The sum of two vectors in the coordinate plane. Coordinates and vectors

16.12.2023

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Sum of two vectors in the coordinate plane

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Vectors on the plane and in space, methods for solving problems, examples, formulas

1 Vectors in space

Vectors in space include 10th grade geometry, 11th grade geometry and analytical geometry. Vectors allow you to effectively solve geometric problems of the second part of the Unified State Exam and analytical geometry in space. Vectors in space are given in the same way as vectors in the plane, but the third coordinate z is taken into account. Exclusion from vectors in third-dimensional space gives vectors on the plane, which are explained by geometry 8th, 9th grade.

1.1 Vector on the plane and in space

A vector is a directed segment with a beginning and an end, depicted in the figure by an arrow. An arbitrary point in space can be considered a zero vector. The zero vector does not have a specific direction, since the beginning and end are the same, so it can be given any direction.

Vector translated from English means vector, direction, course, guidance, direction setting, aircraft course.

The length (modulus) of a non-zero vector is the length of the segment AB, which is denoted
. Vector length denoted by . The null vector has a length equal to zero = 0.

Non-zero vectors lying on the same line or on parallel lines are called collinear.

The null vector is collinear to any vector.

Collinear nonzero vectors that have the same direction are called codirectional. Codirectional vectors are indicated by . For example, if the vector is codirectional with the vector , then the notation is used.

The zero vector is codirectional with any vector.

Oppositely directed are two collinear non-zero vectors that have opposite directions. Oppositely directed vectors are indicated by the sign ↓. For example, if the vector is oppositely directed to the vector, then the notation ↓ is used.

Co-directed vectors of equal length are called equal.

Many physical quantities are vector quantities: force, speed, electric field.

If the point of application (start) of the vector is not specified, then it is chosen arbitrarily.

If the beginning of the vector is placed at point O, then the vector is considered to be delayed from point O. From any point you can plot a single vector equal to a given vector.

1.2 Vector sum

When adding vectors according to the triangle rule, vector 1 is drawn, from the end of which vector 2 is drawn, and the sum of these two vectors is vector 3, drawn from the beginning of vector 1 to the end of vector 2:

For arbitrary points A, B and C, you can write the sum of vectors:

+
=

If two vectors originate from the same point

then it is better to add them according to the parallelogram rule.

When adding two vectors according to the parallelogram rule, the added vectors are laid out from one point, and from the ends of these vectors a parallelogram is completed by applying the beginning of another to the end of one vector. The vector formed by the diagonal of the parallelogram, originating from the point of origin of the vectors being added, will be the sum of the vectors

The parallelogram rule contains a different order of adding vectors according to the triangle rule.

Laws of vector addition:

1. Displacement law + = +.

2. Combination law ( + ) + = + ( + ).

If it is necessary to add several vectors, then the vectors are added in pairs or according to the polygon rule: vector 2 is drawn from the end of vector 1, vector 3 is drawn from the end of vector 2, vector 4 is drawn from the end of vector 3, vector 5 is drawn from the end of vector 4, etc. A vector that is the sum of several vectors is drawn from the beginning of vector 1 to the end of the last vector.

According to the laws of vector addition, the order of vector addition does not affect the resulting vector, which is the sum of several vectors.

Two non-zero oppositely directed vectors of equal length are called opposite. Vector - is the opposite of vector

These vectors are oppositely directed and equal in magnitude.

1.3 Vector difference

The vector difference can be written as a sum of vectors

- = + (-),

where "-" is the vector opposite to the vector .

Vectors and - can be added according to the triangle or parallelogram rule.

Let the vectors and

To find the difference between vectors, we construct a vector -

We add the vectors and - according to the triangle rule, applying the beginning of the vector - to the end of the vector, we get the vector + (-) = -

We add the vectors and - according to the parallelogram rule, setting aside the beginnings of the vectors and - from one point

If the vectors and originate from the same point

then the difference of vectors gives a vector connecting their ends and the arrow at the end of the resulting vector is placed in the direction of the vector from which the second vector is subtracted

The figure below demonstrates addition and vector difference

The figure below demonstrates vector addition and difference in different ways

Task. The vectors and are given.

Draw the sum and difference of vectors in all possible ways in all possible combinations of vectors.

1.4 Lemma on collinear vectors

= k

1.5 Product of a vector and a number

The product of a non-zero vector by the number k gives the vector = k, collinear to the vector. Vector length:

| | = |k |·| |

If k > 0, then the vectors and are codirectional.

If k = 0, then the vector is zero.

If k< 0, то векторы и противоположно направленные.

If | k | = 1, then vectors and are of equal length.

If k = 1, then the vectors are equal.

If k = -1, then the opposite vectors.

If | k | > 1, then the length of the vector is greater than the length of the vector .

If k > 1, then the vectors are both codirectional and the length is greater than the length of the vector.

If k< -1, то векторы и противоположно направленные и длина больше длины вектора .

If | k |< 1, то длина вектора меньше длины вектора .

If 0< k< 1, то векторы и сонаправленные и длина меньше длины вектора .

If -1< k< 0, то векторы и противоположно направленные и длина меньше длины вектора .

The product of a zero vector and a number gives a zero vector.

Task. Given a vector.

Construct vectors 2, -3, 0.5, -1.5.

Task. The vectors and are given.

Construct vectors 3 + 2, 2 - 2, -2 -.

Laws describing multiplication of a vector by a number

1. Combination law (kn) = k (n)

2. The first distribution law k ( + ) = k + k .

3. Second distribution law (k + n) = k + n.

For collinear vectors and , if ≠ 0, there is a single number k that allows you to express the vector in terms of:

= k

1.6 Coplanar vectors

Vectors that lie in the same plane or in parallel planes are called coplanar. If we draw vectors equal to these coplanar vectors from one point, then they will lie in the same plane. Therefore, we can say that vectors are called coplanar if there are equal vectors lying in the same plane.

Two arbitrary vectors are always coplanar. The three vectors may be coplanar or non-coplanar. Three vectors, at least two of which are collinear, are coplanar. Collinear vectors are always coplanar.

1.7 Decomposition of a vector into two non-collinear vectors

Any vector uniquely decomposes on the plane in two non-collinear non-zero vectors And with single expansion coefficients x and y:

= x+y

Any vector coplanar to the non-zero vectors and can be uniquely expanded into two non-collinear vectors and with unique expansion coefficients x and y:

= x+y

Let us expand the given vector on the plane according to the given non-collinear vectors and :

Let us draw the given coplanar vectors from one point

From the end of the vector we draw lines parallel to the vectors and until they intersect with the lines drawn through the vectors and . We get a parallelogram

The lengths of the sides of a parallelogram are obtained by multiplying the lengths of the vectors and by the numbers x and y, which are determined by dividing the lengths of the sides of the parallelogram by the lengths of their corresponding vectors and. We obtain the decomposition of the vector according to the given non-collinear vectors and:

= x+y

In the problem being solved, x ≈ 1.3, y ≈ 1.9, therefore the expansion of the vector in given non-collinear vectors can be written in the form

1,3 + 1,9 .

In the problem being solved, x ≈ 1.3, y ≈ -1.9, therefore the expansion of the vector in given non-collinear vectors can be written in the form

1,3 - 1,9 .

1.8 Parallelepiped rule

A parallelepiped is a three-dimensional figure whose opposite faces consist of two equal parallelograms lying in parallel planes.

The parallelepiped rule allows you to add three non-coplanar vectors, which are plotted from one point, and a parallelepiped is constructed so that the summed vectors form its edges, and the remaining edges of the parallelepiped are respectively parallel and equal to the lengths of the edges formed by the summed vectors. The diagonal of the parallelepiped forms a vector, which is the sum of the given three vectors, which begins from the point of origin of the vectors being added.

1.9 Decomposition of a vector into three non-coplanar vectors

Any vector expands into three given non-coplanar vectors , and with single expansion coefficients x, y, z:

= x + y + z .

1.10 Rectangular coordinate system in space

In three-dimensional space, the rectangular coordinate system Oxyz is defined by the origin O and the intersecting mutually perpendicular coordinate axes Ox, Oy and Oz with selected positive directions indicated by arrows and the unit of measurement of segments. If the scale of the segments is the same on all three axes, then such a system is called a Cartesian coordinate system.

Coordinate x is called the abscissa, y is the ordinate, z is the applicate. The coordinates of point M are written in brackets M (x; y; z).

1.11 Vector coordinates in space

In space we will define a rectangular coordinate system Oxyz. From the origin of coordinates in the positive directions of the axes Ox, Oy, Oz, we draw the corresponding unit vectors , , , which are called coordinate vectors and are non-coplanar. Therefore, any vector is decomposed into three given non-coplanar coordinate vectors, and with unique expansion coefficients x, y, z:

= x + y + z .

The expansion coefficients x, y, z are the coordinates of the vector in a given rectangular coordinate system, which are written in parentheses (x; y; z). The zero vector has coordinates equal to zero (0; 0; 0). Equal vectors have equal corresponding coordinates.

Rules for finding the coordinates of the resulting vector:

1. When summing two or more vectors, each coordinate of the resulting vector is equal to the sum of the corresponding coordinates of the given vectors. If two vectors (x 1 ; y 1 ; z 1) and (x 1 ; y 1 ; z 1) are given, then the sum of the vectors + gives a vector with coordinates (x 1 + x 1 ; y 1 + y 1 ; z 1 + z 1)

+ = (x 1 + x 1 ; y 1 + y 1 ; z 1 + z 1)

2. Difference is a type of sum, so the difference of the corresponding coordinates gives each coordinate of the vector obtained by subtracting two given vectors. If two vectors are given (x a; y a; z a) and (x b; y b; z b), then the difference of the vectors gives a vector with coordinates (x a - x b; y a - y b; z a - z b)

- = (x a - x b; y a - y b; z a - z b)

3. When multiplying a vector by a number, each coordinate of the resulting vector is equal to the product of this number and the corresponding coordinate of the given vector. If a number k and a vector (x; y; z) are given, then multiplying the vector by the number k gives the vector k with coordinates

k = (kx; ky; kz).

Task. Find the coordinates of the vector = 2 - 3 + 4, if the coordinates of the vectors are (1; -2; -1), (-2; 3; -4), (-1; -3; 2).

Solution

2 + (-3) + 4

2 = (2·1; 2·(-2); 2·(-1)) = (2; -4; -2);

3 = (-3·(-2); -3·3; -3·(-4)) = (6; -9; 12);

4 = (4·(-1); 4·(-3); 4·2) = (-4; -12; 8).

= (2 + 6 - 4; -4 - 9 -12; -2 + 12 + 8) = (4; -25; 18).

1.12 Coordinates of a vector, radius vector and point

The coordinates of a vector are the coordinates of the end of the vector if the beginning of the vector is placed at the origin.

A radius vector is a vector drawn from the origin to a given point; the coordinates of the radius vector and the point are equal.

If the vector
is given by points M 1 (x 1 ; y 1 ; z 1) and M 2 (x 2 ; y 2 ; z 2), then each of its coordinates is equal to the difference of the corresponding coordinates of the end and beginning of the vector

For collinear vectors = (x 1 ; y 1 ; z 1) and = (x 2 ; y 2 ; z 2), if ≠ 0, there is a single number k that allows you to express the vector through:

= k

Then the coordinates of the vector are expressed through the coordinates of the vector

= (kx 1 ; ky 1 ; kz 1)

The ratio of the corresponding coordinates of collinear vectors is equal to the singular number k

1.13 Vector length and distance between two points

The length of the vector (x; y; z) is equal to the square root of the sum of the squares of its coordinates

The length of the vector specified by the starting points M 1 (x 1 ; y 1 ; z 1) and the end M 2 (x 2 ; y 2 ; z 2) is equal to the square root of the sum of squares of the difference between the corresponding coordinates of the end of the vector and the beginning

Distance d between two points M 1 (x 1 ; y 1 ; z 1) and M 2 (x 2 ; y 2 ; z 2) is equal to the length of the vector

There is no z coordinate on the plane

Distance between points M 1 (x 1 ; y 1) and M 2 (x 2 ; y 2)

1.14 Coordinates of the middle of the segment

If the point C is the middle of the segment AB, then the radius vector of point C in an arbitrary coordinate system with the origin at point O is equal to half the sum of the radius vectors of points A and B

If the coordinates of the vectors
(x; y; z),
(x 1 ; y 1 ; z 1),
(x 2 ; y 2 ; z 2), then each vector coordinate is equal to half the sum of the corresponding vector coordinates and

,
,

= (x, y, z) =

Each of the coordinates of the middle of the segment is equal to half the sum of the corresponding coordinates of the ends of the segment.

1.15 Angle between vectors

The angle between vectors is equal to the angle between rays drawn from one point and codirected with these vectors. The angle between vectors can be from 0 0 to 180 0 inclusive. The angle between codirectional vectors is 0 0 . If one vector or both are zero, then the angle between the vectors, at least one of which is zero, is equal to 0 0 . The angle between perpendicular vectors is 90 0. The angle between oppositely directed vectors is 180 0.

1.16 Vector projection

1.17 Dot product of vectors

The scalar product of two vectors is a number (scalar) equal to the product of the lengths of the vectors and the cosine of the angle between the vectors

If = 0 0 , then the vectors are codirectional
And
= cos 0 0 = 1, therefore, the scalar product of codirectional vectors is equal to the product of their lengths (modules)

If the angle between the vectors is 0< < 90 0 , то косинус угла между такими векторами больше нуля
, therefore the scalar product is greater than zero
.

If non-zero vectors are perpendicular, then their scalar product is equal to zero
, since cos 90 0 = 0. The scalar product of perpendicular vectors is equal to zero.

If
, then the cosine of the angle between such vectors is less than zero
, therefore the scalar product is less than zero
.

As the angle between vectors increases, the cosine of the angle between them
decreases and reaches a minimum value at = 180 0 when the vectors are oppositely directed
. Since cos 180 0 = -1, then
. The scalar product of oppositely directed vectors is equal to the negative product of their lengths (modules).

The scalar square of a vector is equal to the modulus of the vector squared

The dot product of vectors at least one of which is zero is equal to zero.

1.18 Physical meaning of the scalar product of vectors

From a physics course it is known that the work done by A force when moving the body equal to the product of the lengths of the force and displacement vectors and the cosine of the angle between them, that is, equal to the scalar product of the force and displacement vectors

If the force vector is codirectional with the movement of the body, then the angle between the vectors
= 0 0, therefore the work done by the force on displacement is maximum and equal to A =
.

If 0< < 90 0 , то работа силы на перемещении положительна A > 0.

If = 90 0, then the work done by the force on displacement is zero A = 0.

If 90 0< < 180 0 , то работа силы на перемещении отрицательна A < 0.

If the force vector is directed opposite to the movement of the body, then the angle between the vectors = 180 0, therefore the work of the force on the movement is negative and equal to A = -.

Task. Determine the work done by gravity when lifting a passenger car weighing 1 ton along a 1 km long road with an inclination angle of 30 0 to the horizon. How many liters of water at a temperature of 20 0 can be boiled using this energy?

Solution

Job A gravity when moving a body, it is equal to the product of the lengths of the vectors and the cosine of the angle between them, that is, equal to the scalar product of the vectors of gravity and displacement

Gravity

G = mg = 1000 kg 10 m/s 2 = 10,000 N.

= 1000 m.

Angle between vectors = 120 0 . Then

cos 120 0 = cos (90 0 + 30 0) = - sin 30 0 = - 0.5.

Let's substitute

A = 10,000 N · 1000 m · (-0.5) = - 5,000,000 J = - 5 MJ.

1.19 Dot product of vectors in coordinates

Dot product of two vectors = (x 1 ; y 1 ; z 1) and = (x 2 ; y 2 ; z 2) in a rectangular coordinate system is equal to the sum of the products of coordinates of the same name

= x 1 x 2 + y 1 y 2 + z 1 z 2 .

1.20 Condition of perpendicularity of vectors

If non-zero vectors = (x 1 ; y 1 ; z 1) and = (x 2 ; y 2 ; z 2) are perpendicular, then their scalar product is zero

If one non-zero vector = (x 1 ; y 1 ; z 1) is given, then the coordinates of the vector perpendicular (normal) to it = (x 2 ; y 2 ; z 2) must satisfy the equality

x 1 x 2 + y 1 y 2 + z 1 z 2 = 0.

There are an infinite number of such vectors.

If one non-zero vector = (x 1 ; y 1) is given on the plane, then the coordinates of the vector perpendicular (normal) to it = (x 2 ; y 2) must satisfy the equality

x 1 x 2 + y 1 y 2 = 0.

If a non-zero vector = (x 1 ; y 1) is given on the plane, then it is enough to arbitrarily set one of the coordinates of the vector perpendicular (normal) to it = (x 2 ; y 2) and from the condition of perpendicularity of the vectors

x 1 x 2 + y 1 y 2 = 0

express the second coordinate of the vector.

For example, if you substitute an arbitrary coordinate x 2, then

y 1 y 2 = - x 1 x 2 .

Second vector coordinate

If we give x 2 = y 1, then the second coordinate of the vector

If a non-zero vector = (x 1 ; y 1) is given on the plane, then the vector perpendicular (normal) to it = (y 1 ; -x 1).

If one of the coordinates of a non-zero vector is equal to zero, then the vector has the same coordinate not equal to zero, and the second coordinate is equal to zero. Such vectors lie on the coordinate axes and are therefore perpendicular.

Let's define a second vector perpendicular to the vector = (x 1 ; y 1), but opposite to the vector , that is, the vector - . Then it is enough to change the signs of the vector coordinates

- = (-y 1 ; x 1)

1 = (y 1 ; -x 1),

2 = (-y 1 ; x 1).

Task.

Solution

Coordinates of two vectors perpendicular to vector = (x 1 ; y 1) on the plane

1 = (y 1 ; -x 1),

2 = (-y 1 ; x 1).

Substitute vector coordinates = (3; -5)

1 = (-5; -3),

2 = (-(-5); 3) = (5; 3).

x 1 x 2 + y 1 y 2 = 0

3·(-5) + (-5)·(-3) = -15 + 15 = 0

right!

3·5 + (-5)·3 = 15 - 15 = 0

right!

Answer: 1 = (-5; -3), 2 = (5; 3).

If we assign x 2 = 1, substitute

x 1 + y 1 y 2 = 0.

y 1 y 2 = -x 1

We obtain the coordinate y 2 of the vector perpendicular to the vector = (x 1 ; y 1)

To obtain a second vector perpendicular to the vector = (x 1 ; y 1), but opposite to the vector . Let

Then it is enough to change the signs of the vector coordinates.

Coordinates of two vectors perpendicular to vector = (x 1 ; y 1) on the plane

Task. Given vector = (3; -5). Find two normal vectors with different orientations.

Solution

Coordinates of two vectors perpendicular to vector = (x 1 ; y 1) on the plane

Coordinates of one vector

Coordinates of the second vector

To check the perpendicularity of the vectors, we substitute their coordinates into the condition of the perpendicularity of the vectors

x 1 x 2 + y 1 y 2 = 0

3 1 + (-5) 0.6 = 3 - 3 = 0

right!

3·(-1) + (-5)·(-0.6) = -3 + 3 = 0

right!

Answer: and.

If you assign x 2 = - x 1 , substitute

x 1 (-x 1) + y 1 y 2 = 0.

-x 1 2 + y 1 y 2 = 0.

y 1 y 2 = x 1 2

We get the coordinate of the vector perpendicular to the vector

If you assign x 2 = x 1 , substitute

x 1 x 1 + y 1 y 2 = 0.

x 1 2 + y 1 y 2 = 0.

y 1 y 2 = -x 1 2

We obtain the y coordinate of the second vector perpendicular to the vector

Coordinates of one vector perpendicular to the vector on the plane = (x 1 ; y 1)

Coordinates of the second vector perpendicular to the vector on the plane = (x 1 ; y 1)

Coordinates of two vectors perpendicular to vector = (x 1 ; y 1) on the plane

1.21 Cosine of the angle between vectors

The cosine of the angle between two non-zero vectors = (x 1 ; y 1 ; z 1) and = (x 2 ; y 2 ; z 2) is equal to the scalar product of the vectors divided by the product of the lengths of these vectors

If
= 1, then the angle between the vectors is 0 0, the vectors are co-directional.

If 0< < 1, то 0 0 < < 90 0 .

If = 0, then the angle between the vectors is 90 0, the vectors are perpendicular.

If -1< < 0, то 90 0 < < 180 0 .

If = -1, then the angle between the vectors is 180 0, the vectors are oppositely directed.

If a vector is given by the coordinates of the beginning and end, then subtracting the coordinates of the beginning from the corresponding coordinates of the end of the vector, we obtain the coordinates of this vector.

Task. Find the angle between the vectors (0; -2; 0), (-2; 0; -4).

Solution

Dot product of vectors

= 0·(-2) + (-2)·0 + 0·(-4) = 0,

therefore the angle between the vectors is equal to = 90 0 .

1.22 Properties of the scalar product of vectors

The properties of the scalar product are valid for any , , , k :

1.
, If
, That
, If =, That
= 0.

2. Travel law

3. Distributive law

4. Combination law
.

1.23 Direct vector

The direction vector of a line is a non-zero vector lying on a line or on a line parallel to a given line.

If a straight line is defined by two points M 1 (x 1 ; y 1 ; z 1) and M 2 (x 2 ; y 2 ; z 2), then the guide is the vector
or its opposite vector
= - , whose coordinates

It is advisable to set the coordinate system so that the line passes through the origin of coordinates, then the coordinates of the only point on the line will be the coordinates of the direction vector.

Task. Determine the coordinates of the direction vector of the straight line passing through the points M 1 (1; 0; 0), M 2 (0; 1; 0).

Solution

The direction vector of a straight line passing through the points M 1 (1; 0; 0), M 2 (0; 1; 0) is denoted
. Each of its coordinates is equal to the difference between the corresponding coordinates of the end and beginning of the vector

= (0 - 1; 1 - 0; 0 - 0) = (-1; 1; 0)

Let us depict the directing vector of a straight line in the coordinate system with the beginning at point M 1, with the end at point M 2 and an equal vector
from the origin with the end at point M (-1; 1; 0)

1.24 Angle between two straight lines

Possible options for the relative position of 2 straight lines on a plane and the angle between such straight lines:

1. Straight lines intersect at a single point, forming 4 angles, 2 pairs of vertical angles are equal in pairs. The angle φ between two intersecting lines is the angle not exceeding the other three angles between these lines. Therefore, the angle between the lines is φ ≤ 90 0.

Intersecting lines can be, in particular, perpendicular to φ = 90 0.

Possible options for the relative position of 2 straight lines in space and the angle between such straight lines:

2. The lines are parallel, that is, they do not coincide and do not intersect, φ=0 0 .

3. The lines coincide, φ = 0 0 .

4. Lines intersect, that is, they do not intersect in space and are not parallel. The angle φ between intersecting lines is the angle between lines drawn parallel to these lines so that they intersect. Therefore, the angle between the lines is φ ≤ 90 0.

The angle between 2 straight lines is equal to the angle between straight lines drawn parallel to these straight lines in the same plane. Therefore, the angle between the lines is 0 0 ≤ φ ≤ 90 0.

Angle θ (theta) between vectors and 0 0 ≤ θ ≤ 180 0 .

If the angle φ between lines α and β is equal to the angle θ between the direction vectors of these lines φ = θ, then

cos φ = cos θ.

If the angle between straight lines is φ = 180 0 - θ, then

cos φ = cos (180 0 - θ) = - cos θ.

cos φ = - cos θ.

Therefore, the cosine of the angle between straight lines is equal to the modulus of the cosine of the angle between vectors

cos φ = |cos θ|.

If the coordinates of non-zero vectors = (x 1 ; y 1 ; z 1) and = (x 2 ; y 2 ; z 2) are given, then the cosine of the angle θ between them

The cosine of the angle between the lines is equal to the modulus of the cosine of the angle between the direction vectors of these lines

cos φ = |cos θ| =

The lines are the same geometric objects, therefore the same trigonometric cos functions are present in the formula.

If each of two lines is given by two points, then it is possible to determine the direction vectors of these lines and the cosine of the angle between the lines.

If cos φ = 1, then the angle φ between the lines is equal to 0 0, we can take for these lines one of the direction vectors of these lines, the lines are parallel or coincide. If the lines do not coincide, then they are parallel. If the lines coincide, then any point on one line belongs to the other line.

If 0< cos φ ≤ 1, then the angle between the lines is 0 0< φ ≤ 90 0 , прямые пересекаются или скрещиваются. Если прямые не пересекаются, то они скрещиваются. Если прямые пересекаются, то они имеют общую точку.

If cos φ = 0, then the angle φ between the lines is 90 0 (the lines are perpendicular), the lines intersect or cross.

Task. Determine the angle between straight lines M 1 M 3 and M 2 M 3 with the coordinates of points M 1 (1; 0; 0), M 2 (0; 1; 0) and M 3 (0; 0; 1).

Solution

Let's construct given points and lines in the Oxyz coordinate system.

We direct the direction vectors of the lines so that the angle θ between the vectors coincides with the angle φ between the given lines. Let us represent the vectors =
and =
, as well as angles θ and φ:

Let us determine the coordinates of the vectors and

= = (1 - 0; 0 - 0; 0 - 1) = (1; 0; -1);

= = (0 - 0; 1 - 0; 0 - 1) = (0; 1; -1). d = 0 and ax + by + cz = 0;

The plane is parallel to the coordinate axis, the designation of which is absent in the equation of the plane and, therefore, the corresponding coefficient is zero, for example, at c = 0, the plane is parallel to the Oz axis and does not contain z in the equation ax + by + d = 0;

The plane contains that coordinate axis, the designation of which is missing, therefore, the corresponding coefficient is zero and d = 0, for example, with c = d = 0, the plane is parallel to the Oz axis and does not contain z in the equation ax + by = 0;

The plane is parallel to the coordinate plane, the symbols of which are absent in the equation of the plane and, therefore, the corresponding coefficients are zero, for example, for b = c = 0, the plane is parallel to the coordinate plane Oyz and does not contain y, z in the equation ax + d = 0.

If the plane coincides with the coordinate plane, then the equation of such a plane is the equality to zero of the designation of the coordinate axis perpendicular to the given coordinate plane, for example, when x = 0, the given plane is the coordinate plane Oyz.

Task. The normal vector is given by the equation

Present the equation of the plane in normal form.

Solution

Normal vector coordinates

A; b ; c), then you can substitute the coordinates of the point M 0 (x 0 ; y 0 ; z 0) and the coordinates a, b, c of the normal vector into the general equation of the plane

ax + by + cz + d = 0 (1)

We obtain an equation with one unknown d

ax 0 + by 0 + cz 0 + d = 0

From here

d = -(ax 0 + by 0 + cz 0 )

Plane equation (1) after substituting d

ax + by + cz - (ax 0 + by 0 + cz 0) = 0

We obtain the equation of the plane passing through the point M 0 (x 0 ; y 0 ; z 0) perpendicular to the non-zero vector (a; b; c)

a (x - x 0) + b (y - y 0) + c (z - z 0) = 0

Let's open the brackets

ax - ax 0 + by - by 0 + cz - cz 0 = 0

ax + by + cz - ax 0 - by 0 - cz 0 = 0

Let's denote

d = - ax 0 - by 0 - cz 0

We obtain the general equation of the plane

ax + by + cz + d = 0.

1.29 Equation of a plane passing through two points and the origin

ax + by + cz + d = 0.

It is advisable to set the coordinate system so that the plane passes through the origin of this coordinate system. Points M 1 (x 1 ; y 1 ; z 1) and M 2 (x 2 ; y 2 ; z 2) lying in this plane must be specified so that the straight line connecting these points does not pass through the origin.

The plane will pass through the origin, so d = 0. Then the general equation of the plane takes the form

ax + by + cz = 0.

There are 3 unknown coefficients a, b, c. Substituting the coordinates of two points into the general equation of the plane gives a system of 2 equations. If we take some coefficient in the general equation of the plane equal to one, then the system of 2 equations will allow us to determine 2 unknown coefficients.

If one of the coordinates of a point is zero, then the coefficient corresponding to this coordinate is taken as one.

If some point has two zero coordinates, then the coefficient corresponding to one of these zero coordinates is taken as one.

If a = 1 is accepted, then a system of 2 equations will allow us to determine 2 unknown coefficients b and c:

It is easier to solve a system of these equations by multiplying some equation by such a number that the coefficients for some unknown become equal. Then the difference of the equations will allow us to eliminate this unknown and determine another unknown. Substituting the found unknown into any equation will allow you to determine the second unknown.

1.30 Equation of a plane passing through three points

Let us determine the coefficients of the general equation of the plane

ax + by + cz + d = 0,

passing through the points M 1 (x 1 ; y 1 ; z 1), M 2 (x 2 ; y 2 ; z 2) and M 3 (x 3 ; y 3 ; z 3). Points should not have two identical coordinates.

There are 4 unknown coefficients a, b, c and d. Substituting the coordinates of three points into the general equation of the plane gives a system of 3 equations. Take some coefficient in the general equation of the plane equal to unity, then the system of 3 equations will allow you to determine 3 unknown coefficients. Usually a = 1 is accepted, then a system of 3 equations will allow us to determine 3 unknown coefficients b, c and d:

It is better to solve a system of equations by eliminating the unknowns (Gauss method). You can rearrange the equations in the system. Any equation can be multiplied or divided by any coefficient not equal to zero. Any two equations can be added and the resulting equation can be written in place of either of the two added equations. Unknowns are excluded from the equations by obtaining a zero coefficient in front of them. In one equation, usually the lowest one, there is one variable left that is determined. The found variable is substituted into the second equation from below, which usually leaves 2 unknowns. The equations are solved from bottom to top and all unknown coefficients are determined.

Coefficients are placed in front of the unknowns, and terms free of unknowns are transferred to the right side of the equations

The top line usually contains an equation that has a coefficient of 1 before the first or any unknown, or the entire first equation is divided by the coefficient before the first unknown. In this system of equations, divide the first equation by y 1

Before the first unknown we got a coefficient of 1:

To reset the coefficient in front of the first variable of the second equation, multiply the first equation by -y 2, add it to the second equation, and write the resulting equation instead of the second equation. The first unknown in the second equation will be eliminated because

y 2 b - y 2 b = 0.

Similarly, we eliminate the first unknown in the third equation by multiplying the first equation by -y 3, adding it to the third equation and writing the resulting equation instead of the third equation. The first unknown in the third equation will also be eliminated because

y 3 b - y 3 b = 0.

Similarly, we eliminate the second unknown in the third equation. We solve the system from the bottom up.

Task.

ax + by + cz + d = 0,

passing through points M 1 (0; 0; 0), M 2 (0; 1; 0) and y+ 0 z + 0 = 0

x = 0.

The specified plane is the coordinate plane Oyz.

Task. Determine the general equation of the plane

ax + by + cz + d = 0,

passing through the points M 1 (1; 0; 0), M 2 (0; 1; 0) and M 3 (0; 0; 1). Find the distance from this plane to point M 0 (10; -3; -7).

Solution

Let's construct the given points in the Oxyz coordinate system.

Let's accept a= 1. Substituting the coordinates of three points into the general equation of the plane gives a system of 3 equations

ℓ =

Web pages: 1 2 Vectors on the plane and in space (continued)

Consultations with Andrey Georgievich Olshevsky on Skype da.irk.ru

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10.15.17 Olshevsky Andrey Georgieviche-mail:[email protected]

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 or 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 vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a vector 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 a student saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is mathematically correct - the vector can be attached 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 of the vector 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 by the triangle rule, addition by the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, we 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 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: , . Rearrange the terms and see in the drawing how well 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 corresponding 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 necessary, we can easily move it away from some other point in 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 corresponding 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 .

Standard definition: “A vector is a directed segment.” This is usually the extent of a graduate’s knowledge about vectors. Who needs any “directional segments”?

But really, what are vectors and what are they for?
Weather forecast. “Wind northwest, speed 18 meters per second.” Agree, both the direction of the wind (where it blows from) and the magnitude (that is, the absolute value) of its speed matter.

Quantities that have no direction are called scalar. Mass, work, electric charge are not directed anywhere. They are characterized only by a numerical value - “how many kilograms” or “how many joules”.

Physical quantities that have not only an absolute value, but also a direction, are called vector quantities.

Speed, force, acceleration - vectors. For them, “how much” is important and “where” is important. For example, the acceleration of gravity is directed towards the surface of the Earth, and its value is 9.8 m/s 2. Impulse, electric field strength, magnetic field induction are also vector quantities.

You remember that physical quantities are denoted by letters, Latin or Greek. The arrow above the letter indicates that the quantity is vector:

Here's another example.
A car moves from A to B. The end result is its movement from point A to point B, that is, movement by a vector .

Now it’s clear why a vector is a directed segment. Please note that the end of the vector is where the arrow is. Vector length is called the length of this segment. Indicated by: or

Until now, we have worked with scalar quantities, according to the rules of arithmetic and elementary algebra. Vectors are a new concept. This is another class of mathematical objects. They have their own rules.

Once upon a time we didn’t even know anything about numbers. My acquaintance with them began in elementary school. It turned out that numbers can be compared with each other, added, subtracted, multiplied and divided. We learned that there is a number one and a number zero.
Now we are introduced to vectors.

The concepts of “more” and “less” for vectors do not exist - after all, their directions can be different. Only vector lengths can be compared.

But there is a concept of equality for vectors.
Equal vectors that have the same length and the same direction are called. This means that the vector can be transferred parallel to itself to any point in the plane.
Single is a vector whose length is 1. Zero is a vector whose length is zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in a rectangular coordinate system - the same one in which we draw graphs of functions. Each point in the coordinate system corresponds to two numbers - its x and y coordinates, abscissa and ordinate.
The vector is also specified by two coordinates:

Here the coordinates of the vector are written in parentheses - in x and y.
They are found simply: the coordinate of the end of the vector minus the coordinate of its beginning.

If the vector coordinates are given, its length is found by the formula

Vector addition

There are two ways to add vectors.

1 . Parallelogram rule. To add the vectors and , we place the origins of both at the same point. We build up to a parallelogram and from the same point we draw a diagonal of the parallelogram. This will be the sum of the vectors and .

Remember the fable about the swan, crayfish and pike? They tried very hard, but they never moved the cart. After all, the vector sum of the forces they applied to the cart was equal to zero.

2. The second way to add vectors is the triangle rule. Let's take the same vectors and . We will add the beginning of the second to the end of the first vector. Now let's connect the beginning of the first and the end of the second. This is the sum of the vectors and .

Using the same rule, you can add several vectors. We arrange them one after another, and then connect the beginning of the first to the end of the last.

Imagine that you are going from point A to point B, from B to C, from C to D, then to E and to F. The end result of these actions is movement from A to F.

When adding vectors and we get:

Vector subtraction

The vector is directed opposite to the vector. The lengths of the vectors and are equal.

Now it’s clear what vector subtraction is. The vector difference and is the sum of the vector and the vector .

Multiplying a vector by a number

When a vector is multiplied by the number k, a vector is obtained whose length is k times different from the length . It is codirectional with the vector if k is greater than zero, and opposite if k is less than zero.

Dot product of vectors

Vectors can be multiplied not only by numbers, but also by each other.

The scalar product of vectors is the product of the lengths of the vectors and the cosine of the angle between them.

Please note that we multiplied two vectors, and the result was a scalar, that is, a number. For example, in physics, mechanical work is equal to the scalar product of two vectors - force and displacement:

If the vectors are perpendicular, their scalar product is zero.
And this is how the scalar product is expressed through the coordinates of the vectors and:

From the formula for the scalar product you can find the angle between the vectors:

This formula is especially convenient in stereometry. For example, in Problem 14 of the Profile Unified State Exam in Mathematics, you need to find the angle between intersecting lines or between a straight line and a plane. Problem 14 is often solved several times faster using the vector method than using the classical method.

In the school mathematics curriculum, only the scalar product of vectors is taught.
It turns out that, in addition to the scalar product, there is also a vector product, when the result of multiplying two vectors is a vector. Anyone who takes the Unified State Exam in physics knows what the Lorentz force and the Ampere force are. The formulas for finding these forces include vector products.

Vectors are a very useful mathematical tool. You will see this in your first year.

So, services:

The service for working with vectors allows you to perform actions on vectors.
If you have a task to perform a more complex transformation, then this service should be used as a constructor.
Example. Vector data a And b, we need to find the vector With = a + 3*b,