In this article, we will consider ways to determine the distance from a point to a point theoretically and on the example of specific tasks. Let's start with some definitions.

Definition 1

Distance between points- this is the length of the segment connecting them, in the existing scale. It is necessary to set the scale in order to have a unit of length for measurement. Therefore, basically the problem of finding the distance between points is solved by using their coordinates on the coordinate line, in the coordinate plane or three-dimensional space.

Initial data: the coordinate line O x and an arbitrary point A lying on it. One real number is inherent in any point of the line: let this be a certain number for point A xA, it is the coordinate of point A.

In general, we can say that the estimation of the length of a certain segment occurs in comparison with the segment taken as a unit of length on a given scale.

If point A corresponds to an integer real number, having set aside successively from point O to a point along a straight line O A segments - units of length, we can determine the length of segment O A by the total number of pending unit segments.

For example, point A corresponds to the number 3 - in order to get to it from point O, it will be necessary to set aside three unit segments. If point A has a coordinate of - 4, single segments are plotted in a similar way, but in a different, negative direction. Thus, in the first case, the distance O A is 3; in the second case, O A \u003d 4.

If point A has a rational number as a coordinate, then from the origin (point O) we set aside an integer number of unit segments, and then its necessary part. But geometrically it is not always possible to make a measurement. For example, it seems difficult to put aside the coordinate direct fraction 4 111 .

In the above way, it is completely impossible to postpone an irrational number on a straight line. For example, when the coordinate of point A is 11 . In this case, it is possible to turn to abstraction: if the given coordinate of point A is greater than zero, then O A \u003d x A (the number is taken as a distance); if the coordinate is less than zero, then O A = - x A . In general, these statements are true for any real number x A .

Summarizing: the distance from the origin to the point, which corresponds to a real number on the coordinate line, is equal to:

• 0 if the point is the same as the origin;

• x A if x A > 0 ;

• - x A if x A< 0 .

In this case, it is obvious that the length of the segment itself cannot be negative, therefore, using the modulus sign, we write the distance from the point O to the point A with the coordinate x A: O A = x A

The correct statement would be: the distance from one point to another will be equal to the modulus of the difference in coordinates. Those. for points A and B lying on the same coordinate line at any location and having, respectively, the coordinates x A And x B: A B = x B - x A .

Initial data: points A and B lying on a plane in a rectangular coordinate system O x y with given coordinates: A (x A , y A) and B (x B , y B) .

Let's draw perpendiculars to the coordinate axes O x and O y through points A and B and get the projection points as a result: A x , A y , B x , B y . Based on the location of points A and B, the following options are further possible:

If points A and B coincide, then the distance between them is zero;

If points A and B lie on a straight line perpendicular to the O x axis (abscissa axis), then the points and coincide, and | A B | = | A y B y | . Since the distance between the points is equal to the modulus of the difference between their coordinates, then A y B y = y B - y A , and, therefore, A B = A y B y = y B - y A .

If points A and B lie on a straight line perpendicular to the O y axis (y-axis) - by analogy with the previous paragraph: A B = A x B x = x B - x A

If points A and B do not lie on a straight line perpendicular to one of the coordinate axes, we find the distance between them by deriving the calculation formula:

We see that the triangle A B C is right-angled by construction. In this case, A C = A x B x and B C = A y B y . Using the Pythagorean theorem, we compose the equality: A B 2 = A C 2 + B C 2 ⇔ A B 2 = A x B x 2 + A y B y 2 , and then transform it: A B = A x B x 2 + A y B y 2 = x B - x A 2 + y B - y A 2 = (x B - x A) 2 + (y B - y A) 2

Let's form a conclusion from the result obtained: the distance from point A to point B on the plane is determined by the calculation using the formula using the coordinates of these points

A B = (x B - x A) 2 + (y B - y A) 2

The resulting formula also confirms the previously formed statements for the cases of coincidence of points or situations when the points lie on straight lines perpendicular to the axes. So, for the case of coincidence of points A and B, the equality will be true: A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + 0 2 = 0

For the situation when points A and B lie on a straight line perpendicular to the x-axis:

A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + (y B - y A) 2 = y B - y A

For the case when points A and B lie on a straight line perpendicular to the y-axis:

A B = (x B - x A) 2 + (y B - y A) 2 = (x B - x A) 2 + 0 2 = x B - x A

Initial data: rectangular coordinate system O x y z with arbitrary points lying on it with given coordinates A (x A , y A , z A) and B (x B , y B , z B) . It is necessary to determine the distance between these points.

Consider the general case when points A and B do not lie in a plane parallel to one of the coordinate planes. Draw through points A and B planes perpendicular to the coordinate axes, and get the corresponding projection points: A x , A y , A z , B x , B y , B z

The distance between points A and B is the diagonal of the resulting box. According to the construction of the measurement of this box: A x B x , A y B y and A z B z

From the course of geometry it is known that the square of the diagonal of a parallelepiped is equal to the sum of the squares of its dimensions. Based on this statement, we obtain the equality: A B 2 \u003d A x B x 2 + A y B y 2 + A z B z 2

Using the conclusions obtained earlier, we write the following:

A x B x = x B - x A , A y B y = y B - y A , A z B z = z B - z A

Let's transform the expression:

A B 2 = A x B x 2 + A y B y 2 + A z B z 2 = x B - x A 2 + y B - y A 2 + z B - z A 2 = = (x B - x A) 2 + (y B - y A) 2 + z B - z A 2

Final formula for determining the distance between points in space will look like this:

A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

The resulting formula is also valid for cases where:

The dots match;

They lie on the same coordinate axis or on a straight line parallel to one of the coordinate axes.

Examples of solving problems for finding the distance between points

Example 1

Initial data: a coordinate line and points lying on it with given coordinates A (1 - 2) and B (11 + 2) are given. It is necessary to find the distance from the reference point O to point A and between points A and B.

Solution

1. The distance from the reference point to the point is equal to the module of the coordinate of this point, respectively O A \u003d 1 - 2 \u003d 2 - 1
2. The distance between points A and B is defined as the modulus of the difference between the coordinates of these points: A B = 11 + 2 - (1 - 2) = 10 + 2 2

Answer: O A = 2 - 1, A B = 10 + 2 2

Example 2

Initial data: given a rectangular coordinate system and two points lying on it A (1 , - 1) and B (λ + 1 , 3) ​​. λ is some real number. It is necessary to find all values ​​of this number for which the distance A B will be equal to 5.

Solution

To find the distance between points A and B, you must use the formula A B = (x B - x A) 2 + y B - y A 2

Substituting the real values ​​of the coordinates, we get: A B = (λ + 1 - 1) 2 + (3 - (- 1)) 2 = λ 2 + 16

And also we use the existing condition that A B = 5 and then the equality will be true:

λ 2 + 16 = 5 λ 2 + 16 = 25 λ = ± 3

Answer: A B \u003d 5 if λ \u003d ± 3.

Example 3

Initial data: a three-dimensional space in a rectangular coordinate system O x y z and the points A (1 , 2 , 3) ​​and B - 7 , - 2 , 4 lying in it are given.

Solution

To solve the problem, we use the formula A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

Substituting the real values, we get: A B = (- 7 - 1) 2 + (- 2 - 2) 2 + (4 - 3) 2 = 81 = 9

Answer: | A B | = 9

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Let , (Figure 2.3). Required to find.

Figure 2.3. The distance between two points.

From rectangular by the Pythagorean theorem we have

That is ,

This formula is valid for any arrangement of points and .

II. The division of the segment in this respect:

Let , . It is required to find lying on the segment and dividing it in this ratio (Figure 2.4.).

Figure 2.4. The division of the segment in this respect.

From similarity ~ , that is , , whence . Likewise.

Thus,

- the formula for dividing a segment in relation to .

If , then

are the coordinates of the middle of the segment.

Comment. The derived formulas can also be generalized to the case of a spatial rectangular Cartesian coordinate system. Let points , . Then

- formula for finding the distance between points and .

The formula for dividing a segment in relation to .

In addition to Cartesian on the plane and in space, you can build a large number of other coordinate systems, that is, ways to characterize the position of a point on a plane or in space using two or three numerical parameters (coordinates). Consider some of the existing coordinate systems.

On a plane, one can define polar coordinate system , which is used, in particular, in the study of rotational movements.

Figure 2.5. Polar coordinate system.

We fix a point on the plane and a half-line emerging from it, and also choose a scale unit (Figure 2.5). The point is called pole , half-line - polar axis . Let's assign two numbers to an arbitrary point:

– polar radius , equal to the distance from the point M to the pole O;

– polar angle , equal to the angle between the polar axis and the half-line.

Measured in radians, counting the positive direction of the values ​​​​is from counterclockwise, usually assumed to be .

The pole corresponds to the polar radius, the polar angle for it is not defined.

Let's find the relationship between rectangular and polar coordinates (Figure 2.6).

Figure 2.6. Relationship between rectangular and polar coordinate systems.

We will consider the origin of the rectangular coordinate system as a pole, and we will take the beam as the polar axis. Let - in a rectangular Cartesian coordinate system and - in a polar coordinate system. Find the relationship between rectangular and polar coordinates.

From rectangular, and from rectangular. So the formulas

express the rectangular coordinates of a point in terms of its polar coordinates.

The inverse relationship is expressed by the formulas

Comment. The polar angle can also be determined from the formula, having previously determined by rectangular coordinates in which quarter the point lies.

Example 1 Find the polar coordinates of the point .

Solution. Calculate ; polar angle is found from the conditions:

Therefore, , therefore .

Example 2 Find the rectangular coordinates of the point .

Solution. Calculate

We get .

In three-dimensional space, in addition to the rectangular Cartesian coordinate system, cylindrical and spherical coordinate systems are often used.

Cylindrical coordinate system is a polar coordinate system in the plane , to which the spatial axis is added, perpendicular to this plane (Figure 2.7). The position of any point is characterized by three numbers - its cylindrical coordinates: , where and are the polar coordinates (polar radius and polar angle) of the projection of the point onto the plane in which the polar coordinate system is selected, - the applicate, which is equal to the distance from the point to the specified plane.

Figure 2.7. Cylindrical coordinate system

To establish the relationship between the rectangular Cartesian coordinate system and the cylindrical coordinate system, we will arrange them relative to each other as in Figure 2.8 (we will place the plane in the plane, and the polar axis coincides with the positive direction of the axis, the axis is common in both coordinate systems).

Let be the rectangular coordinates of the point , be the cylindrical coordinates of this point, and be the projection of the point onto the plane . Then

formulas relating rectangular and cylindrical coordinates of a point.

Figure 2.8. Relationship between rectangular cartesian

and cylindrical coordinate systems

Comment. Cylindrical coordinates are often used when considering bodies of revolution, and the axis is located along the axis of rotation.

Spherical coordinate system can be built in the following way. We choose the polar axis in the plane. Through the point we draw a line perpendicular to the plane (normal). Then any point in space can be associated with three real numbers, where is the distance from the point to, is the angle between the axis and the projection of the segment onto the plane, is the angle between the normal and the segment. Notice, that , , .

If we place the plane in the plane , and choose the polar axis coinciding with the positive direction of the axis , choose the axis as the normal (Figure 2.9), then we get formulas connecting these two coordinate systems

Figure 2.9. Relationship between spherical and rectangular cartesian

coordinate systems

scalars, or scalars are fully characterized by their numerical value in the chosen system of units. Vector quantities or vectors, in addition to a numerical value, also have a direction. For example, if we say that the wind is blowing at a speed of 10 m/s, then we will introduce the scalar value of the wind speed, but if we say that the southwest wind is blowing at a speed of 10 m/s, then in this case the wind speed will be already a vector.

Vector a directed segment is called, having a certain length, i.e. a segment of a certain length, in which one of the limiting points is taken as the beginning, and the second - as the end. The vector will be denoted either , or (Figure 2.10).

The length of a vector is denoted by the symbol or and is called the modulus of the vector. A vector whose length is 1 is called single . The vector is called zero , if its beginning and end coincide, and is denoted by θ or . The zero vector has no definite direction and has a length equal to zero. Vectors and located on the same line or on parallel lines are called collinear . Two vectors and are called equal if they are collinear, have the same length and the same direction. All zero vectors are considered equal.

Two non-zero collinear vectors that have equal modulus but opposite direction are called opposite . The vector opposite to is denoted by , for the opposite vector .

To the number line operations over vectors include the operations of addition, subtraction of vectors and multiplication of a vector by a number, i.e. operations that result in a vector.

Let us define these operations on vectors. Let two vectors and be given. Let's take an arbitrary point O and construct a vector , from point A we set aside the vector . Then the vector connecting the beginning of the first term of the vector with the end of the second is called sum of these vectors and is denoted by . The considered rule for finding the sum of vectors is called triangle rules (Figure 2.11).

The same sum of vectors can be obtained in another way (Figure 2.12). Set aside the vector and the vector from the point. Let's build on these vectors as on the sides of a parallelogram. The vector , which is the diagonal of the parallelogram drawn from the vertex , will be the sum . This rule for finding the sum is called parallelogram rules .

The sum of any finite number of vectors can be obtained using the broken line rule (Figure 2.13). From an arbitrary point, we postpone the vector , then we postpone the vector, etc. The vector connecting the beginning of the former to the end of the latter is the sum

data vectors, i.e. . Obviously, if the end of the last term of the vector coincides with the beginning of the first, then the sum of the vectors is equal to the null vector.

difference two vectors and is called such a vector , the sum of which with the subtracted vector gives the vector . From here difference vector construction rule(Figure 2.14). From a point we set aside a vector and a vector . The vector connecting the ends of the reduced vector and the vector to be subtracted and directed from the vector being subtracted to the reduced vector is the difference .

Vector product to a real number λ is called a vector that is collinear to the vector , has a length and the same direction as the vector if , and a direction opposite to the vector if .

Introduced linear operations over vectors have properties :

10 . Commutativity of addition: .

20 . Addition associativity: .

thirty . The existence of a neutral element by addition: .

40 . The existence of the opposite element by addition:

50 . Distributivity of multiplication by a number with respect to vector addition: .

60 . Distributivity of multiplying a vector by the sum of two numbers:

70. Associativity property with respect to multiplication of a vector by a product of numbers: .

Let the system of vectors be given:

The expression , where λ i (i = 1,2,…, n) are some numbers, is called linear combination systems of vectors (2.1). The system of vectors (2.1) is called linearly dependent , if their linear combination is equal to zero, provided that not all numbers λ 1 , λ 2 , …, λ n are equal to zero. The system of vectors (2.1) is called linearly independent , if their linear combination is equal to zero only under the condition that all numbers λ i = 0 (). It is possible to give another definition of the linear dependence of vectors. The system of vectors (2.1) is called linearly dependent , if any vector of this system is linearly expressed in terms of the rest, otherwise the system of vectors (2.1) linearly independent .

For vectors lying in a plane, the following statements are true.

10 . Any three vectors in the plane are linearly dependent.

20 . If the number of these vectors on the plane is more than three, then they are also linearly dependent.

thirty . For two vectors in the plane to be linearly independent, it is necessary and sufficient that they be non-collinear.

Thus, the maximum number of linearly independent vectors in the plane is two.

The vectors are called coplanar if they lie in the same plane or are parallel to the same plane. The following statements are true for space vectors.

10 . Any four space vectors are linearly dependent.

20 . If the number of given vectors in space is greater than four, then they are also linearly dependent.

thirty . For three vectors to be linearly independent, it is necessary and sufficient that they be non-coplanar.

Thus, the maximum number of linearly independent vectors in space is three.

Any maximal subsystem of linearly independent vectors through which any vector of this system is expressed is called basis considered vector systems . It is easy to conclude that the basis on the plane consists of two non-collinear vectors, and the basis in space consists of three non-coplanar vectors. The number of basis vectors is called rank vector systems. The coefficients of the expansion of a vector in terms of basis vectors are called vector coordinates in this basis.

Let the vectors form a basis and let , then the numbers λ 1 , λ 2 , λ 3 are the coordinates of the vector in the basis. In this case, they write down. It can be shown that the expansion of the vector in terms of the basis is unique. The main meaning of the basis is that linear operations on vectors become ordinary linear operations on numbers - the coordinates of these vectors. Using the properties of linear operations on vectors, we can prove the following theorem.

Theorem. When two vectors are added, their corresponding coordinates are added. When a vector is multiplied by a number, all of its coordinates are multiplied by that number.

Thus, if and , then , where , and where , λ is some number.

Usually, the set of all vectors in the plane, reduced to a common origin, with the introduced linear operations, is denoted by V 2 , and the set of all space vectors, reduced to a common origin, is denoted by V 3 . The sets V 2 and V 3 are called spaces of geometric vectors.

Angle between vectors and the smallest angle () is called, by which one of the vectors must be rotated until it coincides with the second after bringing these vectors to a common origin.

Dot product two vectors is called a number equal to the product of the modules of these vectors by the cosine of the angle between them. Dot product of vectors and denote , or

If the angle between the vectors and is equal, then

From a geometric point of view, the scalar product of vectors is equal to the product of the modulus of one vector and the projection of another vector onto it. It follows from equality (2.2) that

From here condition of orthogonality of two vectors: two vectors And are orthogonal if and only if their scalar product is equal to zero, i.e. .

The dot product of vectors is not a linear operation because it results in a number, not a vector.

Properties of the scalar product.

1º. - commutativity.

2º. - distributivity.

3º. – associativity with respect to a numerical factor.

4º. - property of a scalar square.

Property 4º implies the definition vector length :

Let a basis be given in the space V 3 , where the vectors are unit vectors (they are called orts), the direction of each of which coincides with the positive direction of the coordinate axes Ox, Oy, Oz of a rectangular Cartesian coordinate system.

Let's expand the space vector V 3 according to this basis (Figure 2.15):

Vectors are called components of a vector along the coordinate axes, or components, of a number a x , a y , a z are the rectangular Cartesian coordinates of the vector A. The direction of the vector is determined by the angles α, β, γ formed by it with the coordinate lines. The cosine of these angles is called vector guides. Then the direction cosines are determined by the formulas:

It is easy to show that

We express the scalar product in coordinate form.

Let and . Multiplying these vectors as polynomials and considering that we get an expression for finding dot product in coordinate form:

those. the scalar product of two vectors is equal to the sum of the paired products of the coordinates of the same name.

From (2.6) and (2.4) follows the formula for finding vector length :

From (2.6) and (2.7) we obtain a formula for determining angle between vectors:

A triple of vectors is called ordered if it is indicated which of them is considered the first, which is the second, and which is the third.

Ordered trio of vectors called right , if after bringing them to a common beginning from the end of the third vector, the shortest turn from the first to the second vector is counterclockwise. Otherwise, the triple of vectors is called left . For example, in Figure 2.15, the vectors , , form the right triple of vectors, and the vectors , , form the left triple of vectors.

The concept of right and left coordinate systems in three-dimensional space is introduced in a similar way.

vector art vector by vector is called a vector (another notation) that:

1) has length , where is the angle between vectors and ;

2) is perpendicular to the vectors and (), i.e. perpendicular to the plane containing the vectors and ;

By definition, we find the vector product of coordinate orts , , :

If , , then the coordinates of the cross product of a vector and a vector are determined by the formula:

It follows from the definition geometric meaning of the vector product : the modulus of the vector is equal to the area of ​​the parallelogram built on the vectors and .

Vector product properties:

40 . , if the vectors and are collinear, or one of these vectors is zero.

Example 3 The parallelogram is built on the vectors and , where , , . Calculate the length of the diagonals of this parallelogram, the angle between the diagonals, and the area of ​​the parallelogram.

Solution. The construction of vectors and is shown in Figure 2.16, the construction of a parallelogram on these vectors is shown in Figure 2.17.

Let us carry out an analytical solution of this problem. We express the vectors that define the diagonals of the constructed parallelogram through the vectors and , and then through and . We find , . Next, we find the lengths of the diagonals of the parallelogram, as the lengths of the constructed vectors

The angle between the diagonals of the parallelogram is denoted by . Then from the formula for the scalar product of vectors we have:

Hence, .

Using the properties of the cross product, we calculate the area of ​​the parallelogram:

Let there be three vectors , and . Imagine that a vector is multiplied vectorially by and vector and the resulting vector is multiplied scalarly by vector , thereby determining the number . It is called vector-scalar or mixed product three vectors , and . Denoted or .

Let's find out geometric meaning of the mixed product (Figure 2.18). Let , , be not coplanar. Let's construct a parallelepiped on these vectors as on edges. The cross product is a vector whose modulus is equal to the area of ​​the parallelogram (the base of the parallelepiped) built on the vectors and and directed perpendicular to the plane of the parallelogram.

Dot product (equal to the product of the modulus of the vector and the projection on ). The height of the constructed parallelepiped is the absolute value of this projection. Therefore, the absolute value of the mixed product of three vectors is equal to the volume of the parallelepiped built on the vectors , and , i.e. .

Hence the volume of the triangular pyramid built on the vectors , and , is calculated by the formula .

We note some more mixed product properties vectors.

1 o. The sign of the product is positive if the vectors , , form a system with the same name as the main one, and negative otherwise.

Really, the dot product is positive if the angle between and is acute and negative if the angle is obtuse. With an acute angle between and, the vectors and are located on the same side relative to the base of the parallelepiped, and therefore, from the end of the vector, the rotation from to will be seen in the same way as from the end of the vector, i.e. in the positive direction (counterclockwise).

At an obtuse angle, and the vectors and are located on different sides relative to the plane of the parallelogram lying at the base of the parallelepiped, and therefore, from the end of the vector, the rotation from to is visible in the negative direction (clockwise).

2 o The mixed product does not change with a circular permutation of its factors: .

3 o When any two vectors are interchanged, the mixed product changes only the sign. For example, , . , . - unknown systems.

System(3.1) is called homogeneous if all free members are . System (3.1) is called heterogeneous , if at least one of the free members of .

System solution is called a set of numbers, when substituting which into the equations of the system instead of the corresponding unknowns, each equation of the system turns into an identity. A system that has no solution is called incompatible, or controversial . A system that has at least one solution is called joint .

The joint system is called certain if it has a unique solution. If a joint system has more than one solution, then it is called uncertain . A homogeneous system is always consistent, since it has at least the zero solution . The expression for the unknowns, from which any particular solution of the system can be obtained, is called it common solution , and any particular solution of the system is its private decision . Two systems with the same unknowns are equivalent (are tantamount to ) if each solution of one of them is a solution of the other or both systems are inconsistent.

Consider methods for solving systems of linear equations.

One of the main methods for solving systems of linear equations is gauss method, or sequential method exclusion of unknowns. The essence of this method is to reduce the system of linear equations to a stepwise form. In this case, the equations have to carry out the following elementary transformations :

1. Permutation of the equations of the system.

2. Adding another equation to one equation.

3. Multiplying both sides of the equation by a non-zero number.

As a result, the system will take the form:

Continuing this process further, we eliminate the unknown from all equations, starting with the third one. To do this, we multiply the second equation by numbers and add to the 3rd, ..., to the -th equation of the system. The next steps of the Gauss method are carried out in a similar way. If as a result of the transformations an identical equation is obtained, then we delete it from the system. If at some step of the Gauss method an equation of the form is obtained:

then the system under consideration is inconsistent and its further solution stops. If the equation of the form (3.2) does not occur when performing elementary transformations, then in no more than - steps the system (3.1) will be transformed to a step form:

To obtain a particular solution of the system, it will be necessary in (3.4) to assign specific values ​​to the free variables.

Note that since in the Gauss method all transformations are performed on the coefficients of unknown equations and free terms, in practice this method is usually applied to a matrix composed of coefficients of unknowns and a column of free terms. This matrix is ​​called extended. With the help of elementary transformations, this matrix is ​​reduced to a stepped form. After that, the system is restored using the obtained matrix and all previous considerations are applied to it.

Example 1 Solve system:

Solution. We compose the augmented matrix and reduce it to a stepped form:

~ *) ~ **) ~ ***)

*) - the second line is multiplied by and the third line is crossed out.

Solving problems in mathematics for students is often accompanied by many difficulties. To help the student cope with these difficulties, as well as to teach him how to apply his theoretical knowledge in solving specific problems in all sections of the course of the subject "Mathematics" is the main purpose of our site.

Starting to solve problems on the topic, students should be able to build a point on a plane according to its coordinates, as well as find the coordinates of a given point.

The calculation of the distance between two points taken on the plane A (x A; y A) and B (x B; y B) is performed by the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2), where d is the length of the segment that connects these points on the plane.

If one of the ends of the segment coincides with the origin, and the other has coordinates M (x M; y M), then the formula for calculating d will take the form OM = √ (x M 2 + y M 2).

1. Calculating the distance between two points given the coordinates of these points

Example 1.

Find the length of the segment that connects the points A(2; -5) and B(-4; 3) on the coordinate plane (Fig. 1).

Solution.

The condition of the problem is given: x A = 2; x B \u003d -4; y A = -5 and y B = 3. Find d.

Applying the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2), we get:

d \u003d AB \u003d √ ((2 - (-4)) 2 + (-5 - 3) 2) \u003d 10.

2. Calculating the coordinates of a point that is equidistant from three given points

Example 2

Find the coordinates of the point O 1, which is equidistant from the three points A(7; -1) and B(-2; 2) and C(-1; -5).

Solution.

From the formulation of the condition of the problem it follows that O 1 A \u003d O 1 B \u003d O 1 C. Let the desired point O 1 have coordinates (a; b). According to the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2) we find:

O 1 A \u003d √ ((a - 7) 2 + (b + 1) 2);

O 1 V \u003d √ ((a + 2) 2 + (b - 2) 2);

O 1 C \u003d √ ((a + 1) 2 + (b + 5) 2).

We compose a system of two equations:

(√((a - 7) 2 + (b + 1) 2) = √((a + 2) 2 + (b - 2) 2),
(√((a - 7) 2 + (b + 1) 2) = √((a + 1) 2 + (b + 5) 2).

After squaring the left and right sides of the equations, we write:

((a - 7) 2 + (b + 1) 2 \u003d (a + 2) 2 + (b - 2) 2,
((a - 7) 2 + (b + 1) 2 = (a + 1) 2 + (b + 5) 2 .

Simplifying, we write

(-3a + b + 7 = 0,
(-2a - b + 3 = 0.

Having solved the system, we get: a = 2; b = -1.

Point O 1 (2; -1) is equidistant from the three points given in the condition that do not lie on one straight line. This point is the center of a circle passing through three given points. (Fig. 2).

3. Calculation of the abscissa (ordinate) of a point that lies on the abscissa (ordinate) axis and is at a given distance from this point

Example 3

The distance from point B(-5; 6) to point A lying on the x-axis is 10. Find point A.

Solution.

It follows from the formulation of the condition of the problem that the ordinate of point A is zero and AB = 10.

Denoting the abscissa of the point A through a, we write A(a; 0).

AB \u003d √ ((a + 5) 2 + (0 - 6) 2) \u003d √ ((a + 5) 2 + 36).

We get the equation √((a + 5) 2 + 36) = 10. Simplifying it, we have

a 2 + 10a - 39 = 0.

The roots of this equation a 1 = -13; and 2 = 3.

We get two points A 1 (-13; 0) and A 2 (3; 0).

Examination:

A 1 B \u003d √ ((-13 + 5) 2 + (0 - 6) 2) \u003d 10.

A 2 B \u003d √ ((3 + 5) 2 + (0 - 6) 2) \u003d 10.

Both obtained points fit the condition of the problem (Fig. 3).

4. Calculation of the abscissa (ordinate) of a point that lies on the abscissa (ordinate) axis and is at the same distance from two given points

Example 4

Find a point on the Oy axis that is at the same distance from points A (6; 12) and B (-8; 10).

Solution.

Let the coordinates of the point required by the condition of the problem, lying on the Oy axis, be O 1 (0; b) (at the point lying on the Oy axis, the abscissa is equal to zero). It follows from the condition that O 1 A \u003d O 1 V.

According to the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2) we find:

O 1 A \u003d √ ((0 - 6) 2 + (b - 12) 2) \u003d √ (36 + (b - 12) 2);

O 1 V \u003d √ ((a + 8) 2 + (b - 10) 2) \u003d √ (64 + (b - 10) 2).

We have the equation √(36 + (b - 12) 2) = √(64 + (b - 10) 2) or 36 + (b - 12) 2 = 64 + (b - 10) 2 .

After simplification, we get: b - 4 = 0, b = 4.

Required by the condition of the problem point O 1 (0; 4) (Fig. 4).

5. Calculating the coordinates of a point that is at the same distance from the coordinate axes and some given point

Example 5

Find point M located on the coordinate plane at the same distance from the coordinate axes and from point A (-2; 1).

Solution.

The required point M, like point A (-2; 1), is located in the second coordinate corner, since it is equidistant from points A, P 1 and P 2 (Fig. 5). The distances of the point M from the coordinate axes are the same, therefore, its coordinates will be (-a; a), where a > 0.

It follows from the conditions of the problem that MA = MP 1 = MP 2, MP 1 = a; MP 2 = |-a|,

those. |-a| = a.

According to the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2) we find:

MA \u003d √ ((-a + 2) 2 + (a - 1) 2).

Let's make an equation:

√ ((-a + 2) 2 + (a - 1) 2) = a.

After squaring and simplifying, we have: a 2 - 6a + 5 = 0. We solve the equation, we find a 1 = 1; and 2 = 5.

We get two points M 1 (-1; 1) and M 2 (-5; 5), satisfying the condition of the problem.

6. Calculation of the coordinates of a point that is at the same specified distance from the abscissa (ordinate) axis and from this point

Example 6

Find a point M such that its distance from the y-axis and from the point A (8; 6) will be equal to 5.

Solution.

It follows from the condition of the problem that MA = 5 and the abscissa of the point M is equal to 5. Let the ordinate of the point M be equal to b, then M(5; b) (Fig. 6).

According to the formula d \u003d √ ((x A - x B) 2 + (y A - y B) 2) we have:

MA \u003d √ ((5 - 8) 2 + (b - 6) 2).

Let's make an equation:

√((5 - 8) 2 + (b - 6) 2) = 5. Simplifying it, we get: b 2 - 12b + 20 = 0. The roots of this equation are b 1 = 2; b 2 \u003d 10. Therefore, there are two points that satisfy the condition of the problem: M 1 (5; 2) and M 2 (5; 10).

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Let a rectangular coordinate system be given.

Theorem 1.1. For any two points M 1 (x 1; y 1) and M 2 (x 2; y 2) of the plane, the distance d between them is expressed by the formula

Proof. Let us drop from the points M 1 and M 2 the perpendiculars M 1 B and M 2 A, respectively

on the Oy and Ox axes and denote by K the point of intersection of the lines M 1 B and M 2 A (Fig. 1.4). The following cases are possible:

1) Points M 1, M 2 and K are different. Obviously, the point K has coordinates (x 2; y 1). It is easy to see that M 1 K = ôx 2 – x 1 ô, M 2 K = ôy 2 – y 1 ô. Because ∆M 1 KM 2 is rectangular, then by the Pythagorean theorem d = M 1 M 2 = = .

2) Point K coincides with point M 2, but is different from point M 1 (Fig. 1.5). In this case y 2 = y 1

and d \u003d M 1 M 2 \u003d M 1 K \u003d ôx 2 - x 1 ô \u003d =

3) The point K coincides with the point M 1, but is different from the point M 2. In this case x 2 = x 1 and d =

M 1 M 2 \u003d KM 2 \u003d ôy 2 - y 1 ô \u003d = .

4) Point M 2 coincides with point M 1. Then x 1 \u003d x 2, y 1 \u003d y 2 and

d \u003d M 1 M 2 \u003d O \u003d.

The division of the segment in this respect.

Let an arbitrary segment M 1 M 2 be given on the plane and let M be any point of this

segment other than the point M 2 (Fig. 1.6). The number l defined by the equality l = , is called attitude, in which the point M divides the segment M 1 M 2.

Theorem 1.2. If the point M (x; y) divides the segment M 1 M 2 in relation to l, then the coordinates of this are determined by the formulas

x = , y = , (4)

where (x 1; y 1) are the coordinates of the point M 1, (x 2; y 2) are the coordinates of the point M 2.

Proof. Let us prove the first of formulas (4). The second formula is proved similarly. Two cases are possible.

x = x 1 = = = .

2) The straight line M 1 M 2 is not perpendicular to the Ox axis (Fig. 1.6). Let's drop the perpendiculars from the points M 1 , M, M 2 to the axis Ox and denote the points of their intersection with the axis Ox respectively P 1 , P, P 2 . According to the proportional segments theorem =l.

Because P 1 P \u003d ôx - x 1 ô, PP 2 \u003d ôx 2 - xô and the numbers (x - x 1) and (x 2 - x) have the same sign (for x 1< х 2 они положительны, а при х 1 >x 2 are negative), then

l == ,

x - x 1 \u003d l (x 2 - x), x + lx \u003d x 1 + lx 2,

x = .

Corollary 1.2.1. If M 1 (x 1; y 1) and M 2 (x 2; y 2) are two arbitrary points and the point M (x; y) is the midpoint of the segment M 1 M 2, then

x = , y = (5)

Proof. Since M 1 M = M 2 M, then l = 1 and by formulas (4) we obtain formulas (5).

Area of ​​a triangle.

Theorem 1.3. For any points A (x 1; y 1), B (x 2; y 2) and C (x 3; y 3) that do not lie on the same

straight line, the area S of triangle ABC is expressed by the formula

S \u003d ô (x 2 - x 1) (y 3 - y 1) - (x 3 - x 1) (y 2 - y 1) ô (6)

Proof. The area ∆ ABC shown in fig. 1.7, we calculate as follows

S ABC \u003d S ADEC + S BCEF - S ABFD.

Calculate the area of ​​the trapezoid:

S-ADEC=
,

SBCEF=

S ABFD =

Now we have

S ABC \u003d ((x 3 - x 1) (y 3 + y 1) + (x 3 - x 2) (y 3 + y 2) - (x 2 - -x 1) (y 1 + y 2)) \u003d (x 3 y 3 - x 1 y 3 + x 3 y 1 - x 1 y 1 + + x 2 y 3 - -x 3 y 3 + x 2 y 2 - x 3 y 2 - x 2 y 1 + x 1 y 1 - x 2 y 2 + x 1 y 2) \u003d (x 3 y 1 - x 3 y 2 + x 1 y 2 - x 2 y 1 + x 2 y 3 -

X 1 y 3) \u003d (x 3 (y 1 - y 2) + x 1 y 2 - x 1 y 1 + x 1 y 1 - x 2 y 1 + y 3 (x 2 - x 1)) \u003d (x 1 (y 2 - y 1) - x 3 (y 2 - y 1) + + y 1 (x 1 - x 2) - y 3 (x 1 - x 2)) \u003d ((x 1 - x 3) ( y 2 - y 1) + (x 1 - x 2) (y 1 - y 3)) \u003d ((x 2 - x 1) (y 3 - y 1) -

- (x 3 - x 1) (y 2 - y 1)).

For another location ∆ ABC, formula (6) is proved similarly, but it can be obtained with the “-” sign. Therefore, in the formula (6) put the sign of the modulus.

Lecture 2

The equation of a straight line on a plane: the equation of a straight line with the main coefficient, the general equation of a straight line, the equation of a straight line in segments, the equation of a straight line passing through two points. Angle between lines, conditions of parallelism and perpendicularity of lines on a plane.

2.1. Let a rectangular coordinate system and some line L be given on the plane.

Definition 2.1. An equation of the form F(x;y) = 0 relating the variables x and y is called line equation L(in a given coordinate system) if this equation is satisfied by the coordinates of any point lying on the line L, and not by the coordinates of any point not lying on this line.

Examples of equations of lines on a plane.

1) Consider a straight line parallel to the axis Oy of a rectangular coordinate system (Fig. 2.1). Let us denote by the letter A the point of intersection of this line with the axis Ox, (a; o) ─ its or-

dinats. The equation x = a is the equation of the given line. Indeed, this equation is satisfied by the coordinates of any point M(a; y) of this line and not by the coordinates of any point that does not lie on the line. If a = 0, then the line coincides with the Oy axis, which has the equation x = 0.

2) The equation x - y \u003d 0 defines the set of points in the plane that make up the bisectors of I and III coordinate angles.

3) The equation x 2 - y 2 \u003d 0 is the equation of two bisectors of coordinate angles.

4) The equation x 2 + y 2 = 0 defines a single point O(0;0) on the plane.

5) The equation x 2 + y 2 \u003d 25 is the equation of a circle of radius 5 centered at the origin.

Coordinates determine the location of an object on the globe. Coordinates are indicated by latitude and longitude. Latitudes are measured from the equator line on both sides. In the Northern Hemisphere the latitudes are positive, in the Southern Hemisphere they are negative. Longitude is measured from the initial meridian either to the east or to the west, respectively, either eastern or western longitude is obtained.

According to the generally accepted position, the meridian is taken as the initial one, which passes through the old Greenwich Observatory in Greenwich. The geographic coordinates of the location can be obtained using a GPS navigator. This device receives signals from a satellite positioning system in the WGS-84 coordinate system, the same for the whole world.

Navigator models differ in manufacturers, functionality and interface. Currently, built-in GPS-navigators are available in some models of cell phones. But any model can record and save point coordinates.

Distance between GPS coordinates

To solve practical and theoretical problems in some industries, it is necessary to be able to determine the distances between points by their coordinates. To do this, you can use several methods. Canonical representation of geographic coordinates: degrees, minutes, seconds.

For example, you can determine the distance between the following coordinates: point No. 1 - latitude 55°45′07″ N, longitude 37°36′56″ E; point No. 2 - latitude 58°00′02″ N, longitude 102°39′42″ E

The easiest way is to use a -calculator to calculate the distance between two points. In the browser search engine, you must set the following search parameters: online - to calculate the distance between two coordinates. In the online calculator, latitude and longitude values ​​are entered into the query fields for the first and second coordinates. When calculating, the online calculator gave the result - 3,800,619 m.

The next method is more time consuming, but also more visual. It is necessary to use any available mapping or navigation program. The programs in which you can create points by coordinates and measure the distances between them include the following applications: BaseCamp (a modern analogue of the MapSource program), Google Earth, SAS.Planet.

All of the above programs are available to any network user. For example, to calculate the distance between two coordinates in Google Earth, you need to create two labels indicating the coordinates of the first point and the second point. Then, using the Ruler tool, you need to connect the first and second marks with a line, the program will automatically give the measurement result and show the path on the satellite image of the Earth.

In the case of the example above, the Google Earth program returned the result - the length of the distance between point #1 and point #2 is 3,817,353 m.

Why there is an error in determining the distance

All distance calculations between coordinates are based on arc length calculations. The radius of the Earth is involved in the calculation of the arc length. But since the shape of the Earth is close to an oblate ellipsoid, the radius of the Earth at certain points is different. To calculate the distance between the coordinates, the average value of the Earth's radius is taken, which gives an error in the measurement. The greater the measured distance, the greater the error.