Which vector is called unit. Vectors: definition and basic concepts

Such a concept as a vector is considered in almost all natural sciences, and it can have completely different meanings, therefore it is impossible to give an unambiguous definition of a vector for all areas. But let's try to figure it out. So, vector - what is it?

The concept of a vector in classical geometry

A vector in geometry is a segment for which it is indicated which of its points is the beginning and which is the end. That is, to put it simply, a directed segment is called a vector.

Accordingly, a vector is indicated (what it is - discussed above), as well as a segment, that is, two capital letters of the Latin alphabet with the addition of a line or an arrow pointing to the right on top. It can also be signed with a lowercase (small) letter of the Latin alphabet with a dash or an arrow. The arrow always points to the right and does not change depending on the position of the vector.

So a vector has a direction and a length.

The designation of a vector also contains its direction. This is expressed as shown in the figure below.

Changing direction reverses the value of the vector.

The length of a vector is the length of the segment from which it is formed. It is designated as a module from a vector. This is shown in the figure below.

Accordingly, zero is a vector whose length is equal to zero. It follows from this that the zero vector is a point, moreover, the start and end points coincide in it.

The length of a vector is always a non-negative value. In other words, if there is a segment, then it necessarily has a certain length or is a point, then its length is zero.

The very concept of a point is basic and has no definition.

Vector addition

There are special formulas and rules for vectors that can be used to perform addition.

Triangle rule. To add vectors according to this rule, it is enough to combine the end of the first vector and the beginning of the second, using parallel translation, and connect them. The resulting third vector will be equal to the addition of the other two.

parallelogram rule. To add according to this rule, you need to draw both vectors from one point, and then draw another vector from the end of each of them. That is, the second one will be drawn from the first one, and the first one from the second one. As a result, a new intersection point will be obtained and a parallelogram will be formed. If we combine the intersection point of the beginnings and ends of the vectors, then the resulting vector will be the result of addition.

Similarly, it is possible to perform subtraction.

Vector difference

Similarly to the addition of vectors, it is possible to perform their subtraction. It is based on the principle shown in the figure below.

That is, it is enough to represent the vector to be subtracted as a vector opposite to it, and to calculate according to the principles of addition.

Also, absolutely any non-zero vector can be multiplied by any number k, this will change its length by k times.

In addition to these, there are other vector formulas (for example, to express the length of a vector in terms of its coordinates).

Location of vectors

Surely many have come across such a concept as a collinear vector. What is collinearity?

Collinearity of vectors is the equivalent of parallelism of straight lines. If two vectors lie on lines that are parallel to each other, or on the same line, then such vectors are called collinear.

Direction. Relative to each other, collinear vectors can be co-directed or oppositely directed, this is determined by the direction of the vectors. Accordingly, if a vector is co-directed with another, then the vector opposite to it is directed oppositely.

The first figure shows two oppositely directed vectors and a third one that is not collinear with them.

After introducing the above properties, it is also possible to define equal vectors - these are vectors that are directed in the same direction and have the same length of the segments from which they are formed.

In many sciences, the concept of a radius vector is also used. Such a vector describes the position of one point of the plane relative to another fixed point (often this is the origin).

Vectors in physics

Let's assume that when solving the problem, a condition arose: the body moves at a speed of 3 m/s. This means that the body moves with a specific direction in one straight line, so this variable will be a vector quantity. To solve it, it is important to know both the value and the direction, since depending on the consideration, the speed can be either 3 m/s or -3 m/s.

In general, the vector in physics is used to indicate the direction of the force acting on the body, and to determine the resultant.

When these forces are indicated in the figure, they are indicated by arrows with a vector label above it. Classically, the length of the arrow is just as important, with the help of it they indicate which force is stronger, but this property is secondary, you should not rely on it.

Vector in linear algebra and calculus

The elements of linear spaces are also called vectors, but in this case they are an ordered system of numbers that describe some of the elements. Therefore, the direction in this case is no longer important. The definition of a vector in classical geometry and in mathematical analysis are very different.

Vector projection

Projected vector - what is it?

Quite often, for a correct and convenient calculation, it is necessary to decompose a vector located in two-dimensional or three-dimensional space along the coordinate axes. This operation is necessary, for example, in mechanics when calculating the forces acting on the body. The vector in physics is used quite often.

To perform the projection, it is enough to lower the perpendiculars from the beginning and end of the vector to each of the coordinate axes, the segments obtained on them will be called the projection of the vector onto the axis.

To calculate the projection length, it is enough to multiply its initial length by a certain trigonometric function, which is obtained by solving a mini-problem. In fact, there is a right triangle in which the hypotenuse is the original vector, one of the legs is the projection, and the other leg is the dropped perpendicular.

Finally, I got my hands on an extensive and long-awaited topic analytical geometry. First, a little about this section of higher mathematics…. Surely you now remembered the 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. Analytic geometry, oddly enough, may seem more interesting and accessible. What does the adjective "analytical" mean? Two stamped mathematical turns immediately come to mind: “graphic method of solution” and “analytical method of solution”. Graphic method, of course, is associated with the construction of graphs, drawings. Analytical same method involves problem solving predominantly 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 accurately apply the necessary formulas - and the answer is ready! No, of course, it will not do without drawings at all, besides, for a better understanding of the material, I will try to bring them in excess of the need.

The open course of lessons in geometry does not claim to be theoretical completeness, 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 a more complete reference on any subsection, I recommend the following quite accessible literature:

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

2) Geometry in 2 volumes. The authors L.S. Atanasyan, Bazylev V.T.. This is literature for higher education, you will need first volume. Infrequently occurring tasks may fall out of my field of vision, and the tutorial will be of invaluable help.

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

Of the tools, I again offer my own development - software package on 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 repeaters)

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

The concept of a vector. 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 rearrange 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 admit that 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, space as the so-called zero vector. Such a vector has the same end and beginning.

!!! Note: Here and below, 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 drew attention to a stick without an arrow in the designation and said that they also put an arrow at the top! That's right, you can write with an arrow: , but admissible and record that I will use later. Why? Apparently, such a habit has developed from practical considerations, my shooters at school and university turned out to be too diverse and shaggy. In educational literature, sometimes they don’t bother with cuneiform at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was the style, and now about the ways of writing vectors:

1) Vectors can be written in two capital Latin letters:
etc. While the first letter necessarily denotes the start 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 non-zero vector is called the length of the segment. The length of the null vector is zero. Logically.

The length of a vector is denoted by the modulo sign: ,

How to find the length of a vector, we will learn (or repeat, for whom how) a little later.

That was elementary information about the vector, familiar to all schoolchildren. In analytic geometry, the so-called free vector.

If it's quite simple - vector can be drawn from any point:

We used to call such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, this is the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” one or another vector to ANY point of the plane or space you need. This is a very cool property! 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's proverb: Each lecturer in f ** u in the vector. After all, not just a witty rhyme, everything is mathematically correct - a vector can be attached there too. But do not rush to rejoice, students themselves suffer more often =)

So, free vector- This a bunch of identical directional 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 attached to a certain 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 on the forehead is enough to develop my stupid example entails different consequences. However, not free vectors are also found in the course of vyshmat (do not go there :)).

Actions with vectors. Collinearity of vectors

In the school geometry course, a number of actions and rules with vectors are considered: addition according to the triangle rule, addition according to the parallelogram rule, the rule of the difference of vectors, multiplication of a vector by a number, the scalar product of vectors, etc. As a seed, we repeat two rules that are especially relevant for solving problems of analytical geometry.

Rule of addition of vectors according to the rule of triangles

Consider two arbitrary non-zero vectors and :

It is required to find the sum of these vectors. Due to the fact that all vectors are considered free, we postpone 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 make a path along the vector , and then along the vector . Then the sum of the vectors is the vector of the resulting path starting at the point of departure and ending at the point of arrival. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way strongly zigzag, or maybe on autopilot - along the resulting sum vector.

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

First, about the 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-directional. If the arrows look in different directions, then the vectors will be oppositely directed.

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

work of a nonzero vector by 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 a picture:

We understand in more detail:

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

2) Length. If the factor is contained within or , then the length of the vector decreases. So, the length of the vector is twice less than the length of the vector . If the modulo 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 in terms of another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to original) vector.

4) The vectors are codirectional. The vectors and are also codirectional. Any vector of the first group is opposite to any vector of the second group.

What vectors are equal?

Two vectors are equal if they are codirectional and have the same length. Note that co-direction implies that the vectors are collinear. The definition will be inaccurate (redundant) if you say: "Two vectors are equal if they are collinear, co-directed and have the same length."

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

Vector coordinates on the plane and in space

The first point is to consider vectors on a plane. Draw a Cartesian rectangular coordinate system and set aside from the origin single vectors and :

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

Designation: orthogonality of vectors is written with the usual perpendicular sign, for example: .

The considered vectors are called coordinate vectors or orts. These vectors form basis on surface. What is the basis, I think, is intuitively clear to many, more detailed information can be found in the article Linear (non) dependence of vectors. Vector basis.In simple words, the basis and the 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 order basis vectors are listed, for example: . Coordinate vectors it is forbidden swap places.

Any plane vector the only way expressed as:
, where - numbers, which are called vector coordinates in this basis. But the expression itself called vector decompositionbasis .

Dinner served:

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

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

Vectors , illustrate exactly the rule for multiplying a vector by a number, the vector is co-directed with the basis vector , the vector is directed opposite to the basis vector . For these vectors, one of the coordinates is equal to zero, it can be meticulously written as follows:


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 tell you about the subtraction rule? Somewhere in linear algebra, I don't remember where, I noted that subtraction is a special case of addition. So, the expansions of the vectors "de" and "e" are calmly written as a sum: . Rearrange the terms in places and follow the drawing how clearly the good old addition of vectors according to the triangle rule works in these situations.

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

Or with an equals sign:

The basis vectors themselves are written as follows: and

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

I doubted whether to speak, but still I will say: vector coordinates cannot be rearranged. Strictly in first place write down the coordinate that corresponds to the unit vector , strictly in second place 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 consider vectors in three-dimensional space, everything is almost the same here! Only one more coordinate will be added. It is difficult to perform three-dimensional drawings, so I will limit myself to one vector, which for simplicity I will postpone from the origin:

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

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

All vectors of three-dimensional space, of course, are also free, try to mentally postpone the vector from any other point, and you will understand that its expansion "remains with it."

Similarly to the plane 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 instead. Examples:
vector (meticulously ) – write down ;
vector (meticulously ) – write down ;
vector (meticulously ) – write down .

Basis vectors are written as follows:

Here, perhaps, is all the minimum theoretical knowledge necessary for solving problems of analytical geometry. Perhaps there are too many terms and definitions, so I recommend dummies to 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 for better assimilation of the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in what follows. I note that the materials of the site are not enough to pass a theoretical test, a colloquium on geometry, since I carefully encrypt all theorems (besides without proofs) - to the detriment of the scientific style of presentation, but a plus for your understanding of the subject. For detailed theoretical information, I ask you to bow to Professor Atanasyan.

Now let's move on to the practical part:

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

The tasks that will be considered, it is highly desirable to learn how to solve them fully automatically, and the formulas memorize, don't even 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 extra time eating pawns. You do not 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 given 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:

I.e, from the coordinates of the end of the vector you need to subtract the corresponding coordinates vector start.

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 in the plane and . Find vector coordinates

Decision: according to the corresponding formula:

Alternatively, the following notation could be used:

Aesthetes will decide like this:

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

Answer:

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

Must be understood difference between point coordinates and vector coordinates:

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

The coordinates of the same vector is its expansion with respect to the basis , in this case . Any vector is free, therefore, if necessary, we can easily postpone it from some other point in the plane. Interestingly, for vectors, you can not build axes at all, a rectangular coordinate system, you only need a basis, in this case, an orthonormal basis of the plane.

The records of point coordinates and vector coordinates seem to be similar: , and sense of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, is also true for space.

Ladies and gentlemen, we fill our hands:

Example 2

a) Given points and . Find vectors and .
b) Points are given and . Find vectors and .
c) Given points and . Find vectors and .
d) Points are given. Find Vectors .

Perhaps enough. These are examples for an independent decision, try not to neglect them, it will pay off ;-). Drawings are not required. Solutions and answers at the end of the lesson.

What is important in solving problems of analytical geometry? It is important to be EXTREMELY CAREFUL in order to avoid the masterful “two plus two equals zero” error. I apologize in advance if I made a mistake =)

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 and are given, then the length of the segment can be calculated by the formula

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

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

Example 3

Decision: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Line segment - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the 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 of important points in it that I would like to clarify:

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

Secondly, let's repeat the school material, which is useful not only for the considered problem:

pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the factor out from under the root (if possible). The process looks like this in more detail: . Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.

Here are other common cases:

Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, split completely, 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 is clearly not possible. Trying to divide by nine: . As a result:
Ready.

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

In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.

Let's repeat the squaring of the roots and other powers at the same time:

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

Task for an independent solution with a segment in space:

Example 4

Given points and . Find the length of the segment.

Solution and answer 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 .

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, such as force, position, speed, acceleration, torque, momentum, electric and magnetic fields. They can be contrasted with other quantities, such as mass, volume, pressure, temperature and density, which can be described by an ordinary number, and they are called "scalars". Vector notation is used when working with quantities that cannot be fully 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 from a point to an object, 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 a direction. Graphically, vectors are depicted as directed segments of a straight line 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. The numerical value of the vector A is called the modulus or length and is denoted by A or |A|. This quantity is, of course, a scalar. A vector whose beginning and end coincide is called a null vector and is denoted O.

Two vectors are called equal (or free) if their moduli and directions are the same. 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 "linked" or "sliding", as follows: Linked vectors have fixed application points. For example, the radius vector indicates the position of a point relative to some fixed origin. Related vectors are considered equal if not only do they have the same modules and directions, but they also have a common point of application. Sliding vectors are equal vectors located on the same straight line.
Addition of vectors. The idea of ​​vector addition comes from the fact that we can find a single vector that has the same effect as two other vectors together. If, in order to get to some point, we need to walk first A kilometers in one direction and then B kilometers in the other direction, then we could reach our end point by walking C kilometers in a third direction (Fig. 2). In this sense, one can say that

A+B=C.
The vector C is called the "result vector" of A and B and is given by the construction shown in the figure; a parallelogram is built on the vectors A and B as on the sides, and C is a diagonal connecting the beginning of A and the end of B. From fig. 2 it can be seen that the addition of vectors is "commutative", i.e. A + B = B + A. Similarly, you can add several vectors by connecting them in series in a "continuous chain", as shown in fig. 3 for three vectors D, E, and F. From fig. 3 also shows 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 have to lie in the same plane. Subtracting vectors is represented as adding to a negative vector. For example, A - B = A + (-B), where, as previously defined, -B is a vector equal to B in absolute value 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 terms of this rule; the same can be said about speeds; 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 non-zero 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 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 value -1A is a vector equal in length to A 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 is decomposed into two or more "components", i.e. into two or more vectors that, when added together, will give the original vector as a 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 in three mutually perpendicular directions. Let's choose the right Cartesian coordinate system with axes Ox, Oy and Oz as shown in fig. 5. By right coordinate system, we mean that the x, y, and z axes are positioned as the thumb, index, and middle fingers of the right hand, respectively, can be positioned. From one right coordinate system, it is always possible to obtain another right coordinate system by an appropriate rotation. On fig. 5 shows the decomposition of the vector A into three components and They add up to the vector A , since

Hence,

One could also first add and get and then add to The projections of the vector A on the three coordinate axes, denoted Ax, Ay and Az are called the "scalar components" of the vector A:

where a, b and g are the angles between A and the three coordinate axes. Now we introduce three unit length vectors i, j and k (orths) having the same direction as the corresponding x, y and z axes. 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. This is a vector that is 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 over 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 change) moves throughout space. Suppose A, B and C are unit vectors i, j and k. The vector ai lies on the x-axis; the vector ai + bj can move along the entire 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 a four-dimensional vector as the quantity A = Axi + Ayj + Azk + Awl
with length

and one could continue up to five, six, or any number of dimensions. Although it is impossible to represent such a vector visually, there are no mathematical difficulties here. Such a notation 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), whose components 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 three, then 18, and so on. The number of dimensions can be increased indefinitely; however, 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.
Multiplication of two vectors. The vector addition rule was obtained 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 sound; in addition, it is desirable that the product had 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 "scalar product" or "inner product" of two vectors and is written ACHB 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 physical meaning for one, two, or three dimensions, while vector products are only defined for three dimensions.
Scalar products. If, under the action 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 F in the direction 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, rc. 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, Bs.
Since all the 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 the vectors A and B are perpendicular, then cos bA, Bc is equal to zero, and, therefore, A*B = 0, even if neither A nor B are equal to zero. That 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 performed, 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, hence B must not be equal to C. These conflicting results show that vector division is impossible. The scalar product gives another way to write the numerical value (modulus) of the vector: A*A = AA*cos 0° = A2;
That's why

The scalar product can also 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 artwork. A vector or external product of vectors is a vector whose modulus is equal to the product of their moduli and the sine of the angle perpendicular to the original vectors and together with them making up the right 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: choose 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 that this line has rotated per unit of time. If the 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-handed screw 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 on an axis inserted inside another ring, we can give rotation to the body inside the first ring with an angular velocity w1 and then make the inner ring (and body) rotate with an angular velocity w2. Figure 7 explains the essence of the matter; circular arrows show 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 an angular velocity w1 inside the ring BC, which, in turn, rotates inside the ring DE with an angular velocity w2. The sphere rotates with an angular velocity equal to the sum of the angular velocities and all points on the 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 rather difficult to visualize, but it is quite obvious that the body is no longer rotating about a fixed axis. However, you can still say that it rotates. To show this, we choose some 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 become the radii PJ and PK of the circles PQRS and PTUW, respectively. Let's draw a line POPў passing through the center of the sphere. Now the point P, at the considered moment of time, simultaneously moves along the circles that touch at the point P. For a small time interval Dt, P moves to a distance

This distance is zero if

In this case, the point P is in a state of instantaneous rest, and likewise all the points on the line POP. the axis of rotation of the sphere, just as a wheel rolling on a road at each moment of time rotates about its lowest point What is the angular velocity of the sphere? , it moves in time Dt to a distance

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

From this formula and relation (1) we get

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 an angular velocity w. We 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 along a circle with a speed V = w r sin (w, r). The velocity vector V is tangent to the circle and points in the direction shown in fig. eight.

This equation gives the dependence of the speed V of a point on the combination of two vectors w and r. We use this relation to define a new kind of product and write: V = w * r. Since the result of such a 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 the vector C is such that it is perpendicular to the plane passing through A and B and points in the same direction as the direction of movement of the dextrorotatory screw if it is parallel to C and rotates from A to B. In other words, we can say that A, B, and C, in that order, form the right set of coordinate axes. The vector 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 proved 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 the 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 0, or A and B are collinear. Thus, as with the dot product, division by a vector is not possible. The value of A * B is equal to the area of ​​a parallelogram with sides A and B. This is easy to see, since B sin bA, Bc is its height and A is its base. There are many other physical quantities that are vector products. One of the most important vector products appears in the theory of electromagnetism and is called the Poynting vector P. This vector is given as follows: P = E * H, where E and H are the electric and magnetic field vectors, respectively. The vector P can be thought of as a given energy flow in watts per square meter at any point. Here are a few more examples: the moment of force F (torque) relative to the origin, acting on a point whose radius vector is r, is defined as r * F; a particle located at point r, with mass m and velocity V, has an 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 velocity 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 spoken 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 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 x, y, and z coordinates (components) and multiplying, we can show that A * (B * C) = B * (A*C) - C * (A*B). The third type of product that occurs 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 interchanged, and the product 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 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 change U by 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 such a limit exists. On the other hand, one can represent U as the sum of the components along the 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 the acceleration. Suppose the point moves along the curve shown in Fig. 10. Let s be the distance traveled by the point along the curve. During a small time interval Dt, the point will pass the distance Ds along the curve; the position of the radius vector will change to Dr. Hence Dr/Ds is a vector directed like Dr. Further

Dr vector - radius-vector change.

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

Thus, we see that if the 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, one often has to deal with vector or scalar quantities that change from point to point in a given area. Such areas are called "fields". For example, a scalar can be temperature or pressure; the vector can be the velocity of a moving fluid or the electrostatic field of a system of charges. If we have chosen some coordinate system, then any point P (x, y, z) in the given area corresponds to some radius vector r (= xi + yj + zk) and also the value of the vector quantity U (r) or the 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 value U or f, although different points may, 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 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 "gradient". Let we deal with a scalar field f. First, as an example, consider a contour map of an area of ​​a 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 we move perpendicular to these lines, then the rate of change of f will be maximum. We can associate each point with a vector indicating the magnitude and direction of the maximum change in the speed f; such a map and some of these vectors are shown in Fig. 11. If we do this for every point of the field, we get a vector field associated with the 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 dots denote higher order terms. This expression can be written as a dot product

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

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

This means that the operator C exists by itself. 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 defines 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 equal to

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

Where e0 is the dielectric constant of free space. So

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

Here S indicates that these integrals are taken over the entire surface, da is an element of the surface of S. For simplicity, we will choose the convenient form of S 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. We 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 face n = -i; this face contributes to the integral

Using the Taylor theorem, we get the total contribution from the two faces

Note that DxDyDz = DV. Similarly, the contribution from the other two pairs of faces can be calculated. The full 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 proved in the same way. Let's use Fig. 12; then the contribution from the front face to the integral will be equal to

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

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

The rate of expansion of a unit of volume around point P. This is where the 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 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). Consider now 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 curve element as shown in fig. 13. Comparing these equalities with relation (5), we obtain that

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

if DA is not equal to 0. This expression is maximum 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 about P, ​​then rotU is #0 and the U vectors will rotate around P. Hence the name rotor. The divergence theorem (the Ostrogradsky-Gauss theorem) is a generalization of formula (4) for finite volumes. She states that for some volume V bounded by a closed surface S,

A vector is a mathematical object that is characterized by direction and magnitude. In geometry, a vector is a line segment in a plane or in space, which has its own specific direction and length.

Vector notation

To designate a vector, either one lowercase letter or two uppercase letters are used, which correspond to the beginning and end of the vector, while a horizontal dash is displayed above the letters. The first letter indicates the beginning of the vector, the second - the end (see Figure 1). A graphical display of a vector shows an arrow indicating its direction.

What are the coordinates of a vector on the plane and in space?

The vector coordinates are the coefficients of the only possible linear combination of basis vectors in the selected coordinate system. It sounds complicated, but it's actually quite simple. Let's take an example.

Suppose we need to find the coordinates of the vector a. Let's place it in a three-dimensional coordinate system (see Figure 2) and perform projections of the vector on each axis. The vector a in this case will be written as follows: a= a x i+ a y j+ a z k, where i, j, k are basis vectors, a x , a y , a z are the coefficients that determine the coordinates of the vector a. The expression itself will be called a linear combination. On a plane (in a rectangular coordinate system), a linear combination will consist of two bases and coefficients.

Vector relations

In the theory of vectors, there is such a term as the ratio of vectors. This concept defines the location of vectors relative to each other on the plane and in space. The most famous special cases of vector relations are:

• collinearity;
• co-directionality;
• coplanarity;
• equality.

Collinear vectors lie on the same straight line or are parallel to each other, codirectional vectors have the same direction, coplanar vectors are located in the same plane or in parallel planes, equal vectors have the same direction and length.

A vector is a directed segment of a straight line in Euclidean space, in which one end (point A) is called the beginning of the vector, and the other end (point B) is called the end of the vector (Fig. 1). Vectors are denoted:

If the beginning and end of the vector are the same, then the vector is called zero vector and denoted 0 .

Example. Let the beginning of the vector in two-dimensional space have coordinates A(12,6) , and the end of the vector is the coordinates B(12.6). Then the vector is a null vector.

Cut length AB called module (long, the norm) vector and is denoted by | a|. A vector of length equal to one is called unit vector. In addition to the modulus, a vector is characterized by a direction: a vector has a direction from A to B. A vector is called a vector, opposite vector .

The two vectors are called collinear if they lie on the same line or on parallel lines. In Fig. 3 red vectors are collinear since they lie on the same straight line, and the blue vectors are collinear, because they lie on parallel lines. Two collinear vectors are called equally directed if their ends lie on the same side of the line joining their beginnings. Two collinear vectors are called opposite directions if their ends lie on opposite sides of the line joining their beginnings. If two collinear vectors lie on the same line, then they are said to be equally directed if one of the rays formed by one vector completely contains the ray formed by the other vector. Otherwise, the vectors are called oppositely directed. In Figure 3, the blue vectors are in the same direction and the red vectors are in the opposite direction.

The two vectors are called equal if they have equal modules and are equally directed. In Fig.2, the vectors are equal because their moduli are equal and have the same direction.

The vectors are called coplanar if they lie on the same plane or in parallel planes.

AT n In a dimensional vector space, consider the set of all vectors whose starting point coincides with the origin. Then the vector can be written in the following form:

(1)

where x 1 , x 2 , ..., x n vector end point coordinates x.

The vector written in the form (1) is called row vector, and the vector written as

(2)

called column vector.

Number n called dimension (in order) vector. If a then the vector is called zero vector(because the starting point of the vector ). Two vectors x and y are equal if and only if their corresponding elements are equal.