lesson.

What is important when solving analytical geometry problems? It is important to be EXTREMELY CAREFUL to avoid making the masterful “two plus two equals zero” mistake. I apologize right away if I made a mistake somewhere =)

How to find the length of a segment?

The length, as already noted, is indicated by the modulus sign.

If two points of the plane are given and , then the length of the segment can be calculated using the formula

If two points in space and are given, then the length of the segment can be calculated using the formula

Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard

Example 3

Solution: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Line segment - this is not a vector, and, of course, you cannot move it anywhere. In addition, if you draw to scale: 1 unit. = 1 cm (two notebook cells), then the resulting answer can be checked with a regular ruler by directly measuring the length of the segment.

Yes, the solution is short, but there are a couple more important points in it that I would like to clarify:

Firstly, in the answer we put the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, a mathematically correct solution would be the general formulation: “units” - abbreviated as “units.”

Secondly, let us repeat the school material, which is useful not only for the task considered:

pay attention to important technique – removing the multiplier from under the root. As a result of the calculations, we have a result and good mathematical style involves removing the factor from under the root (if possible). In more detail the process looks like this: . Of course, leaving the answer as is would not be a mistake - but it would certainly be a shortcoming and a weighty argument for quibbling on the part of the teacher.

Here are other common cases:

Often the root produces a fairly large number, for example . What to do in such cases? Using the calculator, we check whether the number is divisible by 4: . Yes, it was completely divided, thus: . Or maybe the number can be divided by 4 again? . Thus: . The last digit of the number is odd, so dividing by 4 for the third time will obviously not work. Let's try to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a number that cannot be extracted as a whole, then we try to remove the factor from under the root - using a calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

When solving various problems, roots are often encountered; always try to extract factors from under the root in order to avoid a lower grade and unnecessary problems with finalizing your solutions based on the teacher’s comments.

Let's also repeat squaring roots and other powers:

The rules for operating with powers in general form can be found in a school algebra textbook, but I think from the examples given, everything or almost everything is already clear.

Task for independent solution with a segment in space:

Example 4

Points and are given. Find the length of the segment.

The solution and answer are at the end of the lesson.

How to find the length of a vector?

If a plane vector is given, then its length is calculated by the formula.

If a space vector is given, then its length is calculated by the formula .