Page 1 of 2
Question 1. What is a vector? How are vectors designated?
Answer. We will call a directed segment a vector (Fig. 211). The direction of a vector is determined by indicating its beginning and end. In the drawing, the direction of the vector is indicated by an arrow. To denote vectors we will use lowercase Latin letters a, b, c, .... You can also denote a vector by indicating its beginning and end. In this case, the beginning of the vector is placed in first place. Instead of the word “vector”, an arrow or a line is sometimes placed above the letter designation of the vector. The vector in Figure 211 can be denoted as follows:
\(\overline(a)\), \(\overrightarrow(a)\) or \(\overline(AB)\), \(\overrightarrow(AB)\).
Question 2. What vectors are called identically directed (oppositely directed)?
Answer. Vectors \(\overline(AB)\) and \(\overline(CD)\) are said to be equally directed if the half-lines AB and CD are equally directed.
Vectors \(\overline(AB)\) and \(\overline(CD)\) are said to be oppositely directed if the half-lines AB and CD are oppositely directed.
In Figure 212, the vectors \(\overline(a)\) and \(\overline(b)\) are equally directed, and the vectors \(\overline(a)\) and \(\overline(c)\) are oppositely directed.
Question 3. What is the absolute magnitude of a vector?
Answer. The absolute value (or modulus) of a vector is the length of the segment representing the vector. The absolute value of the vector \(\overline(a)\) is denoted by |\(\overline(a)\)|.
Question 4. What is a null vector?
Answer. The beginning of a vector can coincide with its end. We will call such a vector the zero vector. The zero vector is denoted by a zero with a dash (\(\overline(0)\)). They don't talk about the direction of the zero vector. The absolute value of the zero vector is considered equal to zero.
Question 5. What vectors are called equal?
Answer. Two vectors are said to be equal if they are combined by parallel translation. This means that there is a parallel translation that takes the start and end of one vector to the start and end of another vector, respectively.
Question 6. Prove that equal vectors have the same direction and are equal in absolute value. And vice versa: identically directed vectors that are equal in absolute value are equal.
Answer. During parallel translation, the vector retains its direction, as well as its absolute value. This means that equal vectors have the same directions and are equal in absolute value.
Let \(\overline(AB)\) and \(\overline(CD)\) be identically directed vectors, equal in absolute value (Fig. 213). A parallel translation that moves point C to point A combines the half-line CD with the half-line AB, since they have the same direction. And since the segments AB and CD are equal, then point D coincides with point B, i.e. parallel translation transforms the vector \(\overline(CD)\) into the vector \(\overline(AB)\). This means that the vectors \(\overline(AB)\) and \(\overline(CD)\) are equal, which is what needed to be proved.
Question 7. Prove that from any point you can plot a vector equal to a given vector, and only one.
Answer. Let CD be a line, and the vector \(\overline(CD)\) be part of the line CD. Let AB be the straight line into which the straight line CD goes during parallel transfer, \(\overline(AB)\) be the vector into which the vector \(\overline(CD)\) goes during parallel transfer, and therefore the vectors \(\ overline(AB)\) and \(\overline(CD)\) are equal, and straight lines AB and CD are parallel (see Fig. 213). As we know, through a point not lying on a given line, it is possible to draw on the plane at most one straight line parallel to the given one (axiom of parallel lines). This means that through point A one line can be drawn parallel to line CD. Since the vector \(\overline(AB)\) is part of the line AB, then through point A one can draw one vector \(\overline(AB)\), equal to the vector \(\overline(CD)\).
Question 8. What are vector coordinates? What is the absolute value of the vector with coordinates a 1, a 2?
Answer. Let the vector \(\overline(a)\) have a beginning point A 1 (x 1 ; y 1), and an end point A 2 (x 2 ; y 2). The coordinates of the vector \(\overline(a)\) will be the numbers a 1 = x 2 - x 1 , a 2 = y 2 - y 1 . We will put the coordinates of the vector next to the letter designation of the vector, in this case \(\overline(a)\) (a 1 ; a 2) or simply \((\overline(a 1 ; a 2 ))\). The coordinates of the zero vector are equal to zero.
From the formula expressing the distance between two points through their coordinates, it follows that the absolute value of the vector with coordinates a 1 , a 2 is equal to \(\sqrt(a^2 1 + a^2 2 )\).
Question 9. Prove that equal vectors have respectively equal coordinates, and vectors with respectively equal coordinates are equal.
Answer. Let A 1 (x 1 ; y 1) and A 2 (x 2 ; y 2) be the beginning and end of the vector \(\overline(a)\). Since the vector \(\overline(a)\) equal to it is obtained from the vector \(\overline(a)\) by parallel transfer, its beginning and end will be A" 1 (x 1 + c; y 1 + d) respectively ), A" 2 (x 2 + c; y 2 + d). This shows that both vectors \(\overline(a)\) and \(\overline(a")\) have the same coordinates: x 2 - x 1, y 2 - y 1.
Let us now prove the converse statement. Let the corresponding coordinates of the vectors \(\overline(A 1 A 2 )\) and \(\overline(A" 1 A" 2 )\) be equal. Let us prove that the vectors are equal.
Let x" 1 and y" 1 be the coordinates of point A" 1, and x" 2, y" 2 be the coordinates of point A" 2. According to the conditions of the theorem, x 2 - x 1 = x" 2 - x" 1, y 2 - y 1 = y" 2 - y" 1. Hence x" 2 = x 2 + x" 1 - x 1, y" 2 = y 2 + y" 1 - y 1. Parallel transfer given by formulas
x" = x + x" 1 - x 1 , y" = y + y" 1 - y 1 ,
transfers point A 1 to point A" 1, and point A 2 to point A" 2, i.e. the vectors \(\overline(A 1 A 2 )\) and \(\overline(A" 1 A" 2 )\) are equal, which is what needed to be proved.
Question 10. Define the sum of vectors.
Answer. The sum of vectors \(\overline(a)\) and \(\overline(b)\) with coordinates a 1 , a 2 and b 1 , b 2 is the vector \(\overline(c)\) with coordinates a 1 + b 1, a 2 + b a 2, i.e.
\(\overline(a) (a 1 ; a 2) + \overline(b)(b 1 ; b 2) = \overline(c) (a 1 + b 1 ; a 2 + b 2)\).
What is a vector? The concept of a vector arises where we have to deal with objects that are characterized by magnitude and direction: for example, speed, force, pressure. Such quantities are called vector quantities or vectors. The concept of a vector arises where we have to deal with objects that are characterized by magnitude and direction: for example, speed, force, pressure. Such quantities are called vector quantities or vectors.
Concept of a vector Consider an arbitrary segment. You can indicate two directions on it. To choose one of the directions, we will call one end of the segment the BEGINNING, and the other the END, and we will assume that the segment is directed from the beginning to the end. Definition. Definition. A segment for which it is indicated which of its ends is considered the beginning and which is the end is called a directed segment or vector. A segment for which it is indicated which of its ends is considered the beginning and which is the end is called a directed segment or vector.
The concept of a vector Vectors are often denoted by one lowercase Latin letter with an arrow above it: Vectors are often denoted by one lowercase Latin letter with an arrow above it: Any point on the plane is also a vector, which is called ZERO. The beginning of the zero vector coincides with its end: Any point on the plane is also a vector, which is called ZERO. The beginning of the zero vector coincides with its end: MM = 0. MM = 0. a b c M
Concept of a vector The length or modulus of a non-zero vector AB is the length of the segment AB: The length or modulus of a non-zero vector AB is the length of the segment AB: AB = a = AB = 5 AB = a = AB = 5 s = 17 s = 17 The length of the zero vector is considered equal to zero : The length of the zero vector is considered equal to zero: MM = 0. MM = 0. a M VA s
Collinear vectors Non-zero vectors are called collinear if they lie either on the same line or on parallel lines. Collinear vectors can be codirectional or oppositely directed. Non-zero vectors are called collinear if they lie either on the same line or on parallel lines. Collinear vectors can be codirectional or oppositely directed. The null vector is considered collinear to any vector. The null vector is considered collinear to any vector. аb c d m n s L
Depositing a vector from a given point If point A is the beginning of vector a, then they say that vector a is postponed from point A. If point A is the beginning of vector a, then they say that vector a is postponed from point A. Statement: From any point M you can set aside a vector equal to the given vector a, and only one. Statement: From any point M you can plot a vector equal to a given vector a, and only one. Equal vectors plotted from different points are often denoted by the same letter Equal vectors plotted from different points are often denoted by the same letter A a M a
Sum of two vectors Consider an example: Consider an example: Petya went from home (D) to Vasya (V), and then went to the cinema (K). Petya went from home (D) to Vasya (V), and then went to the cinema (K). As a result of these two movements, which can be represented by the vectors DV and VK, Petya moved from point D to K, i.e. to vector DK: As a result of these two movements, which can be represented by vectors DV and VK, Petya moved from point D to K, i.e. to vector DK: DK=DB+BK. DK=DB+BK. The vector DK is called the sum of the vectors DB and BK. D V K
Sum of two vectors Triangle rule Let a and b be two vectors. Let's mark an arbitrary point A and plot AB = a from this point, then plot the vector BC = b from point B. Let a and b be two vectors. Let's mark an arbitrary point A and plot AB = a from this point, then plot the vector BC = b from point B. AC = a + b AC = a + b a b A a b B C
Opposite vectors Let a be an arbitrary nonzero vector. Let a be an arbitrary non-zero vector. Definition. A vector b is called opposite to a vector if a and b have equal lengths and opposite directions. a = AB, b = BA The vector opposite to vector c is denoted as follows: -c. Obviously, c+(-c)=0 or AB+BA=0 A B a b c -c
Vector Subtraction Definition. The difference of two vectors a and b is a vector whose sum with vector b is equal to vector a. Definition. The difference of two vectors a and b is a vector whose sum with vector b is equal to vector a. Theorem. For any vectors a and b, the equality a - b = a + (-b) is true. Task. Vectors a and b are given. Construct vector a – b. a a b -b a - b
Finally, I got my hands on this extensive and long-awaited topic. analytical geometry. First, a little about this section of higher mathematics... Surely you now remember a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “analytical solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical 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 many 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 at times.
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 a plane. 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 button up 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 on the plane. It’s 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 
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:
Please note 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 
.
1. General provisions
1.1. In order to maintain business reputation and ensure compliance with federal legislation, the Federal State Institution State Research Institute of Technology "Informika" (hereinafter referred to as the Company) considers the most important task to be ensuring the legitimacy of the processing and security of personal data of subjects in the Company's business processes.
1.2. To solve this problem, the Company has introduced, operates and undergoes periodic review (monitoring) of a personal data protection system.
1.3. The processing of personal data in the Company is based on the following principles:
The legality of the purposes and methods of processing personal data and integrity;
Compliance of the purposes of processing personal data with the goals predetermined and stated when collecting personal data, as well as with the powers of the Company;
Correspondence of the volume and nature of the processed personal data, methods of processing personal data to the purposes of processing personal data;
The reliability of personal data, their relevance and sufficiency for the purposes of processing, the inadmissibility of processing personal data that is excessive in relation to the purposes of collecting personal data;
The legitimacy of organizational and technical measures to ensure the security of personal data;
Continuous improvement of the level of knowledge of Company employees in the field of ensuring the security of personal data during their processing;
Striving for continuous improvement of the personal data protection system.
2. Purposes of processing personal data
2.1. In accordance with the principles of processing personal data, the Company has determined the composition and purposes of processing.
Purposes of processing personal data:
Conclusion, support, amendment, termination of employment contracts, which are the basis for the emergence or termination of labor relations between the Company and its employees;
Providing a portal, personal account services for students, parents and teachers;
Storage of learning results;
Fulfillment of obligations provided for by federal legislation and other regulatory legal acts;
3. Rules for processing personal data
3.1. The Company processes only those personal data that are presented in the approved List of personal data processed in the Federal State Autonomous Institution State Scientific Research Institute of Technology "Informika"
3.2. The Company does not allow the processing of the following categories of personal data:
Race;
Political Views;
Philosophical beliefs;
About the state of health;
State of intimate life;
Nationality;
Religious Beliefs.
3.3. The Company does not process biometric personal data (information that characterizes the physiological and biological characteristics of a person, on the basis of which one can establish his identity).
3.4. The Company does not carry out cross-border transfer of personal data (transfer of personal data to the territory of a foreign state to an authority of a foreign state, a foreign individual or a foreign legal entity).
3.5. The Company prohibits making decisions regarding personal data subjects based solely on automated processing of their personal data.
3.6. The Company does not process data on subjects' criminal records.
3.7. The company does not publish the subject’s personal data in publicly available sources without his prior consent.
4. Implemented requirements to ensure the security of personal data
4.1. In order to ensure the security of personal data during its processing, the Company implements the requirements of the following regulatory documents of the Russian Federation in the field of processing and ensuring the security of personal data:
Federal Law of July 27, 2006 No. 152-FZ “On Personal Data”;
Decree of the Government of the Russian Federation of November 1, 2012 N 1119 “On approval of requirements for the protection of personal data during their processing in personal data information systems”;
Decree of the Government of the Russian Federation dated September 15, 2008 No. 687 “On approval of the Regulations on the specifics of processing personal data carried out without the use of automation tools”;
Order of the FSTEC of Russia dated February 18, 2013 N 21 “On approval of the composition and content of organizational and technical measures to ensure the security of personal data during their processing in personal data information systems”;
Basic model of threats to the security of personal data during their processing in personal data information systems (approved by the Deputy Director of the FSTEC of Russia on February 15, 2008);
Methodology for determining current threats to the security of personal data during their processing in personal data information systems (approved by the Deputy Director of the FSTEC of Russia on February 14, 2008).
4.2. The company assesses the harm that may be caused to personal data subjects and identifies threats to the security of personal data. In accordance with identified current threats, the Company applies necessary and sufficient organizational and technical measures, including the use of information security tools, detection of unauthorized access, restoration of personal data, establishment of rules for access to personal data, as well as monitoring and evaluation of the effectiveness of the measures applied.
4.3. The Company has appointed persons responsible for organizing the processing and ensuring the security of personal data.
4.4. The Company’s management is aware of the need and is interested in ensuring an adequate level of security for personal data processed as part of the Company’s core activities, both in terms of the requirements of regulatory documents of the Russian Federation and justified from the point of view of assessing business risks.
G – 9th grade Lesson No. 2
Topic: The concept of a vector. Equality of vectors. Delaying a vector from a given point.
Goals:
introduce the concept of a vector, its length, collinear and equal vectors;
teach students to depict and designate vectors, to plot a vector equal to a given one from any point on the plane;
consolidate students’ knowledge while solving problems;
develop memory, attention, mathematical thinking;
develop diligence and the desire to achieve goals and objectives.
Progress of the lesson.
Organizational aspects.
Communicate the topic and objectives of the lesson.
Updating the knowledge and skills of students.
1. Checking homework completion. Analysis of unsolved tasks.
2. Checking theoretical information:
Isosceles triangle and its properties. Signs of equality of triangles.
Definition of the midline of a triangle and its properties.
Pythagorean theorem and its converse theorem.
Formula for calculating the area of a triangle.
The concept of a parallelogram, properties and characteristics of a parallelogram, rhombus, rectangle.
Definition of trapezoid, types of trapezoids.
Area of a parallelogram, area of a trapezoid.
Learning new material.
Present the material in paragraphs 76–78 in the form of a short lecture using a variety of Vector presentations
1. The concept of vector quantities (or vectors for short).
2. Examples of vector quantities known to students from a physics course: force, displacement of a material point, speed and others (Fig. 240 of the textbook).
3. Determination of the vector (Fig. 241, 242).
4. Vector designation - two capital Latin letters with an arrow above them, for example,, or often denoted by a single lowercase Latin letter with an arrow above it:(Fig. 243, a, b).
5. The concept of a zero vector: any point on the plane is also a vector; in this case the vector is called zero; stand for:
(Fig. 243, a).
6. Determining the length or modulus of a non-zero vector. Designation:
. Zero vector length= 0.
7. Find the lengths of the vectors shown in Figures 243, a and 243, b.
8. Complete practical tasks No. 738, 739.
9. Consider an example of the movement of a body in which all its points move at the same speed and in the same direction (from paragraph 77 of the textbook), fig. 244.
10. Introduce the concept of collinear vectors (Fig. 245).
11. Definition of the concepts of co-directed vectors and oppositely directed vectors, their designation (Fig. 246).
12. The zero vector is codirectional with any vector.
13. Definition of equal vectors: ifAnd, That.
14. Explanation of the meaning of the expression: “Vectordelayed from point A” (Fig. 247).
15. Proof of the statement that from any point you can plot a vector equal to the given one, and only one (Fig. 248).
16. Completing practical task No. 743.
17. Orally solve problem No. 749 using the finished drawing on the board.
Problem solving.
1. Solve problem No. 740 (a) on the board and in notebooks.
2. Orally solve problem No. 744.
3. Solve problem No. 742.
4. Solve problem No. 745 (selectively).
5. Orally solve problem No. 746 using the prepared drawing.
6. Prove the direct statement in problem No. 750:
Proof
By condition, then AB || CD, therefore, according to the property of a parallelogram ABC is a parallelogram, and the diagonals of the parallelogram are divided in half by the intersection point, which means that the midpoints of the segments AD and BC coincide.
Organize repetition while solving the following problems - Tasks for repetition from the OGE (GIA)-2016 task bank:
№ 9, 10, 11, 12, 13 – from the “Geometry” module; No. 24 – from part 2 of the module “Geometry” Option No. 3
Lesson summary.
Summing up the lesson. Making marks.
As a result of studying § 1, students should know the definitions of a vector and equal vectors; be able to depict and designate vectors, plot a vector equal to a given one from a given point; solve problems like Nos. 741–743; 745–752.
Homework: study the material in paragraphs 76–78; answer questions 1–6, p. 213 textbooks; solve problems No. 747, 749, 751.
