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 constructed 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 length unit.
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.
Finally, I got my hands on this vast and long-awaited topic. analytical geometry. First, a little about this section of higher mathematics... Surely you now remember a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “analytical solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical same method involves solving problems mainly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent; often it is enough to carefully apply the necessary formulas - and the answer is ready! No, of course, we won’t be able to do this without drawings at all, and besides, for a better understanding of the material, I will try to cite them beyond necessity.
The newly opened course of lessons on geometry does not pretend to be theoretically complete; it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need more complete help on any subsection, I recommend the following quite accessible literature:
1) A thing that, no joke, several generations are familiar with: School textbook on geometry, authors - L.S. Atanasyan and Company. This school locker room hanger has already gone through 20 (!) reprints, which, of course, is not the limit.
2) Geometry in 2 volumes. Authors L.S. Atanasyan, Bazylev V.T.. This is literature for high school, you will need first volume. Rarely encountered tasks may fall out of my sight, and the tutorial will be of invaluable help.
Both books can be downloaded for free online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download examples in higher mathematics.
Among the tools, I again propose my own development - software package in analytical geometry, which will greatly simplify life and save a lot of time.
It is assumed that the reader is familiar with basic geometric concepts and figures: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello to repeaters)
And now we will consider sequentially: the concept of a vector, actions with vectors, vector coordinates. I recommend reading further the most important article Dot product of vectors, and also Vector and mixed product of vectors. A local task - Division of a segment in this respect - will also not be superfluous. Based on the above information, you can master equation of a line in a plane With simplest examples of solutions, which will allow learn to solve geometry problems. The following articles are also useful: Equation of a plane in space, Equations of a line in space, Basic problems on a straight line and a plane, other sections of analytical geometry. Naturally, standard tasks will be considered along the way.
Vector concept. Free vector
First, let's repeat the school definition of a vector. Vector called directed a segment for which its beginning and end are indicated:
In this case, the beginning of the segment is the point, the end of the segment is the point. The vector itself is denoted by . Direction is essential, if you move the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must agree, entering the doors of an institute or leaving the doors of an institute are completely different things.
It is convenient to consider individual points of a plane or space as the so-called zero vector. For such a vector, the end and beginning coincide.
!!! Note: Here and further, you can assume that the vectors lie in the same plane or you can assume that they are located in space - the essence of the material presented is valid for both the plane and space.
Designations: Many immediately noticed the stick without an arrow in the designation and said, there’s also an arrow at the top! True, you can write it with an arrow: , but it is also possible the entry that I will use in the future. Why? Apparently, this habit developed for practical reasons; my shooters at school and university turned out to be too different-sized and shaggy. In educational literature, sometimes they don’t bother with cuneiform writing at all, but highlight the letters in bold: , thereby implying that this is a vector.
That was stylistics, and now about ways to write vectors:
1) Vectors can be written in two capital Latin letters: 
and so on. In this case, the first letter Necessarily denotes the beginning point of the vector, and the second letter denotes the end point of the vector.
2) Vectors are also written in small Latin letters:
In particular, our vector can be redesignated for brevity by a small Latin letter.
Length or module a non-zero vector is called the length of the segment. The length of the zero vector is zero. Logical.
The length of the vector is indicated by the modulus sign: ,
We will learn how to find the length of a vector (or we will repeat it, depending on who) a little later.
This was basic information about vectors, familiar to all schoolchildren. In analytical geometry, the so-called free vector.
To put it simply - the vector can be plotted from any point:
We are accustomed to calling such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, they are the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” this or that vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a vector of arbitrary length and direction - it can be “cloned” an infinite number of times and at any point in space, in fact, it exists EVERYWHERE. There is such a student saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is mathematically correct - the vector can be attached there too. But don’t rush to rejoice, it’s the students themselves who often suffer =)
So, free vector- This a bunch of identical directed segments. The school definition of a vector, given at the beginning of the paragraph: “A directed segment is called a vector...” implies specific a directed segment taken from a given set, which is tied to a specific point in the plane or space.
It should be noted that from the point of view of physics, the concept of a free vector is generally incorrect, and the point of application of the vector matters. Indeed, a direct blow of the same force on the nose or forehead, enough to develop my stupid example, entails different consequences. However, unfree vectors are also found in the course of vyshmat (don’t go there :)).
Actions with vectors. Collinearity of vectors
A school geometry course covers a number of actions and rules with vectors: addition according to the triangle rule, addition according to the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, let us repeat two rules that are especially relevant for solving problems of analytical geometry.
The rule for adding vectors using the triangle rule
Consider two arbitrary non-zero vectors and : 
You need to find the sum of these vectors. Due to the fact that all vectors are considered free, we will set aside the vector from end vector: 
The sum of vectors is the vector. For a better understanding of the rule, it is advisable to put a physical meaning into it: let some body travel along the vector , and then along the vector . Then the sum of vectors is the vector of the resulting path with the beginning at the departure point and the end at the arrival point. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way very lean along a zigzag, or maybe on autopilot - along the resulting vector of the sum.
By the way, if the vector is postponed from started vector, then we get the equivalent parallelogram rule addition of vectors.
First, about collinearity of vectors. The two vectors are called collinear, if they lie on the same line or on parallel lines. Roughly speaking, we are talking about parallel vectors. But in relation to them, the adjective “collinear” is always used.
Imagine two collinear vectors. If the arrows of these vectors are directed in the same direction, then such vectors are called co-directed. If the arrows point in different directions, then the vectors will be opposite directions.
Designations: collinearity of vectors is written with the usual parallelism symbol: , while detailing is possible: (vectors are co-directed) or (vectors are oppositely directed).
The work a non-zero vector on a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .
The rule for multiplying a vector by a number is easier to understand with the help of a picture: 
Let's look at it in more detail:
1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.
2) Length. If the multiplier is contained within or , then the length of the vector decreases. So, the length of the vector is half the length of the vector. If the modulus of the multiplier is greater than one, then the length of the vector increases in time.
3) Please note that all vectors are collinear, while one vector is expressed through another, for example, . The reverse is also true: if one vector can be expressed through another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to the original) vector.
4) The vectors are co-directed. Vectors and are also co-directed. Any vector of the first group is oppositely directed with respect to any vector of the second group.
Which vectors are equal?
Two vectors are equal if they are in the same direction and have the same length. Note that codirectionality implies collinearity of vectors. The definition would be inaccurate (redundant) if we said: “Two vectors are equal if they are collinear, codirectional, and have the same length.”
From the point of view of the concept of a free vector, equal vectors are the same vector, as discussed in the previous paragraph.
Vector coordinates on the plane and in space
The first point is to consider vectors on the plane. Let us depict a Cartesian rectangular coordinate system and plot it from the origin of coordinates single vectors and :
Vectors and orthogonal. Orthogonal = Perpendicular. I recommend that you slowly get used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity And orthogonality.
Designation: The orthogonality of vectors is written with the usual perpendicularity symbol, for example: .
The vectors under consideration are called coordinate vectors or orts. These vectors form basis on surface. What a basis is, I think, is intuitively clear to many; more detailed information can be found in the article Linear (non) dependence of vectors. Basis of vectors In simple words, the basis and origin of coordinates define the entire system - this is a kind of foundation on which a full and rich geometric life boils.
Sometimes the constructed basis is called orthonormal basis of the plane: “ortho” - because the coordinate vectors are orthogonal, the adjective “normalized” means unit, i.e. the lengths of the basis vectors are equal to one.
Designation: the basis is usually written in parentheses, inside which in strict sequence basis vectors are listed, for example: . Coordinate vectors it is forbidden rearrange.
Any plane vector the only way expressed as: 
, Where - numbers which are called vector coordinates in this basis. And the expression itself 
called vector decompositionby basis .
Dinner served:
Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing a vector into a basis, the ones just discussed are used:
1) the rule for multiplying a vector by a number: and ;
2) addition of vectors according to the triangle rule: .
Now mentally plot the vector from any other point on the plane. It is quite obvious that his decay will “follow him relentlessly.” Here it is, the freedom of the vector - the vector “carries everything with itself.” This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be plotted from the origin; one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change! True, you don’t need to do this, since the teacher will also show originality and draw you a “credit” in an unexpected place.
Vectors illustrate exactly the rule for multiplying a vector by a number, the vector is codirectional with the base vector, the vector is directed opposite to the base vector. For these vectors, one of the coordinates is equal to zero; you can meticulously write it like this:
And the basis vectors, by the way, are like this: (in fact, they are expressed through themselves).
And finally: , . By the way, what is vector subtraction, and why didn’t I talk about the subtraction rule? Somewhere in linear algebra, I don’t remember where, I noted that subtraction is a special case of addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , 
. Rearrange the terms and see in the drawing how well the good old addition of vectors according to the triangle rule works in these situations.
The considered decomposition of the form 
sometimes called vector decomposition in the ort system(i.e. in a system of unit vectors). But this is not the only way to write a vector; the following option is common:
Or with an equal sign:
The basis vectors themselves are written as follows: and
That is, the coordinates of the vector are indicated in parentheses. In practical problems, all three notation options are used.
I doubted whether to speak, but I’ll say it anyway: vector coordinates cannot be rearranged. Strictly in first place we write down the coordinate that corresponds to the unit vector, strictly in second place we write down the coordinate that corresponds to the unit vector. Indeed, and are two different vectors.
We figured out the coordinates on the plane. Now let's look at vectors in three-dimensional space, almost everything is the same here! It will just add one more coordinate. It’s hard to make three-dimensional drawings, so I’ll limit myself to one vector, which for simplicity I’ll set aside from the origin: 
Any 3D space vector the only way expand over an orthonormal basis: 
, where are the coordinates of the vector (number) in this basis.
Example from the picture: 
. Let's see how the vector rules work here. First, multiplying the vector by a number: (red arrow), (green arrow) and (raspberry arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector begins at the initial point of departure (beginning of the vector) and ends at the final point of arrival (end of the vector).
All vectors of three-dimensional space, naturally, are also free; try to mentally set aside the vector from any other point, and you will understand that its decomposition “will remain with it.”
Similar to the flat case, in addition to writing 
versions with brackets are widely used: either .
If one (or two) coordinate vectors are missing in the expansion, then zeros are put in their place. Examples:
vector (meticulously 
) – let’s write ;
vector (meticulously 
) – let’s write ;
vector (meticulously 
) – let’s write .
The basis vectors are written as follows:
This, perhaps, is all the minimum theoretical knowledge necessary to solve problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that teapots re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time to better assimilate the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I note that the materials on the site are not enough to pass the theoretical test or colloquium on geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of presentation, but a plus to your understanding of the subject. To receive detailed theoretical information, please bow to Professor Atanasyan.
And we move on to the practical part:
The simplest problems of analytical geometry.
Actions with vectors in coordinates
It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, you don’t even have to remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on your shirt; many things are familiar to you from school.
The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas... you will see for yourself.
How to find a vector from two points?
If two points of the plane and are given, then the vector has the following coordinates: 
If two points in space and are given, then the vector has the following coordinates:
That is, from the coordinates of the end of the vector you need to subtract the corresponding coordinates beginning of the vector.
Exercise: For the same points, write down the formulas for finding the coordinates of the vector. Formulas at the end of the lesson.
Example 1
Given two points of the plane and . Find vector coordinates
Solution: according to the corresponding formula:
Alternatively, the following entry could be used:
Aesthetes will decide this:
Personally, I'm used to the first version of the recording.
Answer:
According to the condition, it was not necessary to construct a drawing (which is typical for problems of analytical geometry), but in order to clarify some points for dummies, I will not be lazy: 
You definitely need to understand difference between point coordinates and vector coordinates:
Point coordinates– these are ordinary coordinates in a rectangular coordinate system. I think everyone knows how to plot points on a coordinate plane from the 5th-6th grade. Each point has a strict place on the plane, and they cannot be moved anywhere.
The coordinates of the vector– this is its expansion according to the basis, in this case. Any vector is free, so if necessary, we can easily move it away from some other point in the plane. It is interesting that for vectors you don’t have to build axes or a rectangular coordinate system at all; you only need a basis, in this case an orthonormal basis of the plane.
The records of coordinates of points and coordinates of vectors seem to be similar: , and meaning of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, also applies to space.
Ladies and gentlemen, let's fill our hands:
Example 2
a) Points and are given. Find vectors and .
b) Points are given 
And . Find vectors and .
c) Points and are given. Find vectors and .
d) Points are given. Find vectors 
.
Perhaps that's enough. These are examples for you to decide on your own, try not to neglect them, it will pay off ;-). There is no need to make drawings. Solutions and answers at the end of the lesson.
What is important when solving analytical geometry problems? It is important to be EXTREMELY CAREFUL to avoid making the masterful “two plus two equals zero” mistake. I apologize right away if I made a mistake somewhere =)
How to find the length of a segment?
The length, as already noted, is indicated by the modulus sign.
If two points of the plane are given and , then the length of the segment can be calculated using the formula
If two points in space and are given, then the length of the segment can be calculated using the formula
Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard
Example 3
Solution: according to the corresponding formula:
Answer: 
For clarity, I will make a drawing 
Line segment - this is not a vector, and, of course, you cannot move it anywhere. In addition, if you draw to scale: 1 unit. = 1 cm (two notebook cells), then the resulting answer can be checked with a regular ruler by directly measuring the length of the segment.
Yes, the solution is short, but there are a couple more important points in it that I would like to clarify:
Firstly, in the answer we put the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, a mathematically correct solution would be the general formulation: “units” - abbreviated as “units.”
Secondly, let us repeat the school material, which is useful not only for the task considered:
pay attention to important technique – removing the multiplier from under the root. As a result of the calculations, we have a result and good mathematical style involves removing the factor from under the root (if possible). In more detail the process looks like this: 
. Of course, leaving the answer as is would not be a mistake - but it would certainly be a shortcoming and a weighty argument for quibbling on the part of the teacher.
Here are other common cases: 
Often the root produces a fairly large number, for example . What to do in such cases? Using the calculator, we check whether the number is divisible by 4: . Yes, it was completely divided, thus: 
. Or maybe the number can be divided by 4 again? . Thus: 
. The last digit of the number is odd, so dividing by 4 for the third time will obviously not work. Let's try to divide by nine: . As a result:
Ready.
Conclusion: if under the root we get a number that cannot be extracted as a whole, then we try to remove the factor from under the root - using a calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.
When solving various problems, roots are often encountered; always try to extract factors from under the root in order to avoid a lower grade and unnecessary problems with finalizing your solutions based on the teacher’s comments.
Let's also repeat squaring roots and other powers: 
The rules for operating with powers in general form can be found in a school algebra textbook, but I think from the examples given, everything or almost everything is already clear.
Task for independent solution with a segment in space:
Example 4
Points and are given. Find the length of the segment.
The solution and answer are at the end of the lesson.
How to find the length of a vector?
If a plane vector is given, then its length is calculated by the formula.
If a space vector is given, then its length is calculated by the formula 
.
7.1. Definition of cross product
Three non-coplanar vectors a, b and c, taken in the indicated order, form a right-handed triplet if, from the end of the third vector c, the shortest turn from the first vector a to the second vector b is seen to be counterclockwise, and a left-handed triplet if clockwise (see Fig. . 16).
The vector product of vector a and vector b is called vector c, which:
1. Perpendicular to vectors a and b, i.e. c ^ a and c ^ b ;
2. Has a length numerically equal to the area of a parallelogram constructed on vectors a andb as on the sides (see Fig. 17), i.e.
3. Vectors a, b and c form a right-handed triple.
The cross product is denoted a x b or [a,b]. The following relations between the unit vectors i directly follow from the definition of the vector product, j And k(see Fig. 18):
i x j = k, j x k = i, k x i = j.
Let us prove, for example, that i xj =k.
1) k ^ i, k ^ j ;
2) |k |=1, but | i x j| = |i | |J | sin(90°)=1;
3) vectors i, j and k form a right triple (see Fig. 16).
7.2. Properties of a cross product
1. When rearranging the factors, the vector product changes sign, i.e. and xb =(b xa) (see Fig. 19).
Vectors a xb and b xa are collinear, have the same modules (the area of the parallelogram remains unchanged), but are oppositely directed (triples a, b, a xb and a, b, b x a of opposite orientation). That is axb = -(b xa).
2. The vector product has a combining property with respect to the scalar factor, i.e. l (a xb) = (l a) x b = a x (l b).
Let l >0. Vector l (a xb) is perpendicular to vectors a and b. Vector ( l a)x b is also perpendicular to vectors a and b(vectors a, l but lie in the same plane). This means that the vectors l(a xb) and ( l a)x b collinear. It is obvious that their directions coincide. They have the same length:
That's why l(a xb)= l a xb. It is proved in a similar way for l<0.
3. Two non-zero vectors a and b are collinear if and only if their vector product is equal to the zero vector, i.e. a ||b<=>and xb =0.
In particular, i *i =j *j =k *k =0 .
4. The vector product has the distribution property:
(a+b) xc = a xc + b xs.
We will accept without proof.
7.3. Expressing the cross product in terms of coordinates
We will use the cross product table of vectors i, j and k:
if the direction of the shortest path from the first vector to the second coincides with the direction of the arrow, then the product is equal to the third vector; if it does not coincide, the third vector is taken with a minus sign.
Let two vectors a =a x i +a y be given j+a z k and b =b x i+b y j+b z k. Let's find the vector product of these vectors by multiplying them as polynomials (according to the properties of the vector product):
The resulting formula can be written even more briefly:
since the right side of equality (7.1) corresponds to the expansion of the third-order determinant in terms of the elements of the first row. Equality (7.2) is easy to remember.
7.4. Some applications of cross product
Establishing collinearity of vectors
Finding the area of a parallelogram and a triangle
According to the definition of the vector product of vectors A and b |a xb | =|a | * |b |sin g, i.e. S pairs = |a x b |. And, therefore, D S =1/2|a x b |.
Determination of the moment of force about a point
Let a force be applied at point A F =AB let it go ABOUT- some point in space (see Fig. 20).
It is known from physics that moment of force F relative to the point ABOUT called a vector M, which passes through the point ABOUT And:
1) perpendicular to the plane passing through the points O, A, B;
2) numerically equal to the product of force per arm
3) forms a right triple with vectors OA and A B.
Therefore, M = OA x F.
Finding linear rotation speed
Speed v point M of a rigid body rotating with angular velocity w around a fixed axis, is determined by Euler’s formula v =w xr, where r =OM, where O is some fixed point of the axis (see Fig. 21).
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 goes 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 =- .