Give the concept of coordinate vectors. How to find the coordinates of the vector

Coordinates of the vector

The value is called absissue vector , and the number is his ordate

How the basis is formed on the plane

How the basis is formed in space

The basis of the vector space is called an ordered maximum linearly independent system of vectors from this space.

Definition system A1, A2 vectors ,. . . , An from the vector space V is called the system of forming this space if any vector of V is linearly expressed through vectors A1, A2 ,. . . , AN.

An ordered vectors system is the basis of the vector space V if it is only when it is a linearly independent system of forming this space

What is called a decartian basis

If vectors E1, E2, E3 are mutually orthogonal and the module is equal to one, then they are called orts of a rectangular Cartesian coordinate system, and the basis itself is an orthonormal decartular basis.

Formulate the properties of the coordinates of vectors in the Cartesian base

What is called point coordinates

The distances of the point from the coordinate planes are called the coordinates of the point.
The distance AA 1 points from the plane P 1 is called the applicant point and denote by A, the distance AA 2 points from the plane P 2 - the ordinate point and denote - U A, the distance AA 3 points from the plane P 3 - abscissa point and denote X A.
Obviously, the coordinate of the Application point Z A is the height of AA 1, the coordinate of the order of the ordinate in A is the depth of AA 2, the coordinate of the abscissa point X A - latheaa 3.

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

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

You know that AV (X1-X2; Y1-Y2)
The distance between the points is the length of the vector Av.

What is guide cosines

Cosine guides vector - These are cosines of angles, which vector forms with positive coordinate semi-axles.

Guide cosines definitely set the direction of the vector.

What is called the vector projection on the axis, prove the properties of projections.

Projection of vector on the axis l. () It is called the length of its components on the axis L. , taken with a "plus" sign, if the direction of the component coincides with the direction of the axis l., and with the "minus" sign, if the direction of the component is the opposite direction of the axis.

If \u003d. , that is believed = .

Theorem I The projection of the vector on the axis L is equal to the product of its module on the cosine of the angle between this vector and the axis L.

Evidence. Since vector \u003d free, it can be assumed that the beginning of it about lies on the axis L(Fig. 34).

If corner acute, then the direction of the component \u003d, the vector coincides with the axis direction l.(Figure 34, a).

In this case we have = + = . If the corner (Fig. 34, b) , that direction of the component = vector oppositely axis direction l. Then get \u003d \u003d cOS (-) \u003d cos.

The same - on the vector.

What is a scalar product of vectors

Scalar work two nonzero vectors A and B called a number equal to the product of these lengths vectors On the cosine of the corner between them.

Formulate the condition of the orthogonality of vectors

The condition of the orthogonality of vectors. Video of the vector A and B. orthogonal (perpendicular)If their scalar product is zero.

Prove the properties of a scalar product of vectors

Properties of the scalar product of vectors

The scalar product of the vector itself is always greater than or equal to zero:

The scalar product of the vector is equal to zero if and only if the vector is equal to the zero vector:

a · a \u003d 0<=> a \u003d 0.

The scalar product of the vector itself is equal to the square of its module:

Operation of scalar multiplication Communicative:

If the scalar product of two non-zero vectors is zero, then these vector orthogonal:

a ≠ 0, b ≠ 0, a · b \u003d 0<=> A ┴ B.

(αa) · b \u003d α (A · b)
Scalar multiplication operation Distribution:

(A + B) · C \u003d A · C + B · C

Bring the expression of the scalar product through the coordinates

Formulate vector properties

Only 1 formula

This is determined from above.

Analytic geometry

1. Prove the theorems on the overall line equation on the plane

2. Conduct the study of the general equation direct on the plane

3. To derive the direct equation on the plane with the angular coefficient and the equation is straight in segments on the axes

4. Remove the canonical equation to the line on the plane, write parametric equations, output the equation direct passing through two specified points

5. How determine the angle between straight on the plane, if they are set by canonical equations or equations with an angular coefficient?

6. Remove the conditions of parallelism, coincidence and perpendicularity of direct on the plane

7. Get a formula for calculating the distance from the point to direct on the plane

8. Prove the theorems on the overall plane equation

9. Formulate and prove the theorem about the mutual location of the pair of planes

10. Conduct a study of a general plane equation

11. Get the plane equation in segments and the equation of the plane passing through two setpoints

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

13. How is the angle between the planes?

14. Remove the conditions of parallelism and perpendicularity of two planes

15. Record a general view of equations direct in space, to obtain the canonical view of the equations direct in space

16. To derive parametric equations to the line in space, as well as a direct passing through two points of space.

17. How will the angle between two direct in space? Record the conditions of parallelism and perpendicularity of direct in space

18. How is the angle between the straight and plane determine? Record the conditions for perpendicularity and parallelism of the straight and plane

19. Get the condition to belong two direct single plane

Mathematical analysis

1. What is a function, what are the ways to go to her job?

2. What is aware and odd functions, how to build their charts

3. What is periodic and reverse functions, how to build their charts

4. Picture in the charts indicative and logarithmic functions at a\u003e 1, a<1.

5. What is a harmonious dependence, what is the type of graphics?

6. Pictulate the graphs y \u003d arcsinx, y \u003d arccosx, y \u003d arctgx, y \u003d arcctgx

7. What is an elementary function. Graphics of the main elementary functions

8. How to build graphs of the form y \u003d cf (x), y \u003d f (cx), y \u003d f (x) + c, y \u003d f (x + c)

9. What is the numerical sequence, what are the ways to go to her task?

10. What is a monotonous and limited sequence?

11. What is called the sequence limit? Record the definition of the fact that this number is not the limit of this sequence

12. Formulate properties of sequence limits

13. Prove two basic properties of converging sequences

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

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

16. Prove any of the properties of sequence limits

17. What is infinitely small (big) sequence?

18. Formulate the properties of infinitely small sequences

19. What is called the limit of the function?

20. Formulate the properties of the limits of functions

21. What is called one-sided limit?

22. Record the first wonderful limit and withdraw its consequence

23. Record the second wonderful limit and withdraw its investigation

24. What functions are infinitely small, limited, infinitely big?

25. Formulate the properties of infinitely small functions, to prove any of them

26. What concepts are introduced to compare infinitely small functions, give them definitions

27. What function is called continuous at a specified point?

28. Formulate continuity criteria and characterize the types of breaks

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

30. What is called one-sided derivatives?

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

32. The physical meaning of the first and second derivatives

33. What is a derivative function from the function?

34. List the properties of derivatives, to prove two of them (U + V) "and (UV)"

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

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

37. Remove the equation of tangential and normal to the graph

38. Prove the theorem about the derivative complex function

39. Displays a reverse function derivative (give an example of its location)

40. Justify the theorem on the calculation of derivatives

41. Prove all theorems on average for differentiable functions

42. Formulate and prove the Lopital rule

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

44. Prove the theorems on the connection of the derivative with increasing function

45. What is the point of extremum?

46. \u200b\u200bJustify the desired extremum condition

47. To withdraw two types of sufficient conditions of extremum

48. How to find the greatest and smallest values \u200b\u200bof the function on the segment?

49. What is called convex and concave function?

50. How to investigate the function on bulge and concaveness? What is called inflection points?

51. Asymptotes - give definitions, explain how to find ways

52. To derive the formula for finding a derivative (first and second) parametrically specified function.

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

54. It is characterized by the size and direction of the speed and acceleration of the material point with a uniform movement around the circle

55. Specify the speed and speed and acceleration of the material point in size and direction with an uneven movement around the circle

56. Get derived functions y \u003d e x, y \u003d sinx, y \u003d cosx, y \u003d tgx, y \u003d lnx, y \u003d arcsinx, y \u003d arccosx

What is called vector coordinates

Coordinates of the vector They are called projections and this vector on the axis and, accordingly:

The value is called absissue vector , and the number is his ordate. The fact that the vector has coordinates and is written as follows :.

To begin with, we will give the determination of the vector coordinates in a given coordinate system. To introduce this concept, we define that we call the rectangular or decartian coordinate system.

Definition 1.

Rectangular coordinate system It is a rectilinear coordinate system with mutually perpendicular axes on a plane or in space.

Using the introduction of a rectangular coordinate system on a plane or in three-dimensional space, it becomes possible to describe geometric figures along with their properties using equations and inequalities, that is, to use algebraic methods in solving geometric problems.

Thus, we can bind to the specified coordinate system vectors. This will significantly expand our opportunities in solving certain tasks.

The rectangular coordinate system on the plane is usually denoted by O x Y, where O x and o y - the axis of the coordinate. The O X axis is called the abscissa axis, and the OH axis is the ordinate axis (another O Z axis appears in the space, which is perpendicular to and o x and o y).

Example 1.

So, we are given a rectangular decartian coordinate system O xy on the plane if we postpone the coordinates of the ic → and j →, the direction of which respectively coincides with the positive directions of the axes of O X and O Y, and their length will be equal to the conditional unit, we will get coordinate Vectors. That is, in this case, I → and J → are coordinate vectors.

Coordinate vectors

Definition 2.

Vectors I → and J → corn the coordinate vectors for a given coordinate system.

Example 2.

Decoration from the beginning of the coordinates arbitrary vector A →. Relying on the geometric determination of operations over vectors, vector A → can be represented as a → \u003d a x · i → + a y · j →, where coefficients A X. and A Y. - The only one in its kind, their uniqueness is enough just to prove by the method from nasty.

Decomposition of vector

Definition 3.

Decomposition of vector A → By coordinate vectors I → and J → on surface It is called the representation of the form A → \u003d A x · I → + A y · j →.

Definition 4.

Coefficients a x and a y called vector coordinates in this coordinate system on the plane.

The coordinates of the vector in this system of coordinates are taken to record in parentheses, through the comma, while the specified coordinates should be separated from the designation of the vector sign of equality. For example, recording A → \u003d (2; - 3) means that the vector A → has coordinates (2; - 3) in this coordinate system and can be represented as decomposition by coordinate vectors I → and j → as a → \u003d 2 · I → 3 · J →.

Comment

It should be noted that the order of recording coordinates is important if you record the vector coordinates in another order, you will get a completely different vector.

Based on the definition of the coordinates of the vector and their decomposition it becomes obvious that single vectors I → and J → have coordinates (1; 0) and (0; 1), respectively, and they can be represented in the form of the following expansions I → \u003d 1 · I → + 0 · J →; j → \u003d 0 · I → + 1 · j →.

There is also a zero vector 0 → with coordinates (0; 0) and decomposition 0 → \u003d 0 · I → + 0 · J →.

Equal and opposite vectors

Definition 5.

Vectors A → and B → equal Then, when their respective coordinates are equal.

Definition 6.

Opposing vector It is called the vector opposite to this.

It follows that the coordinates of this vector will be opposite to the coordinates of this vector, that is, - a → \u003d (- a x; - a y).

All of the above can be similarly identified for a rectangular coordinate system specified in three-dimensional space. In such a coordinate system, there is a tripler of the coordinate vectors of I →, j →, k →, and an arbitrary vector A → is unfolded by two, but already in three coordinates, and the only way and the form A → \u003d AX · I → + AY · J → + az · k →, and the coefficients of this decomposition (AX; AY; AZ) are called the coordinates of the vector in this (three-dimensional) coordinate system.

Therefore, the coordinate vectors in the three-dimensional space are also taken to take 1 and have coordinates I → \u003d (1; 0; 0), j → \u003d (0; 1; 0), k → \u003d (0; 0; 1), zero vector coordinates Also equal zero 0 → \u003d (0; 0; 0), and in this case two vectors will be considered equal if all three corresponding vectors coordinates are equal to A → \u003d B → ⇔ AX \u003d BX, AY \u003d BY, AZ \u003d BZ , and the coordinates of the opposite vector A → are opposed to the corresponding vector coordinates of the vector A →, that is, - a → \u003d (- ax; - ay; - AZ).

To enter this definition, you need to show the coordinates and coordinate coordinates in this coordinate system.

Let us give some rectangular decartian coordinate system O x Y and it is set to an arbitrary point M with coordinates M (x m; y m).

Definition 7.

Vector O M → called radius point M. .

We define what coordinates in this coordinate system has a radius-vector point

Vector O M → It has the form of the amount of OM → \u003d OM X → + OM Y → \u003d x m · i → + y m · j →, where the points M x and M y are projections of the point M on the coordinate direct OX and OY, respectively (these reasoning follows from the definition Projection of the point to straight), and I → and J → - coordinate vectors, therefore, vector O M → It has coordinates (x m; y m) in this coordinate system.

In other words, the coordinates of the radius-vector point M are equal to the corresponding coordinates of the point M In a rectangular Cartesian coordinate system.

Similarly, in the three-dimensional space, the radius-vector point M (x m; y m; z m) decomposes on the coordinate vectors as OM → \u003d OM X → + OM Y → + OM Z → \u003d X M · I → + Y M · J → + z m · k →, therefore, Om → \u003d (x m; y m; z m).

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It was still believed that the vectors are considered in space. From now on, wake it with that all vectors are considered on the plane. We will also assume that the coordinate system is set on the plane (even if it does not mention about it), representing two mutually perpendicular numeric axes - a horizontal axis and a vertical axis . Then every point
the plane is put in conformity of a pair of numbers.
which are its coordinates. Back, each pair of numbers
corresponds to the point plane such that a pair of numbers
are its coordinates.

From the elementary geometry it is known that if there are two points on the plane
and
, Distance
between these points, it is expressed through their coordinates by the formula

Let the decartian coordinate system be asked on the plane. Ort axis we will denote symbol , and ort axis symbol . Projection arbitrary vector on the axis we will denote symbol
, and projection on the axis symbol
.

Let be - arbitrary vector on the plane. The following theorem takes place.

Theorem 22.

For any vector on the plane there are a couple of numbers

Wherein
,
.

Evidence.

Let vector be given . We write vector from the beginning of the coordinates. Denote by vector projection vector on the axis , and through vector projection vector on the axis . Then, as can be seen from Figure 21, there is equality

According to Theorem 9,

Denote
,
. Then get

So, it is proved that for any vector there is a couple of numbers
such that right equality

With a different vector location regarding axes, proof is similar.

Definition.

A pair of numbers and such that
are called vector coordinates . Number called the ICSO coordinate, and the number the player coordinate.

Definition.

Pair of orts coordinate axes
it is called an orthonormal basis on the plane. Representation of any vector as
called decomposition of vector basisus
.

Directly from the determination of the vector coordinates it follows that if the coordinates of the vectors are equal, the vectors themselves are equal. The opposite statement is also fair.

Theorem.

Equal vectors have equal coordinates.

Evidence.

and
. We prove that
,
.

From the equality of the vectors it follows that

Suppose that
, but
.

Then
and meaning
that is not true. Similarly, if
, but
T.
. From here
that is not true. Finally, if you assume that
and
then we get that

This means that vectors and collineares. But this is not true, as they are perpendicular. Therefore, it remains that
,
As required to prove.

Thus, the coordinates of the vector completely define the vector itself. Knowing coordinates and vector you can build a vector , Buing vectors
and
and folding them. So often the vector denote in the form of a pair of its coordinates and write
. Such a record means that
.

Directly from the determination of the coordinates of the vector follows the following theorem.

Theorem.

When the vectors are added, their coordinates are folded and when the vector is multiplying, its coordinates are multiplied by this number. These statements are recorded in the form of

Evidence.

Theorem.

Let be
, and the beginning of the vector point it has coordinates
, and the end of the vector is point
. Then the coordinates of the vector are associated with the coordinates of its ends by the following relations.

Evidence.

Let be
and let the vector-projection of the vector on the axis sonated with axis (See Fig. 22). Then

t. aK as the length of the segment on the numeric axis equal to the coordinate of the right end minus the coordinate of the left end. If vector

the axis is contaminated (as in fig. 23), then

Fig. 23.

If a
, then in this case
and then get

Thus, with any arrangement of the vector
relative to the coordinate axes of its coordinate equal

Similarly, it is proved that

Example.

Dana coordinates of the ends of the vector
:
. Find the coordinates of the vector
.

Decision.

The following theorem provides an expression of the vector length through its coordinates.

Theorem 15.

Let be
.Then

Evidence.

Let be and - vector projection vector on axis and , respectively. Then, as shown in the proof of Theorem 9, there is equality

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

Pythagore's theorem we have

Hence

Example.

.To find .

We introduce the concept of vector cosine guides.

Definition.

Let the vector
makes up with axis angle , and with the axis angle (See Fig. 25).

Hence,

As for any vector there is equality

Where - ort vector , that is, vector isolated length, coated with vector T.

Vector determines the direction of the vector . His coordinates
and
called vector guide cosines . Vector cosine guides can be expressed through its coordinates by formulas.

There is a ratio

Until now, this paragraph believed that all vectors are located in the same plane. Now make a generalization for vectors in space.

We assume that the Cartian coordinate system with axes is set in the space ,and .

Ots axes ,and we will denote symbols ,and , respectively (Fig. 26).

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

Fig. 26.

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

Let be ,and - vector projection vector on axis ,and , respectively. Then

In turn

If you designate

That we get equality

Coefficients before basic vectors ,and referred to as coordinates of the vector . Thus, for any vector there is a three numbers in space. ,,called vector coordinates such that for this vector is true

Vector in this case, also referred to in the form
. At the same time, the coordinates of the vector are equal to the projections of this vector on the coordinate axes

where - angle between the vector and axis ,- angle between the vector and axis ,- angle between the vector and axis .

Length vector it is expressed through its coordinates by the formula

Fair statements that equal vectors have equal coordinates, when adding the vectors, their coordinates are folded, and when the vector multiplying the vector, its coordinates are multiplied by this number.
,
and
called vector guide cosines . They are associated with vector coordinates formulas

,
,
.

Hence the ratio

If the ends of the vector
have coordinates
,
, then the coordinates of the vector
associated with the coordinates of the ends of the vector by relations

Example.

Points
and
. Find the coordinates of the vector
.

On the axis of the abscissa and ordents are called coordinates vector. Vector coordinates generally accepted in the form (x, y), And the vector itself is like: \u003d (x, y).

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

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

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

The formula for determining the coordinates of the vector for spatial tasks.

In the case of a spatial problem, the vector with known coordinates of the pointA. (x 1; in 1;z. 1 ) and B. (x. 2 ; y. 2 ; z. 2 ) You can calculate applying the formula:

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

The coordinates give a comprehensive characteristic of the vector, since the coordinates have the opportunity to build and the vector itself. Knowing coordinates, easy to calculate and length vector. (Property 3, shown below).

Properties of vector coordinates.

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

2. Coordinates collinear vectors Proportional. Provided that none of the vectors are zero.

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

4. In the operation multiplication of vector on the valid number Each coordinate is multiplied by this number.

5. With the formation of vectors, we calculate the amount corresponding coordinates of vectors.

6. Scalar product Two vectors equals the sum of the products of their respective coordinates.

Finding the coordinates of the vector quite frequently found the condition of many tasks in mathematics. The ability to find the coordinates of the vector will help you in other, more complex tasks with similar subjects. In this article we will look at the formula of finding the coordinates of the vector and several tasks.

Finding the coordinates of the vector in the plane

What is a plane? The plane is considered a two-dimensional space, a space with two dimensions (Measure X and the measurement Y). For example, paper is a plane. Table surface - plane. Some uninstalistic figure (square, triangle, trapezium) is also a plane. Thus, if in the task condition, you need to find the coordinates of the vector, which lies on the plane, immediately remember about X and Y. Find the coordinates of this vector as follows: The coordinates of the vector \u003d (XB - Xa; YB - XA). It can be seen from the formula that the coordinates of the endpoint require the coordinates of the starting point.

Example:

The vector CD has an initial (5; 6) and finite (7; 8) coordinates.
Find the coordinates of the vector itself.
Using the above-mentioned formula, we obtain the following expression: CD \u003d (7-5; 8-6) \u003d (2; 2).
Thus, the coordinates of the CD vector \u003d (2; 2).
Accordingly, X coordinate is two, y coordinate is also two.

Finding vector coordinates in space

What is space? The space is already a three-dimensional measurement, where 3 coordinates are given: x, y, z. In case you need to find a vector that lies in space, the formula is practically not changed. Only one coordinate is added. To find the vector you need from the coordinates of the end to take the beginning coordinates. AB \u003d (XB - Xa; YB - Ya; ZB - ZA)

Example:

The vector DF has an initial (2; 3; 1) and finite (1; 5; 2).
Using the above-mentioned formula, we obtain: the coordinates of the vector df \u003d (1-2; 5-3; 2-1) \u003d (-1; 2; 1).
Remember, the coordinate value may be negative, there is no problem in this.

How to find vector coordinates online?

If for some reasons you do not want to find the coordinates yourself, you can use the online calculator. To begin with, select the dimension of the vector. The dimension of the vector is responsible for its measurements. Dimension 3 means that the vector is in space, dimension 2 is that on the plane. Next, insert the coordinates of the points to 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 gives you the correct answer together with the solutions stages.

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