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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Vector addition

There are two ways to add vectors.

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

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

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

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

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

When adding vectors and we get:

Vector subtraction

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

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

Multiplying a vector by a number

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

Dot product of vectors

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

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

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

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

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

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

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

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

Finally I got my hands on a 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 co-directed 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 .

Definition

Scalar quantity- a quantity that can be characterized by a number. For example, length, area, mass, temperature, etc.

Vector called the directed segment $\overline(A B)$;

point $A$ is the beginning, point $B$ is the end of the vector (Fig. 1).

Definition

A vector is denoted either by two capital letters - its beginning and end: $\overline(A B)$ or by one small letter: $\overline(a)$. If the beginning and end of a vector coincide, then such a vector is called zero

. collinear Most often, the zero vector is denoted as $\overline(0)$.

Definition

The vectors are called co-directed, if their directions coincide: $\overline(a) \uparrow \uparrow \overline(b)$ (Fig. 3, a). Two collinear vectors $\overline(a)$ and $\overline(b)$ are called oppositely directed, if their directions are opposite: $\overline(a) \uparrow \downarrow \overline(b)$ (Fig. 3, b).

Definition

. coplanar, if they are parallel to the same plane or lie in the same plane (Fig. 4).

Two vectors are always coplanar.

Definition

Length (module) vector $\overline(A B)$ is the distance between its beginning and end: $|\overline(A B)|$

Detailed theory about vector length at the link.

The length of the zero vector is zero.

Definition

A vector whose length is equal to one is called unit vector or ortom.

. equal, if they lie on one or parallel lines; their directions coincide and their lengths are equal.

In other words, two vectors equal, if they are collinear, codirectional and have equal lengths:

$\overline(a)=\overline(b)$ if $\overline(a) \uparrow \uparrow \overline(b),|\overline(a)|=|\overline(b)|$

At an arbitrary point $M$ of space, one can construct a single vector $\overline(M N)$ equal to the given vector $\overline(A B)$.

Vector algebra

Definition:

A vector is a directed segment in a plane or in space.

Characteristics:

1) vector length

Definition:

Two vectors are called collinear if they lie on parallel lines.

Definition:

Two collinear vectors are called codirectional if their directions coincide ( ) Otherwise they are called oppositely directed (↓ ).

Definition:

Two vectors are equal if they are co-directional and have the same length.

For example,

Operations:

1. Multiplying a vector by a number

If
, That

↓If < 0

The direction of the zero vector is arbitrary

Properties of multiplication by a number

2. Vector addition

Parallelogram rule:

Addition properties:

- such vectors are called opposite to each other. It's easy to see that

Joint properties:

ABOUT definition:

The angle between two vectors is the angle that is obtained if these vectors are plotted from one point, 0    

3. Dot product of vectors.

, Where - angle between vectors

Properties of the scalar product of vectors:

1) (equalities take place in the case of opposite direction and co-direction of vectors, respectively)

3)

If
, then the sign of the product is positive, If ↓that is negative

)

6), that is
, or any of the vectors is zero

7)

Application of vectors

1.

MN – midline

Prove that

Proof:

, subtract the vector from both sides
:

2.

Prove that the diagonals of a rhombus are perpendicular

Proof:

Find:

Solution:

Decomposition of vectors into bases.

Definition:

A linear combination of vectors (LCV) is a sum of the form

(LKV)

Where 1 ,  2 , …  s – arbitrary set of numbers

Definition:

An LCI is said to be non-trivial if all i = 0, otherwise it is called nontrivial.

Consequence:

A non-trivial LCV has at least one non-zero coefficient To  0

Definition:

Vector system
called linearly independent (LNI),If() = 0  All i  0,

that is, only its trivial LC is equal to zero.

Consequence:

The nontrivial LC of linearly independent vectors is nonzero

Examples:

1)
- LNZ

2) Let And lie in the same plane, then
- LNZ , non-collinear

3) Let , , do not belong to the same plane, then they form a LNZ system of vectors

Theorem:

If a system of vectors is linearly independent, then at least one of them is a linear combination of the others.

Proof:

 Let () = 0 and not all I are equal to zero. Without losing generality, let s  0. Then
, and this is a linear combination.

 Let

Then, that is, LZ.

Theorem:

Any 3 vectors on a plane are linearly dependent.

Proof:

Let the vectors be given
, possible cases:

1)


2) non-collinear

Let's express it through and:
, where
- non-trivial LC.

Theorem:

Let
- LZ

Then any “wider” system is LZ

Proof:

Since - LZ, then there is at least one i  0, and () = 0

Then and () = 0

Definition:

A system of linearly independent vectors is called maximal if, when any other vector is added to it, it becomes linearly dependent.

Definition:

The dimension of space (plane) is the number of vectors in a maximal linearly independent system of vectors.

Definition:

A basis is any ordered maximal linearly independent system of vectors.

Definition:

A basis is called normalized if the vectors included in it have a length equal to one.

Definition:

A basis is called orthogonal if all its elements (vectors) are pairwise perpendicular.

Theorem:

A system of orthogonal vectors is always linearly independent (if there are no zero vectors).

Proof:

Let be a system of orthogonal vectors (non-zero), that is
. Suppose , we multiply this LC scalarly by the vector :

The first bracket is non-zero (the square of the vector length), and all other brackets are equal to zero by condition. Then 1 = 0. Similarly for 2 …  s

Theorem:

Let M = - basis. Then we can represent any vector in the form:

where are the coefficients 2 …  s are determined uniquely (these are the coordinates of the vector relative to the basis M).

Proof:

1)
=
- LZ (according to the basis condition)

then - nontrivial

A) 0 = 0 which is impossible, since it turns out that M – LZ

b) 0  0

divide by 0

those. there is a personal account

2) Let's prove it by contradiction. Let be another representation of the vector (i.e. at least one pair
). Let's subtract the formulas from each other:

- LC is non-trivial.

But according to the condition - basis a contradiction, that is, the decomposition is unique.

Conclusion:

Every basis M determines a one-to-one correspondence between vectors and their coordinates relative to the basis M.

Designations:

M = - arbitrary vector

Then

The uniqueness of the coefficients of a linear combination is proved in the same way as in the previous corollary.

Consequence: Any four vectors are linearly dependent

Chapter 4. The concept of basis. Properties of a vector in a given basis

Definition:Basis in space is any ordered triple of non-coplanar vectors.

Definition:Basis on the plane is any ordered pair of noncollinear vectors.

A basis in space allows each vector to be uniquely associated with an ordered triple of numbers - the coefficients of representing this vector in the form of a linear combination of basis vectors. On the contrary, we associate a vector with each ordered triple of numbers using a basis if we make a linear combination.

Numbers are called components (or coordinates ) vector in a given basis (written ).

Theorem: When adding two vectors, their coordinates are added. When a vector is multiplied by a number, all coordinates of the vector are multiplied by that number.

Indeed, if , That

The definition and properties of vector coordinates on a plane are similar. You can easily formulate them yourself.

Chapter 5. Vector Projection

Under angle between vectors refers to the angle between vectors equal to data and having a common origin. If the angle reference direction is not specified, then the angle between the vectors is considered to be the angle that does not exceed π. If one of the vectors is zero, then the angle is considered equal to zero. If the angle between the vectors is straight, then the vectors are called orthogonal .

Definition:Orthogonal projection vector to the direction of the vector called a scalar quantity , φ – angle between vectors (Fig. 9).

The modulus of this scalar quantity is equal to the length of the segment O.A. 0 .

If the angle φ is acute, the projection is positive; if the angle φ is obtuse, the projection is negative; if the angle φ is straight, the projection is zero.

With an orthogonal projection, the angle between the segments O.A. 0 And A.A. 0 straight. There are projections in which this angle is different from the right angle.

Projections of vectors have the following properties:

The basis is called orthogonal , if its vectors are pairwise orthogonal.

An orthogonal basis is called orthonormal , if its vectors are equal in length to one. For an orthonormal basis in space, the notation is often used.

Theorem: In an orthonormal basis, the coordinates of the vectors are the corresponding orthogonal projections of this vector onto the directions of the coordinate vectors.

Example: Let a vector of unit length form an angle φ with the vector of an orthonormal basis on the plane, then .

Example: Let a vector of unit length form angles α, β, γ with the vectors , and of an orthonormal basis in space, respectively (Fig. 11), then . Moreover. The quantities cosα, cosβ, cosγ are called direction cosines of the vector

Chapter 6. Dot product

Definition: The scalar product of two vectors is a number equal to the product of the lengths of these vectors and the cosine of the angle between them. If one of the vectors is zero, the scalar product is considered equal to zero.

The scalar product of vectors and is denoted by [or ; or ]. If φ is the angle between the vectors and , then .

The scalar product has the following properties:

Theorem: In an orthogonal basis, the components of any vector are found according to the formulas:

Indeed, let , and each term is collinear to the corresponding basis vector. From the theorem of the second section it follows that , where the plus or minus sign is chosen depending on whether the vectors , and are directed in the same or opposite directions. But, , where φ is the angle between the vectors , and . So, . The remaining components are calculated similarly.

The dot product is used to solve the following basic problems:

1. ; 2. ; 3. .

Let vectors be given in a certain basis, and then, using the properties of the scalar product, we can write:

The quantities are called metric coefficients of a given basis. Hence .

Theorem: In an orthonormal basis

;
;
;
.

Comment: All arguments in this section are given for the case of the location of vectors in space. The case of vectors being located on a plane is obtained by removing unnecessary components. The author suggests you do this yourself.

Chapter 7. Vector product

An ordered triple of non-coplanar vectors is called right-oriented (right ), if after application to the common origin from the end of the third vector, the shortest turn from the first vector to the second is visible counterclockwise. Otherwise, an ordered triple of non-coplanar vectors is called left-oriented (left ).

Definition: The cross product of a vector and a vector is a vector that satisfies the conditions:

If one of the vectors is zero, then the cross product is the zero vector.

The cross product of a vector and a vector is denoted (or).

Theorem: A necessary and sufficient condition for the collinearity of two vectors is that their vector product is equal to zero.

Theorem: The length (modulus) of the vector product of two vectors is equal to the area of ​​the parallelogram constructed on these vectors as sides.

Example: If is a right orthonormal basis, then , , .

Example: If is a left orthonormal basis, then , , .

Example: Let a be orthogonal to . Then it is obtained from the vector by rotating it clockwise around the vector (as seen from the end of the vector).