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 provide 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 “school” vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a directed segment 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 almost correct - a directed segment can be added 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 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 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: , 
. Follow the drawing to see how clearly 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, multiply the vector by the 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 would like to note that the site materials are not enough to pass a theoretical test or a colloquium in geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of presentation, but a plus for 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 appropriate formula:
Alternatively, the following entry could be used:
Aesthetes will decide this:
Personally, I'm used to the first version of the recording.
Answer:
According to the condition, it was not necessary to construct a drawing (which is typical for problems of analytical geometry), but in order to clarify some points for dummies, I will not be lazy: 
You definitely need to understand difference between point coordinates and vector coordinates:
Point coordinates– these are ordinary coordinates in a rectangular coordinate system. I think everyone knows how to plot points on a coordinate plane from the 5th-6th grade. Each point has a strict place on the plane, and they cannot be moved anywhere.
The coordinates of the vector– this is its expansion according to the basis, in this case. Any vector is free, so if desired or necessary, we can easily move it away from some other point on the plane. It is interesting that for vectors you don’t have to build axes or a rectangular coordinate system at all; you only need a basis, in this case an orthonormal basis of the plane.
The records of coordinates of points and coordinates of vectors seem to be similar: , and meaning of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, also applies to space.
Ladies and gentlemen, let's fill our hands:
Example 2
a) Points and are given. Find vectors and .
b) Points are given 
And . Find vectors and .
c) Points and are given. Find vectors and .
d) Points are given. Find vectors 
.
Perhaps that's enough. These are examples for you to decide on your own, try not to neglect them, it will pay off ;-). There is no need to make drawings. Solutions and answers at the end of the lesson.
What is important when solving analytical geometry problems? It is important to be EXTREMELY CAREFUL to avoid making the masterful “two plus two equals zero” mistake. I apologize right away if I made a mistake somewhere =)
How to find the length of a segment?
The length, as already noted, is indicated by the modulus sign.
If two points of the plane are given and , then the length of the segment can be calculated using the formula
If two points in space and are given, then the length of the segment can be calculated using the formula
Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard
Example 3
Solution: according to the appropriate formula:
Answer: 
For clarity, I will make a drawing 
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). The process looks like this in more detail: 
. 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.
Assignment 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 
.
Vector module can be found if we know it projections onto coordinate axes.
given on the plane vector A(Fig. 15).
Let us drop perpendiculars from the beginning and end of the vector onto the coordinate axes to find its projections. In accordance with the Pythagorean theorem
. From here
.
You need to know this formula BY HEART.
Remember!
To find vector module it is necessary to extract the square root of the sum of the squares of its projections.
You already know that the projection of a vector onto an axis can be found if you subtract the coordinate of its beginning point from the coordinate of the end point of the vector. Then for our vector, if it is given on the plane, and x = x k − x n,
and y = y k − y n. Hence, vector module can be found using the formula
.
It is not difficult to imagine what the formula will look like if vector given in space.
Also pay attention to this. After all vector module is the length of the segment enclosed between two points: the beginning point of the vector and the end point. And this is nothing more than the distance between these two points. Therefore, to find the distance between any two points, you need to calculate vector module connecting these points.
Let's find the length of a vector from its coordinates (in a rectangular coordinate system), from the coordinates of the beginning and end points of the vector, and from the cosine theorem (given 2 vectors and the angle between them).
Vector is a directed straight line segment. The length of this segment determines the numerical value of the vector and is called the length of the vector or the modulus of the vector.
1. Calculating the length of a vector from its coordinates
If the vector coordinates are given in a flat (two-dimensional) rectangular coordinate system, i.e. a x and a y are known, then the length of the vector can be found using the formula
In the case of a vector in space, a third coordinate is added
In MS EXCEL expression =ROOT(SUMKV(B8:B9)) allows you to calculate the modulus of a vector (it is assumed that the vector coordinators are entered into the cells B8:B9, see example file).
The SUMMQ() function returns the sum of the squares of the arguments, i.e. in this case it is equivalent to the formula =B8*B8+B9*B9.
The example file also calculates the length of the vector in space.
An alternative formula is =ROOT(SUMPRODUCT(B8:B9,B8:B9)).
2. Finding the length of a vector through the coordinates of points
If the vector given through the coordinates of its start and end points, then the formula will be different =ROOT(SUMVARE(C28:C29,B28:B29))
The formula assumes that the coordinates of the start and end points are entered in the ranges C28:C29 And B28:B29 respectively.
Function SUMMQDIFFERENCE() in Returns the sum of the squared differences of the corresponding values in two arrays.
Essentially, the formula first calculates the coordinates of the vector (the difference between the corresponding coordinates of the points), then calculates the sum of their squares.
3. Finding the length of a vector using the cosine theorem
If you need to find the length of a vector using the cosine theorem, then usually 2 vectors are given (their modules and the angle between them).
Let's find the length of the vector c using the formula =ROOT(SUM(B43:C43)-2*B43*C43*COS(B45))
In cells B43:B43 contains the lengths of vectors a and b, and the cell B45 - the angle between them in radians (in fractions of PI()).
If the angle is specified in degrees, the formula will be slightly different =ROOT(B43*B43+C43*C43-2*B43*C43*COS(B46*PI()/180))
Note: for clarity, in a cell with an angle value in degrees, you can use , see, for example, the article
vector module- vector magnitude - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information technology in general Synonyms vector value EN absolute value of a vector ...
vector module- vektoriaus modulis statusas T sritis fizika atitikmenys: engl. absolute value of vector vok. Vektorbetrag, m rus. vector length, f; vector modulus, m pranc. module d'un vecteur, m ... Fizikos terminų žodynas
- (from Latin modulus “small measure”): Wiktionary has an article “module” Mo ... Wikipedia
A module (from Latin modulus “small measure”) is an integral part, separable or at least mentally distinguished from the general. Modular is usually called a thing consisting of clearly defined parts, which can often be removed or added without destroying the thing... ... Wikipedia
The absolute value or modulus of a real or complex number x is the distance from x to the origin. More precisely: The absolute value of a real number x is a non-negative number, denoted by |x| and defined as follows: ... ... Wikipedia
wave vector module- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information technology in general EN magnitude of propagation vector ... Technical Translator's Guide
envelope code vector convolver module- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information technology in general EN shape codevector convolution module ... Technical Translator's Guide
The modulus of a complex number is the length of the vector corresponding to this number: . The modulus of a complex number z is usually denoted | z | or r. Let the real numbers be such that a complex number (usual notation). Then Numbers... Wikipedia
Module in mathematics, 1) M. (or absolute value) of a complex number z = x + iy is the number ═ (the root is taken with a plus sign). When representing a complex number z in trigonometric form z = r(cos j + i sin j), the real number r is equal to... ... Great Soviet Encyclopedia
Abelian group with a ring of operators. M is a generalization of a (linear) vector space over a field K for the case when K is replaced by some ring. Let a ring A be given. An additive Abelian group Mnaz. left A module, if defined... ... Mathematical Encyclopedia
