In this article we will take a closer look at the concept of the cross product of two vectors. We will give the necessary definitions, write a formula for finding the coordinates of a vector product, list and justify its properties. After this, we will dwell on the geometric meaning of the vector product of two vectors and consider solutions to various typical examples.

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Definition of cross product.

Before defining a vector product, let's understand the orientation of an ordered triple of vectors in three-dimensional space.

Let's plot the vectors from one point. Depending on the direction of the vector, the three can be right or left. Let's look from the end of the vector at how the shortest turn from the vector to . If the shortest rotation occurs counterclockwise, then the triple of vectors is called right, otherwise - left.

Now let's take two non-collinear vectors and . Let us plot the vectors and from point A. Let's construct some vector perpendicular to both and and . Obviously, when constructing a vector, we can do two things, giving it either one direction or the opposite (see illustration).

Depending on the direction of the vector, the ordered triplet of vectors can be right-handed or left-handed.

This brings us close to the definition of a vector product. It is given for two vectors defined in a rectangular coordinate system of three-dimensional space.

Definition.

The cross product of two vectors and , specified in a rectangular coordinate system of three-dimensional space, is called a vector such that

The cross product of vectors and is denoted as .

Coordinates of the vector product.

Now we will give the second definition of a vector product, which allows you to find its coordinates from the coordinates of given vectors and.

Definition.

In a rectangular coordinate system of three-dimensional space vector product of two vectors And is a vector , where are the coordinate vectors.

This definition gives us the cross product in coordinate form.

It is convenient to represent the vector product as the determinant of a third-order square matrix, the first row of which is the vectors, the second row contains the coordinates of the vector, and the third contains the coordinates of the vector in a given rectangular coordinate system:

If we expand this determinant into the elements of the first row, we obtain the equality from the definition of the vector product in coordinates (if necessary, refer to the article):

It should be noted that the coordinate form of the vector product is fully consistent with the definition given in the first paragraph of this article. Moreover, these two definitions of a cross product are equivalent. You can see the proof of this fact in the book listed at the end of the article.

Properties of a vector product.

Since the vector product in coordinates can be represented as a determinant of the matrix, the following can easily be justified on the basis properties of the cross product:

As an example, let us prove the anticommutative property of a vector product.

A-priory And . We know that the value of the determinant of a matrix is ​​reversed if two rows are swapped, therefore, , which proves the anticommutative property of a vector product.

Vector product - examples and solutions.

There are mainly three types of problems.

In problems of the first type, the lengths of two vectors and the angle between them are given, and you need to find the length of the vector product. In this case, the formula is used .

Example.

Find the length of the vector product of the vectors and , if known .

Solution.

We know from the definition that the length of the vector product of vectors and is equal to the product of the lengths of vectors and by the sine of the angle between them, therefore, .

Answer:

.

Problems of the second type are related to the coordinates of vectors, in which the vector product, its length or anything else is searched through the coordinates of given vectors And .

There are a lot of different options possible here. For example, not the coordinates of the vectors and can be specified, but their expansions into coordinate vectors of the form and , or vectors and can be specified by the coordinates of their start and end points.

Let's look at typical examples.

Example.

Two vectors are given in a rectangular coordinate system . Find their cross product.

Solution.

According to the second definition, the vector product of two vectors in coordinates is written as:

We would have arrived at the same result if the vector product had been written in terms of the determinant

Answer:

.

Example.

Find the length of the vector product of the vectors and , where are the unit vectors of the rectangular Cartesian coordinate system.

Solution.

First we find the coordinates of the vector product in a given rectangular coordinate system.

Since vectors and have coordinates and respectively (if necessary, see the article coordinates of a vector in a rectangular coordinate system), then by the second definition of a vector product we have

That is, the vector product has coordinates in a given coordinate system.

We find the length of a vector product as the square root of the sum of the squares of its coordinates (we obtained this formula for the length of a vector in the section on finding the length of a vector):

Answer:

.

Example.

In a rectangular Cartesian coordinate system, the coordinates of three points are given. Find some vector that is perpendicular and at the same time.

Solution.

Vectors and have coordinates and respectively (see the article finding the coordinates of a vector through the coordinates of points). If we find the vector product of the vectors and , then by definition it is a vector perpendicular to both to and to , that is, it is a solution to our problem. Let's find him

Answer:

- one of the perpendicular vectors.

In problems of the third type, the skill of using the properties of the vector product of vectors is tested. After applying the properties, the corresponding formulas are applied.

Example.

The vectors and are perpendicular and their lengths are 3 and 4, respectively. Find the length of the cross product .

Solution.

By the distributive property of a vector product, we can write

Due to the combinational property, we take the numerical coefficients out of the sign of the vector products in the last expression:

The vector products and are equal to zero, since And , Then .

Since the vector product is anticommutative, then .

So, using the properties of the vector product, we arrived at the equality .

By condition, the vectors and are perpendicular, that is, the angle between them is equal to . That is, we have all the data to find the required length

Answer:

.

Geometric meaning of a vector product.

By definition, the length of the vector product of vectors is . And from a high school geometry course we know that the area of ​​a triangle is equal to half the product of the lengths of the two sides of the triangle and the sine of the angle between them. Consequently, the length of the vector product is equal to twice the area of ​​a triangle whose sides are the vectors and , if they are plotted from one point. In other words, the length of the vector product of vectors and is equal to the area of ​​a parallelogram with sides and and the angle between them equal to . This is the geometric meaning of the vector product.

Finally, I got my hands on this extensive 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 or 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 appropriate 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 appropriate 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 .

In this lesson we will look at two more operations with vectors: vector product of vectors And mixed product of vectors (immediate link for those who need it). It’s okay, sometimes it happens that for complete happiness, in addition to scalar product of vectors, more and more are required. This is vector addiction. It may seem that we are getting into the jungle of analytical geometry. This is wrong. In this section of higher mathematics there is generally little wood, except perhaps enough for Pinocchio. In fact, the material is very common and simple - hardly more complicated than the same scalar product, there will even be fewer typical tasks. The main thing in analytical geometry, as many will be convinced or have already been convinced, is NOT TO MAKE MISTAKES IN CALCULATIONS. Repeat like a spell and you will be happy =)

If vectors sparkle somewhere far away, like lightning on the horizon, it doesn’t matter, start with the lesson Vectors for dummies to restore or reacquire basic knowledge about vectors. More prepared readers can get acquainted with the information selectively; I tried to collect the most complete collection of examples that are often found in practical work

What will make you happy right away? When I was little, I could juggle two or even three balls. It worked out well. Now you won't have to juggle at all, since we will consider only spatial vectors, and flat vectors with two coordinates will be left out. Why? This is how these actions were born - the vector and mixed product of vectors are defined and work in three-dimensional space. It's already easier!

This operation, just like the scalar product, involves two vectors. Let these be imperishable letters.

The action itself denoted by in the following way: . There are other options, but I’m used to denoting the vector product of vectors this way, in square brackets with a cross.

And right away question: if in scalar product of vectors two vectors are involved, and here two vectors are also multiplied, then what is the difference? The obvious difference is, first of all, in the RESULT:

The result of the scalar product of vectors is NUMBER:

The result of the cross product of vectors is VECTOR: , that is, we multiply the vectors and get a vector again. Closed club. Actually, this is where the name of the operation comes from. In different educational literature, designations may also vary; I will use the letter.

Definition of cross product

First there will be a definition with a picture, then comments.

Definition: Vector product non-collinear vectors, taken in this order, called VECTOR, length which is numerically equal to the area of ​​the parallelogram, built on these vectors; vector orthogonal to vectors, and is directed so that the basis has a right orientation:

Let’s break down the definition piece by piece, there’s a lot of interesting stuff here!

So, the following significant points can be highlighted:

1) The original vectors, indicated by red arrows, by definition not collinear. It will be appropriate to consider the case of collinear vectors a little later.

2) Vectors are taken in a strictly defined order: – "a" is multiplied by "be", not “be” with “a”. The result of vector multiplication is VECTOR, which is indicated in blue. If the vectors are multiplied in reverse order, we obtain a vector equal in length and opposite in direction (raspberry color). That is, the equality is true .

3) Now let's get acquainted with the geometric meaning of the vector product. This is a very important point! The LENGTH of the blue vector (and, therefore, the crimson vector) is numerically equal to the AREA of the parallelogram built on the vectors. In the figure, this parallelogram is shaded black.

Note : the drawing is schematic, and, naturally, the nominal length of the vector product is not equal to the area of ​​the parallelogram.

Let us recall one of the geometric formulas: The area of ​​a parallelogram is equal to the product of adjacent sides and the sine of the angle between them. Therefore, based on the above, the formula for calculating the LENGTH of a vector product is valid:

I emphasize that the formula is about the LENGTH of the vector, and not about the vector itself. What is the practical meaning? And the meaning is that in problems of analytical geometry, the area of ​​a parallelogram is often found through the concept of a vector product:

Let us obtain the second important formula. The diagonal of a parallelogram (red dotted line) divides it into two equal triangles. Therefore, the area of ​​a triangle built on vectors (red shading) can be found using the formula:

4) An equally important fact is that the vector is orthogonal to the vectors, that is . Of course, the oppositely directed vector (raspberry arrow) is also orthogonal to the original vectors.

5) The vector is directed so that basis It has right orientation. In the lesson about transition to a new basis I spoke in sufficient detail about plane orientation, and now we will figure out what space orientation is. I will explain on your fingers right hand. Mentally combine forefinger with vector and middle finger with vector. Ring finger and little finger press it into your palm. As a result thumb– the vector product will look up. This is a right-oriented basis (it is this one in the figure). Now change the vectors ( index and middle fingers) in some places, as a result the thumb will turn around, and the vector product will already look down. This is also a right-oriented basis. You may have a question: which basis has left orientation? “Assign” to the same fingers left hand vectors, and get the left basis and left orientation of space (in this case, the thumb will be located in the direction of the lower vector). Figuratively speaking, these bases “twist” or orient space in different directions. And this concept should not be considered something far-fetched or abstract - for example, the orientation of space is changed by the most ordinary mirror, and if you “pull the reflected object out of the looking glass,” then in the general case it will not be possible to combine it with the “original.” By the way, hold three fingers up to the mirror and analyze the reflection ;-)

...how good it is that you now know about right- and left-oriented bases, because the statements of some lecturers about a change in orientation are scary =)

Cross product of collinear vectors

The definition has been discussed in detail, it remains to find out what happens when the vectors are collinear. If the vectors are collinear, then they can be placed on one straight line and our parallelogram also “folds” into one straight line. The area of ​​such, as mathematicians say, degenerate parallelogram is equal to zero. The same follows from the formula - the sine of zero or 180 degrees is equal to zero, which means the area is zero

Thus, if , then . Strictly speaking, the vector product itself is equal to the zero vector, but in practice this is often neglected and they are written that it is simply equal to zero.

A special case is the cross product of a vector with itself:

Using the vector product, you can check the collinearity of three-dimensional vectors, and we will also analyze this problem, among others.

To solve practical examples you may need trigonometric table to find the values ​​of sines from it.

Well, let's light the fire:

Example 1

a) Find the length of the vector product of vectors if

b) Find the area of ​​a parallelogram built on vectors if

Solution: No, this is not a typo, I deliberately made the initial data in the clauses the same. Because the design of the solutions will be different!

a) According to the condition, you need to find length vector (cross product). According to the corresponding formula:

Answer:

If you were asked about length, then in the answer we indicate the dimension - units.

b) According to the condition, you need to find square parallelogram built on vectors. The area of ​​this parallelogram is numerically equal to the length of the vector product:

Answer:

Please note that the answer does not talk about the vector product at all; we were asked about area of ​​the figure, accordingly, the dimension is square units.

We always look at WHAT we need to find according to the condition, and, based on this, we formulate clear answer. It may seem like literalism, but there are plenty of literalists among teachers, and the assignment has a good chance of being returned for revision. Although this is not a particularly far-fetched quibble - if the answer is incorrect, then one gets the impression that the person does not understand simple things and/or has not understood the essence of the task. This point must always be kept under control when solving any problem in higher mathematics, and in other subjects too.

Where did the big letter “en” go? In principle, it could have been additionally attached to the solution, but in order to shorten the entry, I did not do this. I hope everyone understands that and is a designation for the same thing.

A popular example for a DIY solution:

Example 2

Find the area of ​​a triangle built on vectors if

The formula for finding the area of ​​a triangle through the vector product is given in the comments to the definition. The solution and answer are at the end of the lesson.

In practice, the task is really very common; triangles can generally torment you.

To solve other problems we will need:

Properties of the vector product of vectors

We have already considered some properties of the vector product, however, I will include them in this list.

For arbitrary vectors and an arbitrary number, the following properties are true:

1) In other sources of information, this item is usually not highlighted in the properties, but it is very important in practical terms. So let it be.

2) – the property is also discussed above, sometimes it is called anticommutativity. In other words, the order of the vectors matters.

3) – associative or associative vector product laws. Constants can be easily moved outside the vector product. Really, what should they do there?

4) – distribution or distributive vector product laws. There are no problems with opening the brackets either.

To demonstrate, let's look at a short example:

Example 3

Find if

Solution: The condition again requires finding the length of the vector product. Let's paint our miniature:

(1) According to associative laws, we take the constants outside the scope of the vector product.

(2) We move the constant outside the module, and the module “eats” the minus sign. The length cannot be negative.

(3) The rest is clear.

Answer:

It's time to add more wood to the fire:

Example 4

Calculate the area of ​​a triangle built on vectors if

Solution: Find the area of ​​the triangle using the formula . The catch is that the vectors “tse” and “de” are themselves presented as sums of vectors. The algorithm here is standard and somewhat reminiscent of examples No. 3 and 4 of the lesson Dot product of vectors. For clarity, we will divide the solution into three stages:

1) At the first step, we express the vector product through the vector product, in fact, let's express a vector in terms of a vector. No word yet on lengths!

(1) Substitute the expressions of the vectors.

(2) Using distributive laws, we open the brackets according to the rule of multiplication of polynomials.

(3) Using associative laws, we move all constants beyond the vector products. With a little experience, steps 2 and 3 can be performed simultaneously.

(4) The first and last terms are equal to zero (zero vector) due to the nice property. In the second term we use the property of anticommutativity of a vector product:

(5) We present similar terms.

As a result, the vector turned out to be expressed through a vector, which is what was required to be achieved:

2) In the second step, we find the length of the vector product we need. This action is similar to Example 3:

3) Find the area of ​​the required triangle:

Stages 2-3 of the solution could have been written in one line.

Answer:

The problem considered is quite common in tests, here is an example for solving it yourself:

Example 5

Find if

A short solution and answer at the end of the lesson. Let's see how attentive you were when studying the previous examples ;-)

Cross product of vectors in coordinates

, specified in an orthonormal basis, expressed by the formula:

The formula is really simple: in the top line of the determinant we write the coordinate vectors, in the second and third lines we “put” the coordinates of the vectors, and we put in strict order– first the coordinates of the “ve” vector, then the coordinates of the “double-ve” vector. If the vectors need to be multiplied in a different order, then the rows should be swapped:

Example 10

Check whether the following space vectors are collinear:
A)
b)

Solution: The check is based on one of the statements in this lesson: if the vectors are collinear, then their vector product is equal to zero (zero vector): .

a) Find the vector product:

Thus, the vectors are not collinear.

b) Find the vector product:

Answer: a) not collinear, b)

Here, perhaps, is all the basic information about the vector product of vectors.

This section will not be very large, since there are few problems where the mixed product of vectors is used. In fact, everything will depend on the definition, geometric meaning and a couple of working formulas.

A mixed product of vectors is the product of three vectors:

So they lined up like a train and can’t wait to be identified.

First, again, a definition and a picture:

Definition: Mixed work non-coplanar vectors, taken in this order, called parallelepiped volume, built on these vectors, equipped with a “+” sign if the basis is right, and a “–” sign if the basis is left.

Let's do the drawing. Lines invisible to us are drawn with dotted lines:

Let's dive into the definition:

2) Vectors are taken in a certain order, that is, the rearrangement of vectors in the product, as you might guess, does not occur without consequences.

3) Before commenting on the geometric meaning, I will note an obvious fact: the mixed product of vectors is a NUMBER: . In educational literature, the design may be slightly different; I am used to denoting a mixed product by , and the result of calculations by the letter “pe”.

A-priory the mixed product is the volume of the parallelepiped, built on vectors (the figure is drawn with red vectors and black lines). That is, the number is equal to the volume of a given parallelepiped.

Note : The drawing is schematic.

4) Let’s not worry again about the concept of orientation of the basis and space. The meaning of the final part is that a minus sign can be added to the volume. In simple words, a mixed product can be negative: .

Directly from the definition follows the formula for calculating the volume of a parallelepiped built on vectors.

Definition An ordered collection of (x 1 , x 2 , ... , x n) n real numbers is called n-dimensional vector, and numbers x i (i = ) - components, or coordinates,

Example. If, for example, a certain automobile plant must produce 50 cars, 100 trucks, 10 buses, 50 sets of spare parts for cars and 150 sets for trucks and buses per shift, then the production program of this plant can be written as a vector (50, 100 , 10, 50, 150), having five components.

Notation. Vectors are denoted by bold lowercase letters or letters with a bar or arrow at the top, e.g. a or. The two vectors are called equal, if they have the same number of components and their corresponding components are equal.

Vector components cannot be swapped, for example, (3, 2, 5, 0, 1) and (2, 3, 5, 0, 1) different vectors.
Operations on vectors. The work x= (x 1 , x 2 , ... ,x n) by a real numberλ called a vectorλ x= (λ x 1, λ x 2, ..., λ x n).

Amountx= (x 1 , x 2 , ... ,x n) and y= (y 1 , y 2 , ... ,y n) is called a vector x+y= (x 1 + y 1 , x 2 + y 2 , ... , x n + + y n).

Vector space. N -dimensional vector space R n is defined as the set of all n-dimensional vectors for which the operations of multiplication by real numbers and addition are defined.

Economic illustration. Economic illustration of n-dimensional vector space: space of goods (goods). Under goods we will understand some good or service that went on sale at a certain time in a certain place. Suppose there is a finite number n of available goods; the quantities of each of them purchased by the consumer are characterized by a set of goods

x= (x 1 , x 2 , ..., x n),

where x i denotes the amount of the i-th good purchased by the consumer. We will assume that all goods have the property of arbitrary divisibility, so that any non-negative quantity of each of them can be purchased. Then all possible sets of goods are vectors of the goods space C = ( x= (x 1 , x 2 , ... , x n) x i ≥ 0, i = ).

Linear independence. System e 1 , e 2 , ... , e m n-dimensional vectors are called linearly dependent, if there are such numbersλ 1 , λ 2 , ... , λ m , of which at least one is non-zero, such that the equalityλ 1 e 1 + λ 2 e 2 +... + λ m e m = 0; otherwise, this system of vectors is called linearly independent, that is, the indicated equality is possible only in the case when all . The geometric meaning of the linear dependence of vectors in R 3, interpreted as directed segments, explain the following theorems.

Theorem 1. A system consisting of one vector is linearly dependent if and only if this vector is zero.

Theorem 2. In order for two vectors to be linearly dependent, it is necessary and sufficient that they be collinear (parallel).

Theorem 3 . In order for three vectors to be linearly dependent, it is necessary and sufficient that they be coplanar (lie in the same plane).

Left and right triples of vectors. Triple of non-coplanar vectors a, b, c called right, if the observer from their common origin bypasses the ends of the vectors a, b, c in the order given appears to occur clockwise. Otherwise a, b, c -left three. All right (or left) triples of vectors are called the same oriented.

Basis and coordinates. Troika e 1, e 2 , e 3 non-coplanar vectors in R 3 is called basis, and the vectors themselves e 1, e 2 , e 3 - basic. Any vector a can be uniquely expanded into basis vectors, that is, represented in the form

A= x 1 e 1+x2 e 2 + x 3 e 3, (1.1)

the numbers x 1 , x 2 , x 3 in expansion (1.1) are called coordinatesa in the basis e 1, e 2 , e 3 and are designated a(x 1, x 2, x 3).

Orthonormal basis. If the vectors e 1, e 2 , e 3 are pairwise perpendicular and the length of each of them is equal to one, then the basis is called orthonormal, and the coordinates x 1 , x 2 , x 3 - rectangular. The basis vectors of an orthonormal basis will be denoted by i, j, k.

We will assume that in space R 3 the right system of Cartesian rectangular coordinates is selected (0, i, j, k}.

Vector artwork. Vector artwork A to vector b called a vector c, which is determined by the following three conditions:

1. Vector length c numerically equal to the area of ​​a parallelogram built on vectors a And b, i.e.
c= |a||b| sin( a^b).

2. Vector c perpendicular to each of the vectors a And b.

3. Vectors a, b And c, taken in the indicated order, form a right triple.

For a cross product c the designation is introduced c =[ab] or
c = a × b.

If the vectors a And b are collinear, then sin( a^b) = 0 and [ ab] = 0, in particular, [ aa] = 0. Vector products of unit vectors: [ ij]=k, [jk] = i, [ki]=j.

If the vectors a And b specified in the basis i, j, k coordinates a(a 1 , a 2 , a 3), b(b 1, b 2, b 3), then

Mixed work. If the vector product of two vectors A And b scalarly multiplied by the third vector c, then such a product of three vectors is called mixed work and is indicated by the symbol a b c.

If the vectors a, b And c in the basis i, j, k given by their coordinates
a(a 1 , a 2 , a 3), b(b 1, b 2, b 3), c(c 1, c 2, c 3), then

.

The mixed product has a simple geometric interpretation - it is a scalar, equal in absolute value to the volume of a parallelepiped built on three given vectors.

If the vectors form a right triple, then their mixed product is a positive number equal to the indicated volume; if it's a three a, b, c - left, then a b c<0 и V = - a b c, therefore V =|a b c|.

The coordinates of the vectors encountered in the problems of the first chapter are assumed to be given relative to a right orthonormal basis. Unit vector codirectional with vector A, indicated by the symbol A O. Symbol r=OM denoted by the radius vector of point M, symbols a, AB or|a|, | AB|modules of vectors are denoted A And AB.

Example 1.2. Find the angle between the vectors a= 2m+4n And b= m-n, Where m And n- unit vectors and angle between m And n equal to 120 o.

Solution. We have: cos φ = ab/ab ab =(2m+4n) (m-n) = 2m 2 - 4n 2 +2mn=
= 2 - 4+2cos120 o = - 2 + 2(-0.5) = -3; a = ; a 2 = (2m+4n) (2m+4n) =
= 4m 2 +16mn+16n 2 = 4+16(-0.5)+16=12, which means a = . b = ; b 2 =
= (m-n)(m-n) = m 2 -2mn+n 2 = 1-2(-0.5)+1 = 3, which means b = . Finally we have: cosφ = = -1/2, φ = 120 o.

Example 1.3.Knowing the vectors AB(-3,-2.6) and B.C.(-2,4,4),calculate the length of the altitude AD of triangle ABC.

Solution. Denoting the area of ​​triangle ABC by S, we get:
S = 1/2 BC AD. Then AD=2S/BC, BC= = = 6,
S = 1/2| AB ×AC|. AC=AB+BC, which means vector A.C. has coordinates
. .

Example 1.4 . Two vectors are given a(11,10,2) and b(4,0,3). Find the unit vector c, orthogonal to vectors a And b and directed so that the ordered triple of vectors a, b, c was right.

Solution.Let us denote the coordinates of the vector c with respect to a given right orthonormal basis in terms of x, y, z.

Because the c ⊥ a, c ⊥b, That ca= 0,cb= 0. According to the conditions of the problem, it is required that c = 1 and a b c >0.

We have a system of equations for finding x,y,z: 11x +10y + 2z = 0, 4x+3z=0, x 2 + y 2 + z 2 = 0.

From the first and second equations of the system we obtain z = -4/3 x, y = -5/6 x. Substituting y and z into the third equation, we have: x 2 = 36/125, whence
x =± . Using the condition a b c > 0, we get the inequality

Taking into account the expressions for z and y, we rewrite the resulting inequality in the form: 625/6 x > 0, which implies that x>0. So, x = , y = - , z =- .

Definition. The vector product of vector a (multiplicand) and a non-collinear vector (multiplicand) is the third vector c (product), which is constructed as follows:

1) its module is numerically equal to the area of ​​the parallelogram in Fig. 155), built on vectors, i.e. it is equal to the direction perpendicular to the plane of the mentioned parallelogram;

3) in this case, the direction of the vector c is chosen (from two possible ones) so that the vectors c form a right-handed system (§ 110).

Designation: or

Addition to the definition. If the vectors are collinear, then considering the figure to be (conditionally) a parallelogram, it is natural to assign zero area. Therefore, the vector product of collinear vectors is considered equal to the null vector.

Since the null vector can be assigned any direction, this agreement does not contradict paragraphs 2 and 3 of the definition.

Remark 1. In the term “vector product” the first word indicates that the result of the action is a vector (as opposed to a scalar product; cf. § 104, remark 1).

Example 1. Find the vector product where are the main vectors of the right coordinate system (Fig. 156).

1. Since the lengths of the main vectors are equal to one scale unit, the area of ​​the parallelogram (square) is numerically equal to one. This means that the modulus of the vector product is equal to one.

2. Since the perpendicular to the plane is an axis, the desired vector product is a vector collinear to the vector k; and since both of them have modulus 1, the desired vector product is equal to either k or -k.

3. Of these two possible vectors, the first one must be chosen, since the vectors k form a right-handed system (and the vectors a left-handed one).

Example 2. Find the cross product

Solution. As in example 1, we conclude that the vector is equal to either k or -k. But now we need to choose -k, since the vectors form a right-handed system (and vectors form a left-handed one). So,

Example 3. Vectors have lengths equal to 80 and 50 cm, respectively, and form an angle of 30°. Taking the meter as the unit of length, find the length of the vector product a

Solution. The area of ​​a parallelogram built on vectors is equal to The length of the desired vector product is equal to

Example 4. Find the length of the vector product of the same vectors, taking centimeters as the unit of length.

Solution. Since the area of ​​a parallelogram built on vectors is equal, the length of the vector product is equal to 2000 cm, i.e.

From a comparison of examples 3 and 4 it is clear that the length of the vector depends not only on the lengths of the factors but also on the choice of the unit of length.

Physical meaning of a vector product. Of the numerous physical quantities represented by the vector product, we will consider only the moment of force.

Let A be the point of application of force. The moment of force relative to point O is called a vector product. Since the modulus of this vector product is numerically equal to the area of ​​the parallelogram (Fig. 157), then the modulus of the moment is equal to the product of the base and the height, i.e., the force multiplied by the distance from point O to the straight line along which the force acts.

In mechanics, it is proven that for a rigid body to be in equilibrium, it is necessary that not only the sum of vectors representing the forces applied to the body be equal to zero, but also the sum of the moments of forces. In the case where all forces are parallel to one plane, the addition of vectors representing moments can be replaced by addition and subtraction of their magnitudes. But with arbitrary directions of forces, such a replacement is impossible. In accordance with this, the vector product is defined precisely as a vector, and not as a number.