How to calculate the coordinates of a vector. Vectors for dummies

Finally, I got my hands on an extensive and long-awaited topic analytical geometry. First, a little about this section of higher mathematics…. Surely you now remembered the school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytic geometry, oddly enough, may seem more interesting and accessible. What does the adjective "analytical" mean? Two stamped mathematical turns immediately come to mind: “graphic method of solution” and “analytical method of solution”. Graphic method, of course, is associated with the construction of graphs, drawings. Analytical same method involves problem solving predominantly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent, often it is enough to accurately apply the necessary formulas - and the answer is ready! No, of course, it will not do without drawings at all, besides, for a better understanding of the material, I will try to bring them in excess of the need.

The open course of lessons in geometry does not claim to be theoretical completeness, it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need a more complete reference on any subsection, I recommend the following quite accessible literature:

1) A thing that, no joke, is familiar to several generations: School textbook on geometry, the authors - L.S. Atanasyan and Company. This school locker room hanger has already withstood 20 (!) reissues, which, of course, is not the limit.

2) Geometry in 2 volumes. The authors L.S. Atanasyan, Bazylev V.T.. This is literature for higher education, you will need first volume. Infrequently occurring tasks may fall out of my field of vision, and the tutorial will be of invaluable help.

Both books are free to download online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download higher mathematics examples.

Of the tools, I again offer my own development - software package on analytical geometry, which will greatly simplify life and save a lot of time.

It is assumed that the reader is familiar with basic geometric concepts and figures: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello repeaters)

And now we will sequentially consider: the concept of a vector, actions with vectors, vector coordinates. Further I recommend reading the most important article Dot product of vectors, as well as Vector and mixed product of vectors. The local task will not be superfluous - Division of the segment in this regard. Based on the above information, you can equation of a straight line in a plane With the simplest examples of solutions, which will allow learn how to solve problems in geometry. The following articles are also helpful: Equation of a plane in space, Equations of a straight line in space, Basic problems on the line and plane , other sections of analytic geometry. Naturally, standard tasks will be considered along the way.

The concept of a vector. free vector

First, let's repeat the school definition of a vector. Vector called directed a segment for which its beginning and end are indicated:

In this case, the beginning of the segment is the point , the end of the segment is the point . The vector itself is denoted by . Direction is essential, if you rearrange the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must admit that 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, space as the so-called zero vector. Such a vector has the same end and beginning.

!!! Note: Here and below, you can assume that the vectors lie in the same plane or you can assume that they are located in space - the essence of the material presented is valid for both the plane and space.

Designations: Many immediately drew attention to a stick without an arrow in the designation and said that they also put an arrow at the top! That's right, you can write with an arrow: , but admissible and record that I will use later. Why? Apparently, such a habit has developed from practical considerations, my shooters at school and university turned out to be too diverse and shaggy. In educational literature, sometimes they don’t bother with cuneiform at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was the style, and now about the ways of writing vectors:

1) Vectors can be written in two capital Latin letters:
and so on. While the first letter necessarily denotes the start point of the vector, and the second letter denotes the end point of the vector.

2) Vectors are also written in small Latin letters:
In particular, our vector can be redesignated for brevity by a small Latin letter .

Length or module non-zero vector is called the length of the segment. The length of the null vector is zero. Logically.

The length of a vector is denoted by the modulo sign: ,

How to find the length of a vector, we will learn (or repeat, for whom how) a little later.

That was elementary information about the vector, familiar to all schoolchildren. In analytic geometry, the so-called free vector.

If it's quite simple - vector can be drawn from any point:

We used to call such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, this is the SAME VECTOR or free vector. Why free? Because in the course of solving problems you can “attach” one or another “school” vector to ANY point of the plane or space you need. This is a very cool property! 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's proverb: Each lecturer in f ** u in the vector. After all, it’s not just a witty rhyme, everything is almost correct - a directed segment can be attached there too. But do not rush to rejoice, students themselves suffer more often =)

So, free vector- this is lots of identical directional segments. 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 attached to a certain point in the plane or space.

It should be noted that from the point of view of physics, the concept of a free vector is generally incorrect, and the point of application matters. Indeed, a direct blow of the same force on the nose or on the forehead is enough to develop my stupid example entails different consequences. However, not free vectors are also found in the course of vyshmat (do not go there :)).

Actions with vectors. Collinearity of vectors

In the school geometry course, a number of actions and rules with vectors are considered: addition according to the triangle rule, addition according to the parallelogram rule, the rule of the difference of vectors, multiplication of a vector by a number, the scalar product of vectors, etc. As a seed, we repeat two rules that are especially relevant for solving problems of analytical geometry.

Rule of addition of vectors according to the rule of triangles

Consider two arbitrary non-zero vectors and :

It is required to find the sum of these vectors. Due to the fact that all vectors are considered free, we postpone the vector from end vector :

The sum of vectors is the vector . For a better understanding of the rule, it is advisable to put a physical meaning into it: let some body make a path along the vector , and then along the vector . Then the sum of the vectors is the vector of the resulting path starting at the point of departure and ending at the point of arrival. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way strongly zigzag, or maybe on autopilot - along the resulting sum vector.

By the way, if the vector is postponed from start vector , then we get the equivalent parallelogram rule addition of vectors.

First, about the collinearity of vectors. The two vectors are called collinear if they lie on the same line or on parallel lines. Roughly speaking, we are talking about parallel vectors. But in relation to them, the adjective "collinear" is always used.

Imagine two collinear vectors. If the arrows of these vectors are directed in the same direction, then such vectors are called co-directional. If the arrows look in different directions, then the vectors will be oppositely directed.

Designations: collinearity of vectors is written with the usual parallelism icon: , while detailing is possible: (vectors are co-directed) or (vectors are directed oppositely).

work of a nonzero vector by a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .

The rule for multiplying a vector by a number is easier to understand with a picture:

We understand in more detail:

1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.

2) Length. If the factor is contained within or , then the length of the vector decreases. So, the length of the vector is twice less than the length of the vector . If the modulo multiplier is greater than one, then the length of the vector increases in time.

3) Please note that all vectors are collinear, while one vector is expressed through another, for example, . The reverse is also true: if one vector can be expressed in terms of another, then such vectors are necessarily collinear. In this way: if we multiply a vector by a number, we get collinear(relative to original) vector.

4) The vectors are codirectional. The vectors and are also codirectional. Any vector of the first group is opposite to any vector of the second group.

What vectors are equal?

Two vectors are equal if they are codirectional and have the same length. Note that co-direction implies that the vectors are collinear. The definition will be inaccurate (redundant) if you say: "Two vectors are equal if they are collinear, co-directed and have the same length."

From the point of view of the concept of a free vector, equal vectors are the same vector, which was already discussed in the previous paragraph.

Vector coordinates on the plane and in space

The first point is to consider vectors on a plane. Draw a Cartesian rectangular coordinate system and set aside from the origin single vectors and :

Vectors and orthogonal. Orthogonal = Perpendicular. I recommend slowly getting used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity and orthogonality.

Designation: orthogonality of vectors is written with the usual perpendicular sign, for example: .

The considered vectors are called coordinate vectors or orts. These vectors form basis on surface. What is the basis, I think, is intuitively clear to many, more detailed information can be found in the article Linear (non) dependence of vectors. Vector basis.In simple words, the basis and the origin of coordinates 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 order basis vectors are listed, for example: . Coordinate vectors it is forbidden swap places.

Any plane vector the only way expressed as:
, where - numbers, which are called vector coordinates in this basis. But the expression itself called vector decompositionbasis .

Dinner served:

Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing the vector in terms of the basis, the ones just considered are used:
1) the rule of multiplication of a vector by a number: and ;
2) addition of vectors according to the triangle rule: .

Now mentally set aside the vector from any other point on the plane. It is quite obvious that his corruption will "relentlessly follow him." Here it is, the freedom of the vector - the vector "carries everything with you." This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be set aside from the origin, one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change from this! True, you don’t need to do this, because the teacher will also show originality and draw you a “pass” in an unexpected place.

Vectors , illustrate exactly the rule for multiplying a vector by a number, the vector is co-directed with the basis vector , the vector is directed opposite to the basis vector . For these vectors, one of the coordinates is equal to zero, it can be meticulously written as follows:


And the basis vectors, by the way, are like this: (in fact, they are expressed through themselves).

And finally: , . By the way, what is vector subtraction, and why didn't I tell you about the subtraction rule? Somewhere in linear algebra, I don't remember where, I noted that subtraction is a special case of addition. So, the expansions of the vectors "de" and "e" are calmly written as a sum: . Follow the drawing to see how well the good old addition of vectors according to the triangle rule works in these situations.

Considered decomposition of the form sometimes called a vector decomposition in the system ort(i.e. in the system of unit vectors). But this is not the only way to write a vector, the following option is common:

Or with an equals sign:

The basis vectors themselves are written as follows: and

That is, the coordinates of the vector are indicated in parentheses. In practical tasks, all three recording options are used.

I doubted whether to speak, but still I will say: vector coordinates cannot be rearranged. Strictly in first place write down the coordinate that corresponds to the unit vector , strictly in second place write down the coordinate that corresponds to the unit vector . Indeed, and are two different vectors.

We figured out the coordinates on the plane. Now consider vectors in three-dimensional space, everything is almost the same here! Only one more coordinate will be added. It is difficult to perform three-dimensional drawings, so I will limit myself to one vector, which for simplicity I will postpone from the origin:

Any 3d space vector the only way expand in an orthonormal basis:
, where are the coordinates of the vector (number) in the given basis.

Example from the picture: . Let's see how the vector action rules work here. First, multiplying a vector by a number: (red arrow), (green arrow) and (magenta arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector starts at the starting point of departure (the beginning of the vector ) and ends up at the final point of arrival (the end of the vector ).

All vectors of three-dimensional space, of course, are also free, try to mentally postpone the vector from any other point, and you will understand that its expansion "remains with it."

Similarly to the plane case, in addition to writing versions with brackets are widely used: either .

If one (or two) coordinate vectors are missing in the expansion, then zeros are put instead. Examples:
vector (meticulously ) – write down ;
vector (meticulously ) – write down ;
vector (meticulously ) – write down .

Basis vectors are written as follows:

Here, perhaps, is all the minimum theoretical knowledge necessary for solving problems of analytical geometry. Perhaps there are too many terms and definitions, so I recommend dummies to re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time for better assimilation of the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in what follows. I note that the materials of the site are not enough to pass a theoretical test, a colloquium on geometry, since I carefully encrypt all theorems (besides without proofs) - to the detriment of the scientific style of presentation, but a plus for your understanding of the subject. For detailed theoretical information, I ask you to bow to Professor Atanasyan.

Now let's move on to the practical part:

The simplest problems of analytic geometry.
Actions with vectors in coordinates

The tasks that will be considered, it is highly desirable to learn how to solve them fully automatically, and the formulas memorize, don't even remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend extra time eating pawns. You do not need to fasten the top buttons on your shirt, many things are familiar to you from school.

The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas ... you will see for yourself.

How to find a vector given two points?

If two points of the plane and are given, then the vector has the following coordinates:

If two points in space and are given, then the vector has the following coordinates:

That is, from the coordinates of the end of the vector you need to subtract the corresponding coordinates vector start.

Exercise: For the same points, write down the formulas for finding the coordinates of the vector. Formulas at the end of the lesson.

Example 1

Given two points in the plane and . Find vector coordinates

Solution: according to the corresponding formula:

Alternatively, the following notation could be used:

Aesthetes will decide like this:

Personally, I'm used to the first version of the record.

Answer:

According to the condition, it was not required to build a drawing (which is typical for problems of analytical geometry), but in order to explain some points to dummies, I will not be too lazy:

Must be understood difference between point coordinates and vector coordinates:

Point coordinates are the usual coordinates in a rectangular coordinate system. I think everyone knows how to plot points on the coordinate plane since grade 5-6. Each point has a strict place on the plane, and they cannot be moved anywhere.

The coordinates of the same vector is its expansion with respect to the basis , in this case . Any vector is free, therefore, if desired or necessary, we can easily postpone it from some other point in the plane. Interestingly, for vectors, you can not build axes at all, a rectangular coordinate system, you only need a basis, in this case, an orthonormal basis of the plane.

The records of point coordinates and vector coordinates seem to be similar: , and sense of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, is also true for space.

Ladies and gentlemen, we fill our hands:

Example 2

a) Given points and . Find vectors and .
b) Points are given and . Find vectors and .
c) Given points and . Find vectors and .
d) Points are given. Find Vectors .

Perhaps enough. These are examples for an independent decision, try not to neglect them, it will pay off ;-). Drawings are not required. Solutions and answers at the end of the lesson.

What is important in solving problems of analytical geometry? It is important to be EXTREMELY CAREFUL in order to avoid the masterful “two plus two equals zero” error. I apologize in advance if I made a mistake =)

How to find the length of a segment?

The length, as already noted, is indicated by the modulus sign.

If two points of the plane and are given, then the length of the segment can be calculated by the formula

If two points in space and are given, then the length of the segment can be calculated by the formula

Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard

Example 3

Solution: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Line segment - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the answer can be checked with a regular ruler by directly measuring the length of the segment.

Yes, the solution is short, but there are a couple of important points in it that I would like to clarify:

First, in the answer we set the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, the general formulation will be a mathematically competent solution: “units” - abbreviated as “units”.

Secondly, let's repeat the school material, which is useful not only for the considered problem:

pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the factor out from under the root (if possible). The process looks like this in more detail: . Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.

Here are other common cases:

Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, split completely, thus: . Or maybe the number can be divided by 4 again? . In this way: . The last digit of the number is odd, so dividing by 4 for the third time is clearly not possible. Trying to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a completely non-extractable number, then we try to take out the factor from under the root - on the calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.

Let's repeat the squaring of the roots and other powers at the same time:

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

Task for an independent solution with a segment in space:

Example 4

Given points and . Find the length of the segment.

Solution and answer at the end of the lesson.

How to find the length of a vector?

If a plane vector is given, then its length is calculated by the formula.

If a space vector is given, then its length is calculated by the formula .

On the abscissa and ordinate axes are called coordinates vector. The vector coordinates are usually indicated in the form (x, y), and the vector itself as: = (x, y).

The formula for determining the coordinates of a vector for two-dimensional problems.

In the case of a two-dimensional problem, a vector with known point coordinates A(x 1; y 1) and B(x 2 ; y 2 ) can be calculated:

\u003d (x 2 - x 1; y 2 - y 1).

The formula for determining the coordinates of a vector for spatial problems.

In the case of a spatial problem, a vector with known point coordinates A (x 1; y 1;z 1 ) and B (x 2 ; y 2 ; z 2 ) can be calculated using the formula:

= (x 2 - x 1 ; y 2 - y 1 ; z 2 - z 1 ).

The coordinates give a comprehensive description of the vector, since it is possible to construct the vector itself from the coordinates. Knowing the coordinates, it is easy to calculate and vector length. (Property 3 below).

Vector coordinate properties.

1. Any equal vectors in a single coordinate system have equal coordinates.

2. Coordinates collinear vectors proportional. Provided that none of the vectors is equal to zero.

3. The square of the length of any vector is equal to the sum of its squares coordinates.

4.When the operation vector multiplications on the real number each of its coordinates is multiplied by this number.

5. During the operation of vector addition, we calculate the sum of the corresponding vector coordinates.

6. Scalar product of two vectors is equal to the sum of the products of their respective coordinates.

Finding the coordinates of a vector is a fairly common condition for many problems in mathematics. The ability to find the coordinates of a vector will help you in other, more complex problems with similar topics. In this article, we will consider the formula for finding the coordinates of a vector and several tasks.

Finding the coordinates of a vector in a plane

What is a plane? A plane is a two-dimensional space, a space with two dimensions (dimension x and dimension y). For example, paper is flat. The surface of the table is flat. Any non-volumetric figure (square, triangle, trapezium) is also a plane. Thus, if in the condition of the problem it is necessary to find the coordinates of a vector that lies on a plane, we immediately recall x and y. You can find the coordinates of such a vector as follows: AB coordinates of the vector = (xB - xA; yB - xA). It can be seen from the formula that the coordinates of the starting point must be subtracted from the coordinates of the end point.

Example:

• The CD vector has start (5; 6) and end (7; 8) coordinates.
• Find the coordinates of the vector itself.
• Using the above formula, we get the following expression: CD = (7-5; 8-6) = (2; 2).
• Thus, the coordinates of the CD vector = (2; 2).
• Accordingly, the x coordinate is equal to two, the y coordinate is also two.

Finding the coordinates of a vector in space

What is space? Space is already a three-dimensional dimension, where 3 coordinates are given: x, y, z. If you need to find a vector that lies in space, the formula practically does not change. Only one coordinate is added. To find the vector, you need to subtract the start coordinates from the end coordinates. AB = (xB - xA; yB - yA; zB - zA)

Example:

• Vector DF has initial (2; 3; 1) and final (1; 5; 2).
• Applying the above formula, we get: Vector coordinates DF = (1-2; 5-3; 2-1) = (-1; 2; 1).
• Remember, the value of the coordinates can be negative, there is no problem with that.

How to find vector coordinates online?

If for some reason you do not want to find the coordinates yourself, you can use the online calculator. First, choose the dimension of the vector. The dimension of a vector is responsible for its dimensions. Dimension 3 means that the vector is in space, dimension 2 means that it is on the plane. Next, insert the coordinates of the points into the appropriate fields and the program will determine the coordinates of the vector itself. Everything is very simple.

By clicking on the button, the page will automatically scroll down and give you the correct answer along with the solution steps.

It is recommended to study this topic well, because the concept of a vector is found not only in mathematics, but also in physics. Students of the Faculty of Information Technology also study the topic of vectors, but at a more complex level.

Analytic geometry

		Week	Grade for the module in points
		module control
			Maximum		Minimum
Semester 1
DZ №1, part 1
DZ №1, part 2
Modulo control No. 1
Reward Points
Modulo control No. 2
Reward Points

Control activities and timing of their implementation Module 1

1. DZ No. 1 part 1 "Vector Algebra" Issue deadline 2 weeks, deadline - 7 weeks

2. DZ No. 1 part 2 "Lines and planes"

Delivery period 1 week, delivery period - 9 weeks

3. Modulo control No. 1 (RK No. 1) "Vector algebra, lines and planes." Deadline - 10 weeks

1. DZ No. 2 "Curves and surfaces 2nd order "Issuance period 6 weeks, delivery period - 13 weeks

5. Test "Curves and surfaces 2nd order. Deadline - 14 weeks

6. Modulo control No. 2 (RK No. 2) "Matrices and systems of linear algebraic equations"

Deadline - 16 weeks

Typical tasks used in the formation of current control options

1. Homework number 1. "Vector Algebra and Analytic Geometry"

Given: points A (0;3;2) , B (1;4;2) , D (0;1;2) ,			A(1;2;0) ; numbers a 30 ,				b1; corner
1. Find the length of the vector |		n | , if			p aq ,	n bp q	and p, q are unit

vectors, the angle between which is equal.

2. Find the coordinates of the point M dividing the vector AB with respect to a :1 .

3. Check if it is possible on vectors AB and AD construct a parallelogram. If yes, then find the lengths of the sides of the parallelogram.

4. Find the angles between the diagonals of the parallelogram ABCD.

5. Find the area of ​​parallelogram ABCD.

6. Make sure the vectors AB , AD , AA 1 you can build a parallelepiped. Find the volume of this parallelepiped and the length of its height.

7. Find vector coordinates AH , directed along the height of the parallelepiped ABCDA 1 B 1 C 1 D 1 , drawn from point A to the base plane A 1 B 1 C 1 D 1 ,

the coordinates of the point H and the coordinates of the unit vector coinciding in direction with the vector AH .

8. Find the decomposition of the vector AH by vectors AB , AD , AA 1 .

9. Find the projection of a vector AH to vector AA 1 .

10. Write the equations of the planes: a) P passing through the points A, B, D;

b) P1 passing through point A and line A1 B1 ;

c) P2 passing through the point A1 parallel to the plane P; d) P3 containing lines AD and AA1 ;

e) P4 passing through points A and C1 perpendicular to the plane P.

11. Find the distance between the lines on which the edges AB and CC lie one ; write the canonical and parametric equations of the common perpendicular to them.

12. Find point A 2, symmetrical to point A1 with respect to the plane of the base

13. Find the angle between the line on which the diagonal A lies 1 C, and base plane ABCD.

14. Find acute angle between planes ABC 1 D (plane P) and ABB1 A1 (plane P1 ).

2. Homework #2. "Curves and surfaces of the second order"

In Problems 1–2, the given equation of the second-order line is reduced to canonical form and the curve is constructed in the OXY coordinate system.

AT task 3, using the given data, find the equation of the curve in the OXY coordinate system. For tasks 1–3 indicate:

1) canonical form of the line equation;

2) parallel transfer transformation leading to canonical form;

3) in the case of an ellipse: semiaxes, eccentricity, center, vertices, foci, distances from point C to foci; in the case of a hyperbola: semiaxes, eccentricity, center, vertices, foci, distances from point C to foci, asymptote equations; in the case of a parabola: parameter, vertex, focus, directrix equation, distances from point C to focus and directrix;

4) for point C, check the property characterizing the given type of curves as the locus of points.

AT In problem 4, indicate the parallel translation transformation that reduces the given surface equation to the canonical form, the canonical form of the surface equation, and the type of the surface. Construct a surface in the canonical coordinate system OXYZ.

	5x 2 y 2 20x 2y 4 , C (0;1				2) 5x 2 4y 2 20x 8y 64 , C (12;14) .
		5) ;
	The parabola is symmetrical with respect to the straight line y 1 0 , has a focus							; 1 ,
	crosses the OX axis at point C				; 0 , and its branches lie in the half-plane	x 0 .
	4y 2 z 2 8y 4z 1 0 .

Modulo control No. 1 “Vector algebra. Analytic geometry"

1. Right and left triples of vectors. Definition of the cross product of vectors. Formulate properties of the vector product of vectors. Derive a formula for calculating the cross product of two vectors given by their coordinates in an orthonormal basis.

vectors

a m n ,

m n ,

1, m, n

Maybe,

vector decomposition

c 3 i

12j6k

vectors

3 j 2 k and b 2 i 3 j 4 k .

Write an equation for the plane

passing through the points M 1 5, 1, 4 ,

M 2 2, 3.1 and

perpendicular to the plane

6x 5y 4z 1 0. Set up canonical equations

a straight line passing through the point M 0 0, 2,1 and orthogonal to the found plane.

Test "Curves and surfaces of the second order"

1. Definition of an ellipse as a locus of points. Derivation of the canonical equation of an ellipse in a rectangular Cartesian coordinate system. The main parameters of the curve.

2. Surface equation x 2 4y 2 z 2 8x 4y 6z 17 0 lead to the canonical

mind. Make a drawing in the canonical coordinate system. Specify the name of this surface.

3. Write an equation for an equiaxed hyperbola if its center O 1 1, 1 and one of its foci F 1 3, 1 are known. Make a drawing.

Modulo control No. 2 “Curves and surfaces of the second order. Matrices and systems of linear algebraic equations»

1. Homogeneous systems of linear algebraic equations (SLAE). Forms of writing a homogeneous SLAE. Proof of a Criterion for the Existence of Nonzero Solutions of a Homogeneous SLAE.

2. Solve the matrix equation AX B ,

Make a check.

3. a) Solve the SLAE. b) Find a normal fundamental system of solutions of the corresponding homogeneous system, a particular solution of the inhomogeneous system; write through them the general solution of this inhomogeneous system:

x 1 2 x 2 3 x 3 4 x 4 4 x 2 x 3 x 4 3

x1 3x2 3x4 1

7 x 2 3 x 3 x 4 3

Questions to prepare for module controls, tests, tests and exams

1. Geometric vectors. Free vectors. Definition of collinear and coplanar vectors. Linear operations on vectors and their properties.

2. Definition of linear dependence and linear independence of vectors. Evidence for the conditions of linear dependence 2 and 3 vectors.

3. Definition of a basis in spaces of vectors V1, V2, V3. Proof of the theorem on the existence and uniqueness of the expansion of a vector in terms of a basis. Linear operations on vectors given by their coordinates in the basis.

4. Definition of the scalar product of vectors, its connection with the orthogonal projection of a vector onto an axis. Properties of the scalar product, their proof. Derivation of the formula for calculating the scalar product of vectors in an orthonormal basis.

5. Definition of an orthonormal basis. Relationship between the coordinates of a vector in an orthonormal basis and its orthogonal projections onto the vectors of this basis. Derivation of formulas for calculating the length of a vector, its direction cosines, the angle between two vectors in an orthonormal basis.

6. Right and left triples of vectors. Definition of the cross product of vectors, its mechanical and geometric meaning. Cross product properties (without doc-va). Derivation of the formula for calculating the cross product in an orthonormal basis.

7. Definition of the mixed product of vectors. The volume of the parallelepiped and the volume of the pyramid, built on non-coplanar vectors. Complanarity condition for three vectors. Properties of a mixed product. Derivation of the formula for calculating the mixed product in an orthonormal basis.

8. Definition of a rectangular Cartesian coordinate system. Solution of the simplest problems of analytical geometry.

9. Various types of equation of a straight line on a plane: vector, parametric, canonical. The direction vector is straight.

10. Derivation of the equation of a straight line passing through two given points.

11. Proof of the theorem that in a rectangular Cartesian coordinate system on a plane, an equation of the first degree defines a straight line. Definition of the normal vector of a straight line.

12. Equation with slope coefficient, equation of a straight line “in segments”. The geometric meaning of the parameters included in the equations. Angle between two lines. Conditions of parallelism and perpendicularity of two lines given by their general or canonical equations.

13. Derivation of the formula for the distance from a point to a line on a plane.

14. Proof of the theorem that in a rectangular Cartesian coordinate system in space, an equation of the first degree defines a plane. General equation of the plane. Definition of the normal vector of the plane. Derivation of the equation of a plane passing through three given points. Equation of the plane “in segments”.

15. Angle between planes. Conditions of parallelism and perpendicularity of two planes.

16. Derivation of the formula for the distance from a point to a plane.

17. General equations of a straight line in space. Derivation of vector, canonical and parametric equations of a straight line in space.

18. Angle between two straight lines in space, conditions of parallelism and perpendicularity of two straight lines. Conditions for two lines to belong to the same plane.

19. Angle between a straight line and a plane, conditions of parallelism and perpendicularity of a straight line and a plane. The condition of belonging to a straight line of a given plane.

20. The problem of finding the distance between intersecting or parallel lines.

21. Definition of an ellipse as a locus of points. Derivation of the canonical equation of the ellipse.

22. Definition of a hyperbola as a locus of points. Derivation of the canonical equation of a hyperbola.

23. Definition of a parabola as a locus of points. Derivation of the canonical parabola equation.

24. Definition of a cylindrical surface. Canonical equations of cylindrical surfaces 2nd order.

25. The concept of a surface of revolution. Canonical equations of surfaces formed by the rotation of an ellipse, hyperbola and parabola.

26. Canonical equations of an ellipsoid and a cone. Investigation of the shape of these surfaces by the section method.

27. Canonical equations of hyperboloids. Investigation of the shape of hyperboloids by the method of sections.

28. Canonical equations of paraboloids. Investigation of the shape of paraboloids by the method of sections.

29. The concept of a matrix. Types of matrices. Matrix equality. Linear operations on matrices and their properties. Matrix transposition.

30. Matrix multiplication. Properties of the operation of matrix multiplication.

31. Definition of an inverse matrix. Proof of the uniqueness of the inverse matrix. Proof of the inverse matrix theorem for the product of two invertible matrices.

32. Criterion for the existence of an inverse matrix. The concept of the associated matrix, its connection with the inverse matrix.

33. Derivation of Cramer's formulas for solving a system of linear equations with a nondegenerate square matrix.

34. Linear dependence and linear independence of rows (columns) of a matrix. Proof of the criterion for linear dependence of rows (columns).

35. Definition of a matrix minor. Basic minor. Basis minor theorem (without doqua). Proof of its corollary for square matrices.

36. Fringing minors method for finding the rank of a matrix.

37. Elementary transformations of rows (columns) of a matrix. Finding the inverse matrix by the method of elementary transformations.

38. Matrix rank invariance theorem under elementary transformations. Finding the rank of a matrix by the method of elementary transformations.

39. Systems of linear algebraic equations (SLAE). Various forms of writing SLAE. Joint and non-joint SLAE. Proof of the Kronecker-Kapeli criterion of SLAE compatibility.

40. Homogeneous systems of linear algebraic equations (SLAE). Properties of their solutions.

41. Definition of a fundamental system of solutions (FSR) of a homogeneous system of linear algebraic equations (SLAE). Theorem on the structure of the general solution of a homogeneous SLAE. Construction of the FSR.

42. Inhomogeneous systems of linear algebraic equations (SLAE). Proof of the theorem on the structure of the general solution of an inhomogeneous SLAE.

Control event	Number of tasks	Points for the task
DZ №1, part 1
Points scored
Control event	Number of tasks	Points for the task
DZ №1, part 2
Points scored

Control event				Number of tasks			Points for the task
Modulo control No. 1				1 theory and 3 problems			theory - 0; 3; 6
							tasks - 0; one; 2
			Points scored
Control event				Number of tasks			Points for the task
	Points scored
Control event				Number of tasks			Points for the task
				1 theory and 3 problems			theory - 0; 3; 6
							tasks - 0; one; 2
			Points scored
						01 theory and 3 problems			theory - 0; 3; 6
		tasks - 0; one; 2
			Points scored

Journal scoring rules

1. Points for DZ. Points for DZ are set the next week after the due date, according to the corresponding table. The student has the right to submit individual assignments for verification before the deadline and correct the errors noted by the teacher, while receiving the necessary advice. If by the deadline for submitting the DZ the student brings the solution to the problem to the correct option, then the maximum score is given to him for this task. After the deadline for submitting the DZ, a student who has not scored the minimum score for the DZ can continue to work on the assignment. At the same time, in case of successful work, the student is awarded the minimum score for the DZ.

2. Points for CR. If a student does not reach the minimum score for the CR on time, then during the semester he can rewrite this work twice. With a positive result (a set of points not less than the established minimum), the student is given the minimum score for the KR.

3. Points for "modulo control". As a “modulo control”, a written work is proposed, consisting of theoretical and practical parts. Each part of the control modulo is evaluated separately. A student who has received a mark not lower than the minimum in one of the parts of the control is considered to have passed this part and is released from its implementation in the future. At the discretion of the teacher, an interview can be conducted on the theoretical part of the assignment. If a student does not score the minimum for each part of the work, then during the semester he has two attempts for each part to correct the situation. With a positive

As a result (a set of points not less than the established minimum), the student is given the minimum score for "module control".

4. Grade per module. If the student has completed all the current control activities of the module (scored at least the established minimum score),

then the assessment for the module is the sum of points for all control activities of the module (in this case, the student automatically scores at least the minimum threshold). The final points for the module are entered in the journal after the completion of all control activities.

5. Total score. Sum of points for two modules.

6. Evaluation. The final certification (exam, differentiated test, test) is carried out based on the results of work in the semester after the student has completed the planned amount of study work and received an assessment for each module that is not lower than the minimum established. The maximum score for all modules, including scores for diligence, is 100, the minimum is 60. The sum of scores for all modules forms a rating score for the discipline for the semester. A student who has passed all control measures receives a final grade in the discipline for the semester in accordance with the scale:

		Exam grade,	Assessment on offset
		differentiated standings
		satisfactorily
		unsatisfactory

You can increase your rating, and, therefore, the examination grade at the final exam (written work on the material of the discipline as a whole is carried out during the examination session), the maximum score is 30, the minimum is 16. These points are summed up with the points obtained for all modules in the discipline. At the same time, in order to increase the grade to “good” for the exam, the student must score at least 21 points, to “excellent” ─ at least 26 points. For specialties where credit is provided for by discipline, the rating is not increased. Students who have a rating in the range of 0-59 by the beginning of the examination session gain the minimum required to obtain a positive grade in the discipline by retaking control events that were not credited earlier for individual modules. At the same time, students who do not have a good reason can eventually (by the end of the examination session) receive a grade no higher than “satisfactory”.