A vector whose length is 1 is called. A vector is a directed segment in a plane or in space

Definition

Scalar quantity- a quantity that can be characterized by a number. For example, length, area, mass, temperature, etc.

Vector called the directed segment $\overline(A B)$; point $A$ is the beginning, point $B$ is the end of the vector (Fig. 1).

A vector is denoted either by two capital letters - its beginning and end: $\overline(A B)$ or by one small letter: $\overline(a)$.

Definition

If the beginning and end of a vector coincide, then such a vector is called zero. Most often, the zero vector is denoted as $\overline(0)$.

The vectors are called collinear, if they lie either on the same line or on parallel lines (Fig. 2).

Definition

Two collinear vectors $\overline(a)$ and $\overline(b)$ are called co-directed, if their directions coincide: $\overline(a) \uparrow \uparrow \overline(b)$ (Fig. 3, a). Two collinear vectors $\overline(a)$ and $\overline(b)$ are called oppositely directed, if their directions are opposite: $\overline(a) \uparrow \downarrow \overline(b)$ (Fig. 3, b).

Definition

The vectors are called coplanar, if they are parallel to the same plane or lie in the same plane (Fig. 4).

Two vectors are always coplanar.

Definition

Length (module) vector $\overline(A B)$ is the distance between its beginning and end: $|\overline(A B)|$

Detailed theory about vector length at the link.

The length of the zero vector is zero.

Definition

A vector whose length is equal to one is called unit vector or ortom.

The vectors are called equal, if they lie on one or parallel lines; their directions coincide and their lengths are equal.

In other words, two vectors equal, if they are collinear, codirectional and have equal lengths:

$\overline(a)=\overline(b)$ if $\overline(a) \uparrow \uparrow \overline(b),|\overline(a)|=|\overline(b)|$

At an arbitrary point $M$ of space, one can construct a single vector $\overline(M N)$ equal to the given vector $\overline(A B)$.

Vector algebra

Definition:

A vector is a directed segment in a plane or in space.

Characteristics:

1) vector length

Definition:

Two vectors are called collinear if they lie on parallel lines.

Definition:

Two collinear vectors are called codirectional if their directions coincide ( ) Otherwise they are called oppositely directed (↓ ).

Definition:

Two vectors are equal if they are co-directional and have the same length.

For example,

Operations:

1. Multiplying a vector by a number

If
, That

↓If < 0

The direction of the zero vector is arbitrary

Properties of multiplication by a number

2. Vector addition

Parallelogram rule:

Addition properties:

- such vectors are called opposite to each other. It's easy to see that

Joint properties:

ABOUT definition:

The angle between two vectors is the angle that is obtained if these vectors are plotted from one point, 0    

3. Dot product of vectors.

, Where - angle between vectors

Properties of the scalar product of vectors:

1) (equalities take place in the case of opposite direction and co-direction of vectors, respectively)

If
, then the sign of the product is positive, If ↓that is negative

)

6), that is
, or any of the vectors is zero

Application of vectors

MN – midline

Prove that

Proof:

, subtract the vector from both sides
:

Prove that the diagonals of a rhombus are perpendicular

Proof:

Find:

Solution:

Decomposition of vectors into bases.

Definition:

A linear combination of vectors (LCV) is a sum of the form

(LKV)

Where 1 ,  2 , …  s – arbitrary set of numbers

Definition:

An LCI is said to be non-trivial if all i = 0, otherwise it is called nontrivial.

Consequence:

A non-trivial LCV has at least one non-zero coefficient To  0

Definition:

Vector system
called linearly independent (LNI),If() = 0  All i  0,

that is, only its trivial LC is equal to zero.

Consequence:

The nontrivial LC of linearly independent vectors is nonzero

Examples:

1)
- LNZ

2) Let And lie in the same plane, then
- LNZ , non-collinear

3) Let , , do not belong to the same plane, then they form a LNZ system of vectors

Theorem:

If a system of vectors is linearly independent, then at least one of them is a linear combination of the others.

Proof:

 Let () = 0 and not all I are equal to zero. Without losing generality, let s  0. Then
, and this is a linear combination.

 Let

Then, that is, LZ.

Theorem:

Any 3 vectors on a plane are linearly dependent.

Proof:

Let the vectors be given
, possible cases:

1)


2) non-collinear

Let's express it through and:
, where
- non-trivial LC.

Theorem:

Let
- LZ

Then any “wider” system is LZ

Proof:

Since - LZ, then there is at least one i  0, and () = 0

Then and () = 0

Definition:

A system of linearly independent vectors is called maximal if, when any other vector is added to it, it becomes linearly dependent.

Definition:

The dimension of space (plane) is the number of vectors in a maximal linearly independent system of vectors.

Definition:

A basis is any ordered maximal linearly independent system of vectors.

Definition:

A basis is called normalized if the vectors included in it have a length equal to one.

Definition:

A basis is called orthogonal if all its elements (vectors) are pairwise perpendicular.

Theorem:

A system of orthogonal vectors is always linearly independent (if there are no zero vectors).

Proof:

Let be a system of orthogonal vectors (non-zero), that is
. Suppose , we multiply this LC scalarly by the vector :

The first bracket is non-zero (the square of the vector length), and all other brackets are equal to zero by condition. Then 1 = 0. Similarly for 2 …  s

Theorem:

Let M = - basis. Then we can represent any vector in the form:

where are the coefficients 2 …  s are determined uniquely (these are the coordinates of the vector relative to the basis M).

Proof:

1)
=
- LZ (according to the basis condition)

then - nontrivial

A) 0 = 0 which is impossible, since it turns out that M – LZ

b) 0  0

divide by 0

those. there is a personal account

2) Let's prove it by contradiction. Let be another representation of the vector (i.e. at least one pair
). Let's subtract the formulas from each other:

- LC is non-trivial.

But according to the condition - basis a contradiction, that is, the decomposition is unique.

Conclusion:

Every basis M determines a one-to-one correspondence between vectors and their coordinates relative to the basis M.

Designations:

M = - arbitrary vector

Then

2018 Olshevsky Andrey Georgievich

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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, 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)

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The uniqueness of the coefficients of a linear combination is proved in the same way as in the previous corollary.

Consequence: Any four vectors are linearly dependent

Chapter 4. The concept of basis. Properties of a vector in a given basis

Definition:Basis in space is any ordered triple of non-coplanar vectors.

Definition:Basis on the plane is any ordered pair of noncollinear vectors.

A basis in space allows each vector to be uniquely associated with an ordered triple of numbers - the coefficients of representing this vector in the form of a linear combination of basis vectors. On the contrary, we associate a vector with each ordered triple of numbers using a basis if we make a linear combination.

Numbers are called components (or coordinates ) vector in a given basis (written ).

Theorem: When adding two vectors, their coordinates are added. When a vector is multiplied by a number, all coordinates of the vector are multiplied by that number.

Indeed, if , That

The definition and properties of vector coordinates on a plane are similar. You can easily formulate them yourself.

Chapter 5. Vector Projection

Under angle between vectors refers to the angle between vectors equal to data and having a common origin. If the angle reference direction is not specified, then the angle between the vectors is considered to be the angle that does not exceed π. If one of the vectors is zero, then the angle is considered equal to zero. If the angle between the vectors is straight, then the vectors are called orthogonal .

Definition:Orthogonal projection vector to the direction of the vector called a scalar quantity , φ – angle between vectors (Fig. 9).

The modulus of this scalar quantity is equal to the length of the segment O.A. 0 .

If the angle φ is acute, the projection is positive; if the angle φ is obtuse, the projection is negative; if the angle φ is straight, the projection is zero.

With an orthogonal projection, the angle between the segments O.A. 0 And A.A. 0 straight. There are projections in which this angle is different from the right angle.

Projections of vectors have the following properties:

The basis is called orthogonal , if its vectors are pairwise orthogonal.

An orthogonal basis is called orthonormal , if its vectors are equal in length to one. For an orthonormal basis in space, the notation is often used.

Theorem: In an orthonormal basis, the coordinates of the vectors are the corresponding orthogonal projections of this vector onto the directions of the coordinate vectors.

Example: Let a vector of unit length form an angle φ with the vector of an orthonormal basis on the plane, then .

Example: Let a vector of unit length form angles α, β, γ with the vectors , and of an orthonormal basis in space, respectively (Fig. 11), then . Moreover. The quantities cosα, cosβ, cosγ are called direction cosines of the vector

Chapter 6. Dot product

Definition: The scalar product of two vectors is a number equal to the product of the lengths of these vectors and the cosine of the angle between them. If one of the vectors is zero, the scalar product is considered equal to zero.

The scalar product of vectors and is denoted by [or ; or ]. If φ is the angle between the vectors and , then .

The scalar product has the following properties:

Theorem: In an orthogonal basis, the components of any vector are found according to the formulas:

Indeed, let , and each term is collinear to the corresponding basis vector. From the theorem of the second section it follows that , where the plus or minus sign is chosen depending on whether the vectors , and are directed in the same or opposite directions. But, , where φ is the angle between the vectors , and . So, . The remaining components are calculated similarly.

The scalar product is used to solve the following basic problems:

1. ; 2. ; 3. .

Let vectors be given in a certain basis, and then, using the properties of the scalar product, we can write:

The quantities are called metric coefficients of a given basis. Hence .

Theorem: In an orthonormal basis

;
;
;
.

Comment: All arguments in this section are given for the case of the location of vectors in space. The case of vectors being located on a plane is obtained by removing unnecessary components. The author suggests you do this yourself.

Chapter 7. Vector product

An ordered triple of non-coplanar vectors is called right-oriented (right ), if after application to the common origin from the end of the third vector, the shortest turn from the first vector to the second is visible counterclockwise. Otherwise, an ordered triple of non-coplanar vectors is called left-oriented (left ).

Definition: The cross product of a vector and a vector is a vector that satisfies the conditions:

If one of the vectors is zero, then the cross product is the zero vector.

The cross product of a vector and a vector is denoted (or).

Theorem: A necessary and sufficient condition for the collinearity of two vectors is that their vector product is equal to zero.

Theorem: The length (modulus) of the vector product of two vectors is equal to the area of the parallelogram constructed on these vectors as sides.

Example: If is a right orthonormal basis, then , , .

Example: If is a left orthonormal basis, then , , .

Example: Let a be orthogonal to . Then it is obtained from the vector by rotating it clockwise around the vector (as seen from the end of the vector).