How are vector coordinates determined? §3

Vector coordinates

The quantity is called abscissa of the vector, and the number is his ordinate

How a basis is formed on a plane

How a basis is formed in space

The basis of a vector space is an ordered maximal linearly independent system of vectors from this space.

Definition System of vectors a1, a2, . . . , an from a vector space V is called a system of generators of this space if any vector from V is linearly expressed through the vectors a1, a2, . . . , an.

An ordered system of vectors is a basis of a vector space V if and only if it is a linearly independent system of generators of this space

What is a Cartesian basis?

If the vectors e1, e2, e3 are mutually orthogonal and modulo equal to one, then they are called the orts of a rectangular Cartesian coordinate system, and the basis itself is an orthonormal Cartesian basis.

Formulate the properties of coordinates of vectors in a Cartesian basis

What are the coordinates of a point?

The distances of a point from coordinate planes are called point coordinates.
The distance AA 1 point from the plane P 1 is called the applicate of the point and is denoted y A, the distance AA 2 points from the plane P 2 is the ordinate of the point and is denoted y A, the distance AA 3 points from the plane P 3 is the abscissa of the point and is denoted x A.
Obviously, the coordinate of the applicate point z A is the height AA 1, the coordinate of the ordinate point y A is the depth AA 2, the coordinate of the abscissa point x A is the latitude AA 3.

How are the coordinates of a vector calculated if the coordinates of its end and beginning are known?

How to calculate the distance between two points if their coordinates are known

You yourself know that AB (x1-x2;y1-y2)
The distance between points is the length of the vector AB.

What are direction cosines

Direction cosines of a vector are the cosines of the angles that the vector forms with the positive semi-axes of coordinates.

Direction cosines uniquely specify the direction of the vector.

What is called the projection of a vector onto an axis, prove the properties of projections.

Vector projection per axis l() is the length of its component per axis l, taken with a plus sign if the direction of the component coincides with the direction of the axis l, and with a minus sign if the direction of the component is opposite to the direction of the axis.

If = , then they believe = .

Theorem I The projection of a vector onto the l axis is equal to the product of its modulus and the cosine of the angle between this vector and the l axis.

Proof. Since the vector = free, we can assume that its origin O lies on the l axis(Fig. 34).

If the angle sharp, then the direction of the component = , vector coincides with the direction of the axis l(Figure 34,a).

In this case we have = + = . If the angle (Fig. 34, b) , then the direction of the component = vector opposite to the direction of the axis l. Then we get = = cos( - ) = cos

The same goes for the vector.

What is the scalar product of vectors

Dot product two non-zero vectors a and b is a number equal to the product of the lengths of these vectors by the cosine of the angle between them.

Formulate the condition for the orthogonality of vectors

Condition for orthogonality of vectors. Two vectors a and b orthogonal (perpendicular), if their scalar product is equal to zero.

Prove the properties of the scalar product of vectors

Properties of the scalar product of vectors

1. The scalar product of a vector with itself is always greater than or equal to zero:

1. The scalar product of a vector with itself is equal to zero if and only if the vector is equal to the zero vector:

a · a = 0<=>a = 0

1. The scalar product of a vector with itself is equal to the square of its modulus:

1. The operation of scalar multiplication is communicative:

1. If the scalar product of two non-zero vectors is equal to zero, then these vectors are orthogonal:

a ≠ 0, b ≠ 0, a b = 0<=>a ┴ b

1. (αa) b = α(a b)
2. The operation of scalar multiplication is distributive:

(a + b) c = a c + b c

Derive the scalar product expression in terms of coordinates

Formulate the properties of a vector product

ONLY 1 FORMULA

From above it is a determinant.

Analytic geometry

1. Prove theorems about the general equation of a line on a plane

2. Conduct a study of the general equation of a line on a plane

3. Derive the equation of a straight line on a plane with an angular coefficient and the equation of a straight line in segments on the axes

4. Derive the canonical equation of a line on a plane, write parametric equations, derive the equation of a line passing through two given points

5. How is the angle between straight lines on a plane determined if they are given by canonical equations or equations with an angular coefficient?

6. Derive conditions for parallelism, coincidence and perpendicularity of lines on a plane

7. Obtain a formula for calculating the distance from a point to a straight line on a plane

8. Prove theorems about the general equation of the plane

9. Formulate and prove a theorem about the relative position of a pair of planes

10. Conduct a study of the general equation of the plane

11. Obtain the equation of a plane in segments and the equation of a plane passing through two given points

12. Get a formula for calculating the distance from a point to a plane

13. How is the angle between planes calculated?

14. Derive the conditions for parallelism and perpendicularity of two planes

15. Write down the general form of the equations of a line in space, obtain the canonical form of the equations of a line in space

16. Derive parametric equations of a line in space, as well as a line passing through two points in space.

17. How is the angle between two straight lines in space determined? Write down the conditions for parallelism and perpendicularity of lines in space

18. How is the angle between a straight line and a plane determined? Write down the conditions for perpendicularity and parallelism of a line and a plane

19. Obtain the condition for two straight lines to belong to the same plane

Mathematical analysis

1. What is a function, what are the ways to define it?

2. What are even and odd functions, how to build their graphs

3. What are periodic and inverse functions, how to build their graphs

4. Draw exponential and logarithmic functions in graphs for a>1, a<1.

5. What is a harmonic dependence, what is the type of its graph?

6. Draw graphs y=arcsinx, y=arccosx, y=arctgx, y=arcctgx

7. What is an elementary function. Graphs of basic elementary functions

8. How to build graphs like y=cf(x), y=f(cx), y=f(x)+c, y=f(x+c)

9. What is a number sequence, what are the methods for defining it?

10. What is a monotonic and bounded sequence?

11. What is called the limit of a sequence? Write down the definition that a given number is not the limit of a given sequence

12. Formulate the properties of sequence limits

13. Prove two main properties of convergent sequences

14. Which of them provides the necessary condition for convergence?

15. Formulate a theorem that gives a sufficient condition for the convergence of the sequence

16. Prove any of the properties of sequence limits

17. What is an infinitesimal (large) sequence?

18. Formulate the properties of infinitesimal sequences

19. What is called the limit of a function?

20. Formulate the properties of function limits

21. What is called a one-sided limit?

22. Write down the first remarkable limit and derive its consequence

23. Write down the second remarkable limit and derive its consequences

24. What functions are called infinitesimal, limited, infinitely large?

25. Formulate the properties of infinitesimal functions, prove any of them

26. What concepts are introduced to compare infinitesimal functions, give their definitions

27. Which function is called continuous at a given point?

28. Formulate a criterion of continuity and characterize the types of discontinuities

29. What is the derivative of a function at a fixed point?

30. What are called one-sided derivatives?

31. What is the differential of a function and how is it related to the increment of a function?

32. Physical meaning of the first and second derivatives

33. What is the derivative of a function?

34. List the properties of derivatives, prove two of them (u+v)" and (uv)"

35. Write down a table of derivatives, prove any two formulas

36. What is the geometric meaning of derivative and differential?

37. Derive the equation of the tangent and normal to the graph of the function

38. Prove the theorem about the derivative of a complex function

39. Derive the derivative of the inverse function (give an example of finding it)

40. Justify the theorem on the calculus of derivatives

41. Prove all mean value theorems for differentiable functions

42. Formulate and prove L'Hopital's rule

43. What functions are called increasing and decreasing on an interval?

44. Prove theorems about the connection between the derivative and the increase of the function

45. What are extremum points?

46. ​​Justify the necessary condition for an extremum

47. Derive two types of sufficient conditions for an extremum

48. How to find the largest and smallest values ​​of a function on a segment?

49. What are called convex and concave functions?

50. How to examine a function for convexity and concavity? What are inflection points?

51. Asymptotes - give definitions, explain methods of finding

52. Derive a formula for finding the derivative (first and second) of a parametrically defined function

53. What is a vector function, its hodograph and its mechanical meaning?

54. Characterize in magnitude and direction the speed and acceleration of a material point with uniform motion in a circle

55. Characterize in magnitude and direction the speed and acceleration of a material point with uneven motion in a circle

56. Obtain derivatives of the function y=e x , y=sinx, y=cosx, y=tgx, y=lnx, y=arcsinx, y=arccosx

What are vector coordinates?

Vector coordinates are called the projections and of a given vector on the and axis, respectively:

The quantity is called abscissa of the vector, and the number is his ordinate. The fact that the vector has coordinates and is written as follows: .

Rectangular coordinate system

To define the concept of coordinates of points, we need to introduce a coordinate system in which we will determine its coordinates. The same point in different coordinate systems can have different coordinates. Here we will consider a rectangular coordinate system in space.

Let's take a point $O$ in space and introduce coordinates $(0,0,0)$ for it. Let's call it the origin of the coordinate system. Let us draw three mutually perpendicular axes $Ox$, $Oy$ and $Oz$ through it, as in Figure 1. These axes will be called the abscissa, ordinate and applicate axes, respectively. All that remains is to enter the scale on the axes (unit segment) - the rectangular coordinate system in space is ready (Fig. 1)

Figure 1. Rectangular coordinate system in space. Author24 - online exchange of student work

Point coordinates

Now let’s look at how the coordinates of any point are determined in such a system. Let's take an arbitrary point $M$ (Fig. 2).

Let's construct a rectangular parallelepiped on the coordinate axes, so that the points $O$ and $M$ are opposite its vertices (Fig. 3).

Figure 3. Construction of a rectangular parallelepiped. Author24 - online exchange of student work

Then point $M$ will have coordinates $(X,Y,Z)$, where $X$ is the value on the number axis $Ox$, $Y$ is the value on the number axis $Oy$, and $Z$ is the value on number axis $Oz$.

Example 1

It is necessary to find a solution to the following problem: write the coordinates of the vertices of the parallelepiped shown in Figure 4.

Solution.

Point $O$ is the origin of coordinates, therefore $O=(0,0,0)$.

Points $Q$, $N$ and $R$ lie on the axes $Ox$, $Oz$ and $Oy$, respectively, which means

$Q=(2,0,0)$, $N=(0,0,1.5)$, $R=(0,2.5,0)$

Points $S$, $L$ and $M$ lie in the planes $Oxz$, $Oxy$ and $Oyz$, respectively, which means

$S=(2,0,1.5)$, $L=(2,2.5,0)$, $R=(0,2.5,1.5)$

Point $P$ has coordinates $P=(2,2.5,1.5)$

Vector coordinates based on two points and the formula for finding

To find out how to find a vector from the coordinates of two points, you need to consider the coordinate system we introduced earlier. In it, from the point $O$ in the direction of the $Ox$ axis we plot the unit vector $\overline(i)$, in the direction of the $Oy$ axis - the unit vector $\overline(j)$, and the unit vector $\overline(k) $ must be directed along the $Oz$ axis.

In order to introduce the concept of vector coordinates, we introduce the following theorem (we will not consider its proof here).

Theorem 1

An arbitrary vector in space can be expanded into any three vectors that do not lie in the same plane, and the coefficients in such an expansion will be uniquely determined.

Mathematically it looks like this:

$\overline(δ)=m\overline(α)+n\overline(β)+l\overline(γ)$

Since the vectors $\overline(i)$, $\overline(j)$ and $\overline(k)$ are constructed on the coordinate axes of a rectangular coordinate system, they obviously will not belong to the same plane. This means that any vector $\overline(δ)$ in this coordinate system, according to Theorem 1, can take the following form

$\overline(δ)=m\overline(i)+n\overline(j)+l\overline(k)$ (1)

where $n,m,l∈R$.

Definition 1

The three vectors $\overline(i)$, $\overline(j)$ and $\overline(k)$ will be called coordinate vectors.

Definition 2

The coefficients in front of the vectors $\overline(i)$, $\overline(j)$ and $\overline(k)$ in expansion (1) will be called the coordinates of this vector in the coordinate system given by us, that is

$\overline(δ)=(m,n,l)$

Linear operations on vectors

Theorem 2

Sum Theorem: The coordinates of the sum of any number of vectors are determined by the sum of their corresponding coordinates.

Proof.

We will prove this theorem for 2 vectors. For 3 or more vectors, the proof is constructed in a similar way. Let $\overline(α)=(α_1,α_2,α_3)$, $\overline(β)=(β_1,β_2 ,β_3)$.

These vectors can be written as follows

$\overline(α)=α_1\overline(i)+ α_2\overline(j)+α_3\overline(k)$, $\overline(β)=β_1\overline(i)+ β_2\overline(j)+ β_3\overline(k)$

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

Finding the coordinates of a vector in a plane

What is a plane? A plane is considered to be a two-dimensional space, a space with two dimensions (the x dimension and the y dimension). For example, paper is flat. The surface of the table is flat. Any non-volumetric figure (square, triangle, trapezoid) is also a plane. Thus, if in the problem statement you need to find the coordinates of a vector that lies on a plane, we immediately remember about x and y. You can find the coordinates of such a vector as follows: Coordinates AB of the vector = (xB – xA; yB – xA). The formula shows that you need to subtract the coordinates of the starting point from the coordinates of the end point.

Example:

• Vector CD has initial (5; 6) and final (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 a vector, you need to subtract the coordinates of the beginning 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 coordinate value can be negative, there is no problem.

How to find vector coordinates online?

If for some reason you don’t want to find the coordinates yourself, you can use an online calculator. To begin, select the vector dimension. 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 for you 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.

Finally, I got my hands on this vast and long-awaited topic. analytical geometry. First, a little about this section of higher mathematics... Surely you now remember a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “analytical solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical same method involves solving problems mainly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent; often it is enough to carefully apply the necessary formulas - and the answer is ready! No, of course, we won’t be able to do this without drawings at all, and besides, for a better understanding of the material, I will try to cite them beyond necessity.

The newly opened course of lessons on geometry does not pretend to be theoretically complete; it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need more complete help on any subsection, I recommend the following quite accessible literature:

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

2) Geometry in 2 volumes. Authors L.S. Atanasyan, Bazylev V.T.. This is literature for high school, you will need first volume. Rarely encountered tasks may fall out of my sight, and the tutorial will be of invaluable help.

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

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

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

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

Vector concept. Free vector

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

In this case, the beginning of the segment is the point, the end of the segment is the point. The vector itself is denoted by . Direction is essential, if you move the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must agree, entering the doors of an institute or leaving the doors of an institute are completely different things.

It is convenient to consider individual points of a plane or space as the so-called zero vector. For such a vector, the end and beginning coincide.

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

Designations: Many immediately noticed the stick without an arrow in the designation and said, there’s also an arrow at the top! True, you can write it with an arrow: , but it is also possible the entry that I will use in the future. Why? Apparently, this habit developed for practical reasons; my shooters at school and university turned out to be too different-sized and shaggy. In educational literature, sometimes they don’t bother with cuneiform writing at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was stylistics, and now about ways to write vectors:

1) Vectors can be written in two capital Latin letters:
and so on. In this case, the first letter Necessarily denotes the beginning point of the vector, and the second letter denotes the end point of the vector.

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

Length or module a non-zero vector is called the length of the segment. The length of the zero vector is zero. Logical.

The length of the vector is indicated by the modulus sign: ,

We will learn how to find the length of a vector (or we will repeat it, depending on who) a little later.

This was basic information about vectors, familiar to all schoolchildren. In analytical geometry, the so-called free vector.

To put it simply - the vector can be plotted from any point:

We are accustomed to calling such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, they are the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” this or that “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 a bunch of identical directed segments. The school definition of a vector, given at the beginning of the paragraph: “A directed segment is called a vector...” implies specific a directed segment taken from a given set, which is tied to a specific point in the plane or space.

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

Actions with vectors. Collinearity of vectors

A school geometry course covers a number of actions and rules with vectors: addition according to the triangle rule, addition according to the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, let us repeat two rules that are especially relevant for solving problems of analytical geometry.

The rule for adding vectors using the triangle rule

Consider two arbitrary non-zero vectors and :

You need to find the sum of these vectors. Due to the fact that all vectors are considered free, we will set aside the vector from end vector:

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

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

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

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

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

The work a non-zero vector on a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .

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

Let's look at it in more detail:

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

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

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

4) The vectors are co-directed. Vectors and are also co-directed. Any vector of the first group is oppositely directed with respect to any vector of the second group.

Which vectors are equal?

Two vectors are equal if they are in the same direction and have the same length. Note that codirectionality implies collinearity of vectors. The definition would be inaccurate (redundant) if we said: “Two vectors are equal if they are collinear, codirectional, and have the same length.”

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

Vector coordinates on the plane and in space

The first point is to consider vectors on the plane. Let us depict a Cartesian rectangular coordinate system and plot it from the origin of coordinates single vectors and :

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

Designation: The orthogonality of vectors is written with the usual perpendicularity symbol, for example: .

The vectors under consideration are called coordinate vectors or orts. These vectors form basis on surface. What a basis is, I think, is intuitively clear to many; more detailed information can be found in the article Linear (non) dependence of vectors. Basis of vectors In simple words, the basis and origin of coordinates define the entire system - this is a kind of foundation on which a full and rich geometric life boils.

Sometimes the constructed basis is called orthonormal basis of the plane: “ortho” - because the coordinate vectors are orthogonal, the adjective “normalized” means unit, i.e. the lengths of the basis vectors are equal to one.

Designation: the basis is usually written in parentheses, inside which in strict sequence basis vectors are listed, for example: . Coordinate vectors it is forbidden rearrange.

Any plane vector the only way expressed as:
, Where - numbers which are called vector coordinates in this basis. And the expression itself called vector decompositionby basis .

Dinner served:

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

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

Vectors illustrate exactly the rule for multiplying a vector by a number, the vector is codirectional with the base vector, the vector is directed opposite to the base vector. For these vectors, one of the coordinates is equal to zero; you can meticulously write it like this:


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

And finally: , . By the way, what is vector subtraction, and why didn’t I talk about the subtraction rule? Somewhere in linear algebra, I don’t remember where, I noted that subtraction is a special case of addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , . 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, multiplying the vector by a number: (red arrow), (green arrow) and (raspberry arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector begins at the initial point of departure (beginning of the vector) and ends at the final point of arrival (end of the vector).

All vectors of three-dimensional space, naturally, are also free; try to mentally set aside the vector from any other point, and you will understand that its decomposition “will remain with it.”

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

If one (or two) coordinate vectors are missing in the expansion, then zeros are put in their place. Examples:
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write .

The basis vectors are written as follows:

This, perhaps, is all the minimum theoretical knowledge necessary to solve problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that teapots re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time to better assimilate the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I note that the materials on the site are not enough to pass the theoretical test or colloquium on geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of presentation, but a plus to your understanding of the subject. To receive detailed theoretical information, please bow to Professor Atanasyan.

And we move on to the practical part:

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

It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, you don’t even have to remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on your shirt; many things are familiar to you from school.

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

How to find a vector from two points?

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

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

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

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

Example 1

Given two points of the plane and . Find vector coordinates

Solution: according to the 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

Line segment - this is not a vector, and, of course, you cannot move it anywhere. In addition, if you draw to scale: 1 unit. = 1 cm (two notebook cells), then the resulting answer can be checked with a regular ruler by directly measuring the length of the segment.

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

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

Secondly, let us repeat the school material, which is useful not only for the task considered:

pay attention to important technique – removing the multiplier from under the root. As a result of the calculations, we have a result and good mathematical style involves removing the factor from under the root (if possible). In more detail the process looks like this: . Of course, leaving the answer as is would not be a mistake - but it would certainly be a shortcoming and a weighty argument for quibbling on the part of the teacher.

Here are other common cases:

Often the root produces a fairly large number, for example . What to do in such cases? Using the calculator, we check whether the number is divisible by 4: . Yes, it was completely divided, thus: . Or maybe the number can be divided by 4 again? . Thus: . The last digit of the number is odd, so dividing by 4 for the third time will obviously not work. Let's try to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a number that cannot be extracted as a whole, then we try to remove the factor from under the root - using a calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

When solving various problems, roots are often encountered; always try to extract factors from under the root in order to avoid a lower grade and unnecessary problems with finalizing your solutions based on the teacher’s comments.

Let's also repeat squaring roots and other powers:

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

Task for independent solution with a segment in space:

Example 4

Points and are given. Find the length of the segment.

The solution and answer are at the end of the lesson.

How to find the length of a vector?

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

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

Until now, it was believed that vectors are considered in space. From this moment on, we assume that all vectors are considered on the plane. We will also assume that a Cartesian coordinate system is specified on the plane (even if this is not stated), representing two mutually perpendicular numerical axes - the horizontal axis and the vertical axis . Then each point
a pair of numbers is assigned on the plane
, which are its coordinates. Conversely, each pair of numbers
corresponds to a point on the plane such that a pair of numbers
are its coordinates.

From elementary geometry it is known that if there are two points on a plane
And
, then the distance
between these points is expressed through their coordinates according to the formula

Let a Cartesian coordinate system be specified on the plane. Orth axis we will denote by the symbol , and the unit vector of the axis symbol . Projection of an arbitrary vector per axis we will denote by the symbol
, and the projection onto the axis symbol
.

Let - an arbitrary vector on the plane. The following theorem holds.

Theorem 22.

For any vector there is a pair of numbers on the plane

.

Wherein
,
.

Proof.

Let a vector be given . Let's put aside the vector from the origin. Let us denote by vector-projection vector per axis , and through vector-projection vector per axis . Then, as can be seen from Figure 21, the equality holds

.

According to Theorem 9,

,

.

Let's denote
,
. Then we get

.

So, it has been proven that for any vector there is a pair of numbers
such that the equality is true

,

,

.

With a different vector location The proof is similar with respect to the axes.

Definition.

Pair of numbers And such that
, are called vector coordinates . Number is called the x-coordinate, and the number game coordinate.

Definition.

A pair of unit vectors of coordinate axes
is called an orthonormal basis on the plane. Representation of any vector as
called vector decomposition by basis
.

It follows directly from the definition of vector coordinates that if the coordinates of the vectors are equal, then the vectors themselves are equal. The converse is also true.

Theorem.

Equal vectors have equal coordinates.

Proof.

,

And
. Let's prove that
,
.

From the equality of vectors it follows that

.

Let's assume that
, A
.

Then
and that means
, which is not true. Likewise, if
, But
, That
. From here
, which is not true. Finally, if we assume that
And
, then we get that

.

This means that the vectors And collinears. But this is not true, since they are perpendicular. Therefore, it remains that
,
, which was what needed to be proven.

Thus, the coordinates of the vector completely determine the vector itself. Knowing the coordinates And vector you can build the vector itself , having constructed the vectors
And
and folding them. Therefore, often the vector itself denoted as a pair of its coordinates and written
. This entry means that
.

The following theorem follows directly from the definition of vector coordinates.

Theorem.

When adding vectors, their coordinates are added, and when multiplying a vector by a number, its coordinates are multiplied by this number. These statements are written in the form

.

Proof.

,

Theorem.

Let
, and the beginning of the vector is point has coordinates
, and the end of the vector is a point
. Then the coordinates of the vector are related to the coordinates of its ends by the following relations

,

.

Proof.

Let
and let the vector be the projection of the vector per axis aligned with the axis (see Fig. 22). Then

T as the length of a segment on the number axis equal to the coordinate of the right end minus the coordinate of the left end. If the vector

opposite to the axis (as in Fig. 23), then

Rice. 23.

If
, then in this case
and then we get

.

Thus, for any location of the vector
relative to the coordinate axes its coordinate equal to

.

Similarly, it is proved that

.

Example.

The coordinates of the ends of the vector are given
:
. Find vector coordinates
.

Solution.

The following theorem provides an expression for the length of a vector in terms of its coordinates.

Theorem 15.

Let
.Then

.

Proof.

Let And - vector projection vector on the axis And , respectively. Then, as shown in the proof of Theorem 9, the equality holds

.

At the same time, vectors And mutually perpendicular. When adding these vectors according to the triangle rule, we obtain a right triangle (see Fig. 24).

By the Pythagorean theorem we have

.

,

.

Hence

,

.

.

.

Example.

.Find .

Let us introduce the concept of direction cosines of a vector.

Definition.

Let the vector
is with the axis corner , and with the axis corner (See Fig. 25).

,

.

Hence,

Since for any vector there is equality

,

Where - unit vector , that is, a vector of unit length, codirectional with the vector , That

Vector determines the direction of the vector . Its coordinates
And
are called direction cosines of the vector . The direction cosines of a vector can be expressed through its coordinates using the formulas

,

.

There is a relationship

.

Until now in this section, it was assumed that all vectors are located in the same plane. Now let's generalize for vectors in space.

We will assume that a Cartesian coordinate system with axes is given in space ,And .

Axis unit vectors ,And we will denote by symbols ,And , respectively (Fig. 26).

It can be shown that all the concepts and formulas that were obtained for vectors on the plane are generalized for

Rice. 26.

vectors in space. Troika of vectors
is called an orthonormal basis in space.

Let ,And - vector projection vector on the axis ,And , respectively. Then

.

In its turn

,

,

.

If we designate

,

,

,

Then we get the equality

.

Coefficients before basis vectors ,And are called vector coordinates . Thus, for any vector there is a triple of numbers in space ,,, called vector coordinates such that for this vector the following representation is valid:

.

Vector in this case also denoted in the form
. In this case, the coordinates of the vector are equal to the projections of this vector onto the coordinate axes

,

,

,

Where - angle between vector and axis ,- angle between vector and axis ,- angle between vector and axis .

Vector length expressed through its coordinates using the formula

.

The statements are true that equal vectors have equal coordinates; when adding vectors, their coordinates are added, and when multiplying a vector by a number, its coordinates are multiplied by this number.
,
And
are called direction cosines of the vector . They are related to vector coordinates by the formulas

,
,
.

This implies the relation

If the ends of the vector
have coordinates
,
, then the coordinates of the vector
are related to the coordinates of the ends of the vector by the relations

,

,

.

Example.

Points are given
And
. Find vector coordinates
.