What is the product of vector coordinates. Vectors for dummies

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

Example. If, for example, some car factory should produce 50 cars, 100 trucks, 10 buses, 50 sets of spare parts for cars and 150 sets for trucks and buses per shift, then production program this plant can be written as a vector (50, 100, 10, 50, 150) with five components.

Notation. Vectors are denoted by bold lowercase letters or letters with a bar or arrow at the top, for example, 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 interchanged, e.g. (3, 2, 5, 0, 1) and (2, 3, 5, 0, 1) different vectors.
Operations on vectors. work x= (x 1 , x 2 , ... ,x n) to a real numberλ called vectorλ x= (λ x 1 , λ x 2 , ... , λ x n).

sumx= (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).

The space of vectors. 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. An economic illustration of an n-dimensional vector space: space of goods (goods). Under commodity we will understand some good or service that went on sale at a certain time in a certain place. Assume that there is a finite number of goods available n; 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 bought. Then all possible sets of goods are vectors of the space of goods C = ( x= (x 1 , x 2 , ... , x n) x i ≥ 0, i = ).

Linear independence. System e 1 , e 2 , ... , e m n-dimensional vectors is called linearly dependent if there are such numbersλ 1 , λ 2 , ... , λ m , of which at least one is nonzero, which satisfies the equalityλ1 e 1 + λ2 e 2+...+λm e m = 0; otherwise this system vectors is called linearly independent, that is, this 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 a single vector is linearly dependent if and only if this vector is zero.

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

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

Left and right triples of vectors. A 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 that order seems to proceed clockwise. Otherwise a, b, c -left triple. All right (or left) triples of vectors are called equally oriented.

Basis and coordinates. Troika e 1, e 2 , e 3 non-coplanar vectors in R 3 called basis, and the vectors themselves e 1, e 2 , e 3 - basic. Any vector a can be expanded in a unique way in terms of basis vectors, that is, it can be 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 basis e 1, e 2 , e 3 and are denoted 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 i, j, k.

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

Vector product. vector art a per vector b called vector c, which is determined by the following three conditions:

1. Vector length c numerically equal to the area of ​​the parallelogram built on the 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 that order, form a right triple.

For vector 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 orts: [ ij]=k, [jk] = i, [ki]=j.

If the vectors a and b given 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 cross product of two vectors a and b scalar multiplied by the third vector c, then such a product of three vectors is called mixed product and is denoted by the symbol a bc.

If the vectors a, b and c in basis i, j, k set 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, in absolute value equal 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 specified volume; if the 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 the right orthonormal basis. Unit vector codirectional to vector a, denoted by the symbol a about. Symbol r=OM denoted by the radius vector of the point M, the symbols a, AB or|a|, | AB |the modules of vectors are denoted a and AB.

Example 1.2. Find the angle between 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, so a = . b= ; b 2 =
= (m-n)(m-n) = m 2 -2mn+n 2 = 1-2(-0.5)+1 = 3, so b = . Finally we have: cosφ \u003d -1/2, φ \u003d 120 o.

Example 1.3.Knowing vectors AB(-3,-2.6) and BC(-2,4,4), calculate the height AD of triangle ABC.

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

Example 1.4 . Given two vectors 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 the given right orthonormal basis in terms of x, y, z.

Because the c ⊥ a, c ⊥b, then ca= 0, cb= 0. By the condition 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 get z = -4/3 x, y = -5/6 x. Substituting y and z into the third equation, we will have: x 2 = 36/125, whence
x=± . Using 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, whence it follows that x>0. So x = , y = - , z = - .

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 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 high school, you will need first volume. Rarely occurring tasks may fall out of my field of vision, and tutorial will provide invaluable assistance.

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 motion physical body: agree, to enter the doors of the institute or to leave the doors of the 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. AT 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:
and so on. 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 nonzero 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 is lots of identical directional segments. school definition 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

AT school course geometry considers a number of actions and rules with vectors: 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 invest in it physical meaning: 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. In this way: 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 set 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 in geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of scientific style presentation, but a plus to 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:

That is, 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

Solution: 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 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

Solution: 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 it has a couple more important points 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 multiplier 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? . In this way: . 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.

During the decision various tasks roots are common, 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:

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

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 .

In this lesson, we will look at two more operations with vectors: cross product of vectors and mixed product of vectors (immediate link for those who need it). It's okay, it sometimes happens that for complete happiness, in addition to dot product of vectors, more and more is needed. Such is vector addiction. One may get the impression that we are getting into the jungle of analytic geometry. This is not true. In this section of higher mathematics, there is generally little firewood, except perhaps enough for Pinocchio. In fact, the material is very common and simple - hardly more difficult than the same scalar product, even typical tasks will be less. The main thing in analytic geometry, as many will see or have already seen, is NOT TO MISTAKE CALCULATIONS. Repeat like a spell, and you will be happy =)

If the 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? When I was little, I could juggle two and even three balls. It worked out well. Now there is no need to juggle at all, since we will consider only space 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. Already easier!

In this operation, in the same way as in the scalar product, two vectors. Let it be imperishable letters.

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

And immediately question: if in dot product of vectors two vectors are involved, and here two vectors are also multiplied, then what is the difference? A clear difference, first of all, in the RESULT:

The result of the scalar product of vectors is a NUMBER:

The result of the cross product of vectors is a VECTOR: , that is, we multiply the vectors and get a vector again. Closed club. Actually, hence the name of the operation. In various educational literature, the 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: cross product non-collinear vectors , taken in this order, is 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:

We analyze the definition by bones, there is a lot of interesting things!

So, we can highlight the following significant points:

1) Source 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 taken in a strict order: – "a" is multiplied by "be", not "be" to "a". The result of vector multiplication is VECTOR , which is denoted in blue. If the vectors are multiplied in reverse order, then we get a vector equal in length and opposite in direction (crimson color). That is, the equality .

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 in black.

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

We 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 foregoing, the formula for calculating the LENGTH of a vector product is valid:

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

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

4) Not less than important fact is that the vector is orthogonal to the vectors , that is, . Of course, the oppositely directed vector (crimson arrow) is also orthogonal to the original vectors .

5) The vector is directed so that basis It has right orientation. In a lesson about transition to a new basis I have spoken in detail about plane orientation, and now we will figure out what the orientation of space 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 into your palm. As a result thumb - the vector product will look up. This is the right-oriented basis (it is in the figure). Now swap 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. Perhaps you have a question: what basis has a left orientation? "Assign" the same fingers left hand vectors , and get the left basis and left space orientation (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 most ordinary mirror changes the orientation of space, and if you “pull the reflected object out of the mirror”, then in general it will not be possible to combine it with the “original”. By the way, bring three fingers 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 the change of orientation are terrible =)

Vector product of collinear vectors

The definition has been worked out 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 zero. The same follows from the formula - the sine of zero or 180 degrees is equal to zero, which means that the area is zero

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

special case is the cross product of a vector and itself:

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

To solve practical examples, it may be necessary trigonometric table to find the values ​​of the sines from it.

Well, let's start a 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 intentionally made the initial data in the condition items the same. Because the design of the solutions will be different!

a) According to the condition, it is required to find length vector (vector product). According to the corresponding formula:

Answer:

Since it was asked about the length, then in the answer we indicate the dimension - units.

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

Answer:

Please note that in the answer about the vector product there is no talk at all, we were asked about figure area, respectively, the dimension is square units.

We always look at WHAT is required to be found by the condition, and, based on this, we formulate clear answer. It may seem like literalism, but there are enough literalists among the teachers, and the task with good chances will be returned for revision. Although this is not a particularly strained nitpick - 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 moment should always be kept under control, solving any problem in higher mathematics, and in other subjects too.

Where did the big letter "en" go? In principle, it could be additionally stuck to the solution, but in order to shorten the record, I did not. I hope everyone understands that and is the designation of the same thing.

A popular example for a do-it-yourself 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. Solution and answer at the end of the lesson.

In practice, the task is really very common, triangles can generally be tortured.

To solve other problems, we need:

Properties of the cross 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 distinguished 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) - combination or associative vector product laws. The constants are easily taken out of the limits of the vector product. Really, what are they doing there?

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

As a demonstration, consider a short example:

Example 3

Find if

Solution: By condition, it is again required to find the length of the vector product. Let's paint our miniature:

(1) According to the associative laws, we take out the constants beyond the limits of the vector product.

(2) We take the constant out of the module, while the module “eats” the minus sign. The length cannot be negative.

(3) What follows is clear.

Answer:

It's time to throw wood on the fire:

Example 4

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

Solution: Find the area of ​​a triangle using the formula . The snag is that the vectors "ce" and "te" are themselves represented as sums of vectors. The algorithm here is standard and is somewhat reminiscent of examples No. 3 and 4 of the lesson. Dot product of vectors. Let's break it down into three steps for clarity:

1) At the first step, we express the vector product through the vector product, in fact, express the vector in terms of the vector. No word on length yet!

(1) We substitute expressions of vectors .

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

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

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

(5) We present similar terms.

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

2) At 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 desired triangle:

Steps 2-3 of the solution could be arranged in one line.

Answer:

The considered problem is quite common in control work, here's an example for a do-it-yourself solution:

Example 5

Find if

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

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

Example 10

Check if the following space vectors are collinear:
a)
b)

Solution: Validation based on one of the assertions this lesson: if the vectors are collinear, then their vector product is zero (zero vector): .

a) Find the vector product:

So 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 rest on the definition, geometric meaning and a couple of working formulas.

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

This is how they lined up like a train and wait, they can’t wait until they are calculated.

First again the definition and picture:

Definition: Mixed product non-coplanar vectors , taken in this order, is called volume of the parallelepiped, 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 by a dotted line:

Let's dive into the definition:

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

3) Before commenting on the geometric meaning, I note obvious fact: the mixed product of vectors is a NUMBER: . In educational literature, the design may be somewhat different, I used to designate a mixed product through, and the result of calculations with the letter "pe".

By definition 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 the given parallelepiped.

Note : The drawing is schematic.

4) Let's not bother again with the concept of the 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 terms, the mixed product can be negative: .

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

Unit vector- this is vector, the absolute value (modulus) of which is equal to one. To denote a unit vector, we will use the subscript e. So, if a vector is given a, then its unit vector will be the vector a e. This unit vector points in the same direction as the vector itself a, and its modulus is equal to one, that is, a e \u003d 1.

Obviously, a= a a e (a - vector modulus a). This follows from the rule by which the operation is performed multiplication of a scalar by a vector.

Unit vectors often associated with the coordinate axes of the coordinate system (in particular, with the axes of the Cartesian coordinate system). Directions of these vectors coincide with the directions of the corresponding axes, and their origins are often combined with the origin of the coordinate system.

Let me remind you that Cartesian coordinate system in space is traditionally called a triple of mutually perpendicular axes intersecting at a point called the origin. The coordinate axes are usually denoted by the letters X, Y, Z and are called the abscissa axis, the ordinate axis, and the applicate axis, respectively. Descartes himself used only one axis, on which the abscissas were plotted. merit of use systems axes belongs to his students. Therefore the phrase cartesian system coordinates historically wrong. Better talk rectangular coordinate system or orthogonal coordinate system. Nevertheless, we will not change traditions and in the future we will assume that the Cartesian and rectangular (orthogonal) coordinate systems are one and the same.

Unit vector, directed along the X axis, is denoted i, unit vector, directed along the Y axis, is denoted j, a unit vector, directed along the Z axis, is denoted k. Vectors i, j, k called orts(Fig. 12, left), they have single modules, that is
i = 1, j = 1, k = 1.

axes and orts rectangular coordinate system in some cases they have other names and designations. So, the abscissa axis X can be called the tangent axis, and its unit vector is denoted τ (Greek small letter tau), the y-axis is the normal axis, its unit vector is denoted n, the applicate axis is the axis of the binormal, its unit vector is denoted b. Why change the names if the essence remains the same?

The fact is that, for example, in mechanics, when studying the motion of bodies, a rectangular coordinate system is used very often. So, if the coordinate system itself is motionless, and the change in the coordinates of a moving object is tracked in this motionless system, then usually the axes denote X, Y, Z, and their orts respectively i, j, k.

But often, when an object moves along some kind of curvilinear trajectory (for example, along a circle), it is more convenient to consider mechanical processes in a coordinate system moving with this object. It is for such a moving coordinate system that other names of the axes and their unit vectors are used. It's just accepted. In this case, the X-axis is directed tangentially to the trajectory at the point where this moment this object is located. And then this axis is no longer called the X axis, but the tangent axis, and its unit vector is no longer denoted i, a τ . The Y axis is directed along the radius of curvature of the trajectory (in the case of movement in a circle - to the center of the circle). And since the radius is perpendicular to the tangent, the axis is called the axis of the normal (perpendicular and normal are the same thing). The ort of this axis is no longer denoted j, a n. The third axis (the former Z) is perpendicular to the two previous ones. This is a binormal with a vector b(Fig. 12, right). By the way, in this case rectangular coordinate system often referred to as "natural" or natural.