Definition 1. On a plane, the parallel projection of point A onto the l-axis is a point - the point of intersection of the l-axis with a straight line drawn through point A parallel to the vector that specifies the design direction.

Definition 2. The parallel projection of a vector onto the l-axis (on a vector) is the coordinate of the vector, relative to the basis the l axis, where the points and are parallel projections of points A and B, respectively, onto the l axis (Fig. 1).

By definition, we have

Definition 3. if and the basis of the l axis cartesian, that is, then the projection of the vector onto the l-axis is called orthogonal (Fig. 2).

In space, definition 2 of the projection of a vector onto an axis remains valid, only the projection direction is given by two non-collinear vectors (Fig. 3).

From the definition of the projection of a vector onto an axis, it follows that each coordinate of the vector is the projection of this vector onto the axis determined by the corresponding basis vector. In this case, the design direction is set by two other basis vectors, if the design is carried out (considered) in space, or by another basis vector, if the design is considered on a plane (Fig. 4).

Theorem 1. The orthogonal projection of a vector onto the l-axis is equal to the product of the modulus of the vector and the cosine of the angle between the positive direction of the l-axis and, i.e.

On the other hand

From we find

Substituting AC into equality (2), we obtain

Since the numbers x and of the same sign in both considered cases ((Fig. 5, a) ; (Fig. 5, b) , then equality (4) implies

Comment. In the future, we will consider only the orthogonal projection of the vector onto the axis, and therefore the word "orth" (orthogonal) in the notation will be omitted.

We present a number of formulas that are used in the future when solving problems.

a) Projection of a vector onto an axis.

If, then the orthogonal projection onto the vector according to formula (5) has the form

c) Distance from a point to a plane.

Let b be a given plane with a normal vector, M be a given point,

d - distance from point M to plane b (Fig. 6).

If N is an arbitrary point of the plane b, and and are the projections of the points M and N onto the axis, then

• G) Distance between intersecting lines.

Let a and b be given intersecting lines, be a vector perpendicular to them, A and B be arbitrary points of lines a and b, respectively (Fig. 7), and be projections of points A and B onto, then

e) Distance from a point to a line.

Let l- given line with direction vector, M - given point,

N - its projection onto the line l, then - the desired distance (Fig. 8).

If A is an arbitrary point on the line l, then in the right triangle MNA the hypotenuse MA and the legs can be found. Means,

e) Angle between a line and a plane.

Let be the direction vector of the given line l, - normal vector of the given plane b, - projection of a straight line l to plane b (Fig. 9).

As you know, the angle q between the line l and its projection onto the plane b is called the angle between the line and the plane. We have

Let us give examples of solving metric problems by the vector-coordinate method.

Solving problems on the equilibrium of converging forces by constructing closed force polygons is associated with cumbersome constructions. A universal method for solving such problems is the transition to determining the projections of given forces on the coordinate axes and operating with these projections. The axis is called a straight line, which is assigned a certain direction.

The projection of a vector onto an axis is a scalar value, which is determined by the segment of the axis cut off by the perpendiculars dropped onto it from the beginning and end of the vector.

The projection of a vector is considered positive if the direction from the beginning of the projection to its end coincides with the positive direction of the axis. The projection of a vector is considered negative if the direction from the beginning of the projection to its end is opposite to the positive direction of the axis.

Thus, the projection of the force on the coordinate axis is equal to the product of the modulus of the force and the cosine of the angle between the force vector and the positive direction of the axis.

Consider a number of cases of projecting forces onto an axis:

Force vector F(Fig. 15) makes an acute angle with the positive direction of the x-axis.

To find the projection, from the beginning and end of the force vector we lower the perpendiculars to the axis oh; we get

1. F x = F cosα

The projection of the vector in this case is positive

Strength F(Fig. 16) is with the positive direction of the axis X obtuse angle α.

Then F x= F cos α, but since α = 180 0 - φ,

F x= F cosα = F cos180 0 - φ =- F cos phi.

Force projection F per axle oh in this case is negative.

Strength F(Fig. 17) perpendicular to the axis oh.

Projection of force F on the axis X zero

F x= F cos 90° = 0.

Force located on a plane howe(Fig. 18), can be projected onto two coordinate axes oh and OU.

Strength F can be broken down into components: F x and F y . Vector modulus F x is equal to the vector projection F per axle ox, and the modulus of the vector F y is equal to the projection of the vector F per axle oy.

From Δ OAB: F x= F cosα, F x= F sinα.

From Δ SLA: F x= F cos phi, F x= F sin phi.

The modulus of force can be found using the Pythagorean theorem:

The projection of the vector sum or the resultant on any axis is equal to the algebraic sum of the projections of the terms of the vectors on the same axis.

Consider converging forces F 1 , F 2 , F 3 , and F 4, (Fig. 19, a). The geometric sum, or resultant, of these forces F determined by the closing side of the force polygon

Drop from the vertices of the force polygon onto the axis x perpendiculars.

Considering the obtained projections of forces directly from the completed construction, we have

F= F 1x+ F 2x+ F 3x+ F 4x

where n is the number of terms of vectors. Their projections enter the above equation with the appropriate sign.

In a plane, the geometric sum of forces can be projected onto two coordinate axes, and in space, respectively, onto three.

Introduction……………………………………………………………………………3

1. The value of a vector and a scalar…………………………………………….4

2. Definition of projection, axis and coordinate of a point………………...5

3. Vector projection onto the axis………………………………………………...6

4. The basic formula of vector algebra……………………………..8

5. Calculation of the module of the vector from its projections…………………...9

Conclusion……………………………………………………………………...11

Literature……………………………………………………………………...12

Introduction:

Physics is inextricably linked with mathematics. Mathematics gives physics the means and techniques of a general and precise expression of the relationship between physical quantities that are discovered as a result of experiment or theoretical research. After all, the main method of research in physics is experimental. This means that the scientist reveals the calculations with the help of measurements. Denotes the relationship between different physical quantities. Then, everything is translated into the language of mathematics. A mathematical model is being formed. Physics is a science that studies the simplest and at the same time the most general laws. The task of physics is to create in our minds such a picture of the physical world that most fully reflects its properties and provides such relationships between the elements of the model that exist between the elements.

So, physics creates a model of the world around us and studies its properties. But any model is limited. When creating models of a particular phenomenon, only properties and connections that are essential for a given range of phenomena are taken into account. This is the art of a scientist - from all the variety to choose the main thing.

Physical models are mathematical, but mathematics is not their basis. Quantitative relationships between physical quantities are clarified as a result of measurements, observations and experimental studies and are only expressed in the language of mathematics. However, there is no other language for constructing physical theories.

1. The value of a vector and a scalar.

In physics and mathematics, a vector is a quantity that is characterized by its numerical value and direction. In physics, there are many important quantities that are vectors, such as force, position, speed, acceleration, torque, momentum, electric and magnetic fields. They can be contrasted with other quantities, such as mass, volume, pressure, temperature and density, which can be described by an ordinary number, and they are called " scalars".

They are written either in letters of a regular font, or in numbers (a, b, t, G, 5, -7 ....). Scalars can be positive or negative. At the same time, some objects of study may have such properties, for a complete description of which the knowledge of only a numerical measure is insufficient, it is also necessary to characterize these properties by a direction in space. Such properties are characterized by vector quantities (vectors). Vectors, unlike scalars, are denoted by bold letters: a, b, g, F, C ....
Often, a vector is denoted by a regular (non-bold) letter, but with an arrow above it:

The module of the vector, that is, the length of the directed straight line segment, is denoted by the same letters as the vector itself, but in the usual (non-bold) writing and without an arrow above them, or just like the vector (that is, in bold or regular, but with arrow), but then the vector designation is enclosed in vertical dashes.
A vector is a complex object that is characterized by both magnitude and direction at the same time.

There are also no positive and negative vectors. But the vectors can be equal to each other. This is when, for example, a and b have the same modules and are directed in the same direction. In this case, the record a= b. It should also be borne in mind that the vector symbol can be preceded by a minus sign, for example, -c, however, this sign symbolically indicates that the vector -c has the same modulus as the vector c, but is directed in the opposite direction.

The vector -c is called the opposite (or inverse) of the vector c.
In physics, however, each vector is filled with specific content, and when comparing vectors of the same type (for example, forces), the points of their application can also be of significant importance.

2.Determination of the projection, axis and coordinate of the point.

Axis is a straight line that is given a direction.
The axis is indicated by any letter: X, Y, Z, s, t ... Usually, a point is (arbitrarily) chosen on the axis, which is called the origin and, as a rule, is indicated by the letter O. Distances to other points of interest to us are measured from this point.

point projection on the axis is called the base of the perpendicular dropped from this point to the given axis. That is, the projection of a point onto the axis is a point.

point coordinate on a given axis is called a number whose absolute value is equal to the length of the segment of the axis (in the selected scale) enclosed between the beginning of the axis and the projection of the point onto this axis. This number is taken with a plus sign if the projection of the point is located in the direction of the axis from its beginning and with a minus sign if in the opposite direction.

3.Projection of a vector onto an axis.

The projection of a vector onto an axis is a vector that is obtained by multiplying the scalar projection of a vector onto this axis and the unit vector of this axis. For example, if a x is the scalar projection of the vector a onto the X axis, then a x i is its vector projection onto this axis.

Let's denote the vector projection in the same way as the vector itself, but with the index of the axis on which the vector is projected. So, the vector projection of the vector a on the X axis is denoted by a x (bold letter denoting the vector and the subscript of the axis name) or

Scalar projection vector per axis is called number, the absolute value of which is equal to the length of the segment of the axis (in the selected scale) enclosed between the projections of the start point and the end point of the vector. Usually instead of the expression scalar projection simply say - projection. The projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a subscript (usually) of the name of the axis on which this vector is projected. For example, if a vector is projected onto the x-axis a, then its projection is denoted a x . When projecting the same vector onto another axis, if the axis is Y , its projection will be denoted as y .

To calculate projection vector on an axis (for example, the X axis) it is necessary to subtract the coordinate of the start point from the coordinate of its end point, that is

and x \u003d x k - x n.

The projection of a vector onto an axis is a number. Moreover, the projection can be positive if the value of x k is greater than the value of x n,

negative if the value of x k is less than the value of x n

and equal to zero if x k is equal to x n.

The projection of a vector onto an axis can also be found by knowing the modulus of the vector and the angle it makes with that axis.

It can be seen from the figure that a x = a Cos α

That is, the projection of the vector onto the axis is equal to the product of the modulus of the vector and the cosine of the angle between the direction of the axis and vector direction. If the angle is acute, then
Cos α > 0 and a x > 0, and if obtuse, then the cosine of an obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

Angles counted from the axis counterclockwise are considered to be positive, and in the direction - negative. However, since the cosine is an even function, that is, Cos α = Cos (− α), when calculating projections, the angles can be counted both clockwise and counterclockwise.

To find the projection of a vector onto an axis, the module of this vector must be multiplied by the cosine of the angle between the direction of the axis and the direction of the vector.

4. Basic formula of vector algebra.

We project a vector a on the X and Y axes of a rectangular coordinate system. Find the vector projections of the vector a on these axes:

and x = a x i, and y = a y j.

But according to the vector addition rule

a \u003d a x + a y.

a = a x i + a y j.

Thus, we have expressed a vector in terms of its projections and orts of a rectangular coordinate system (or in terms of its vector projections).

The vector projections a x and a y are called components or components of the vector a. The operation that we have performed is called the decomposition of the vector along the axes of a rectangular coordinate system.

If the vector is given in space, then

a = a x i + a y j + a z k.

This formula is called the basic formula of vector algebra. Of course, it can also be written like this.

Let two vectors and be given in space. Set aside from an arbitrary point O vectors and . corner between the vectors and is called the smallest of the angles. Denoted .

Consider the axis l and plot a unit vector on it (that is, a vector whose length is equal to one).

Angle between vector and axis l understand the angle between the vectors and .

So let l is some axis and is a vector.

Denote by A 1 and B1 projections on the axis l points A and B. Let's pretend that A 1 has a coordinate x 1, a B1- coordinate x2 on axle l.

Then projection vector per axis l is called difference x 1 – x2 between the coordinates of the projections of the end and beginning of the vector onto this axis.

Projection of a vector onto an axis l we will denote .

It is clear that if the angle between the vector and the axis l sharp then x2> x 1, and the projection x2 – x 1> 0; if this angle is obtuse, then x2< x 1 and projection x2 – x 1< 0. Наконец, если вектор перпендикулярен оси l, then x2= x 1 and x2– x 1=0.

Thus, the projection of the vector onto the axis l is the length of the segment A 1 B 1 taken with a certain sign. Therefore, the projection of a vector onto an axis is a number or a scalar.

The projection of one vector onto another is defined similarly. In this case, the projections of the ends of this vector are found on the line on which the 2nd vector lies.

Let's look at some of the main projection properties.

LINEARLY DEPENDENT AND LINEARLY INDEPENDENT SYSTEMS OF VECTORS

Let's consider several vectors.

Linear combination of these vectors is any vector of the form , where are some numbers. The numbers are called the coefficients of the linear combination. It is also said that in this case is linearly expressed in terms of given vectors , i.e. obtained from them by linear operations.

For example, if three vectors are given, then vectors can be considered as their linear combination:

If a vector is represented as a linear combination of some vectors, then it is said to be decomposed along these vectors.

The vectors are called linearly dependent, if there are such numbers, not all equal to zero, that . It is clear that the given vectors will be linearly dependent if any of these vectors is linearly expressed in terms of the others.

Otherwise, i.e. when the ratio performed only when , these vectors are called linearly independent.

Theorem 1. Any two vectors are linearly dependent if and only if they are collinear.

Proof:

The following theorem can be proved similarly.

Theorem 2. Three vectors are linearly dependent if and only if they are coplanar.

Proof.

BASIS

Basis is the collection of non-zero linearly independent vectors. The elements of the basis will be denoted by .

In the previous subsection, we saw that two non-collinear vectors in the plane are linearly independent. Therefore, according to Theorem 1 from the previous paragraph, a basis on a plane is any two non-collinear vectors on this plane.

Similarly, any three non-coplanar vectors are linearly independent in space. Therefore, three non-coplanar vectors are called a basis in space.

The following assertion is true.

Theorem. Let a basis be given in space. Then any vector can be represented as a linear combination , where x, y, z- some numbers. Such a decomposition is unique.

Proof.

Thus, the basis allows you to uniquely associate each vector with a triple of numbers - the coefficients of the expansion of this vector in terms of the vectors of the basis: . The converse is also true, each triple of numbers x, y, z using the basis, you can match the vector if you make a linear combination .

If the basis and , then the numbers x, y, z called coordinates vectors in the given basis. The vector coordinates denote .

CARTESIAN COORDINATE SYSTEM

Let a point be given in space O and three non-coplanar vectors.

Cartesian coordinate system in space (on a plane) is called the set of a point and a basis, i.e. set of a point and three non-coplanar vectors (2 non-collinear vectors) outgoing from this point.

Dot O called the origin; straight lines passing through the origin in the direction of the basis vectors are called coordinate axes - the abscissa, ordinate and applicate axis. The planes passing through the coordinate axes are called coordinate planes.

Consider an arbitrary point in the chosen coordinate system M. Let us introduce the concept of a point coordinate M. The vector that connects the origin to the point M. called radius vector points M.

A vector in the selected basis can be associated with a triple of numbers - its coordinates: .

Point radius vector coordinates M. called coordinates of point M. in the considered coordinate system. M(x,y,z). The first coordinate is called the abscissa, the second is the ordinate, and the third is the applicate.

The Cartesian coordinates on the plane are defined similarly. Here the point has only two coordinates - the abscissa and the ordinate.

It is easy to see that for a given coordinate system, each point has certain coordinates. On the other hand, for each triplet of numbers, there is a single point that has these numbers as coordinates.

If the vectors taken as a basis in the chosen coordinate system have unit length and are pairwise perpendicular, then the coordinate system is called Cartesian rectangular.

It is easy to show that .

The direction cosines of a vector completely determine its direction, but say nothing about its length.

BASIC CONCEPTS OF VECTOR ALGEBRA

Scalar and vector quantities

From the elementary physics course, it is known that some physical quantities, such as temperature, volume, body mass, density, etc., are determined only by a numerical value. Such quantities are called scalars, or scalars.

To determine some other quantities, such as force, speed, acceleration, and the like, in addition to numerical values, it is also necessary to set their direction in space. Quantities that, in addition to absolute magnitude, are also characterized by direction are called vector.

Definition A vector is a directed segment, which is defined by two points: the first point defines the beginning of the vector, and the second - its end. Therefore, they also say that a vector is an ordered pair of points.

In the figure, the vector is depicted as a straight line segment, on which the arrow marks the direction from the beginning of the vector to its end. For example, fig. 2.1.

If the beginning of the vector coincides with the point , and end with a dot , then the vector is denoted
. In addition, vectors are often denoted by one small letter with an arrow above it. . In books, sometimes the arrow is omitted, then bold type is used to indicate the vector.

Vectors are null vector which has the same start and end. It is denoted or simply .

The distance between the start and end of a vector is called its length, or module. The vector modulus is indicated by two vertical bars on the left:
, or without arrows
or .

Vectors that are parallel to one line are called collinear.

Vectors lying in the same plane or parallel to the same plane are called coplanar.

The null vector is considered collinear to any vector. Its length is 0.

Definition Two vectors
and
are called equal (Fig. 2.2) if they:
1)collinear; 2) co-directed 3) equal in length.

It is written like this:
(2.1)

From the definition of equality of vectors, it follows that with a parallel transfer of a vector, a vector is obtained that is equal to the initial one, therefore the beginning of the vector can be placed at any point in space. Such vectors (in theoretical mechanics, geometry), the beginning of which can be placed at any point in space, are called free. And it is these vectors that we will consider.

Definition Vector system
is called linearly dependent if there are such constants
, among which there is at least one other than zero, and for which equality holds.

Definition An arbitrary three non-coplanar vectors, which are taken in a certain sequence, are called a basis in space.

Definition If a
- basis and vector, then the numbers
are called the coordinates of the vector in this basis.

We will write the vector coordinates in curly brackets after the vector designation. For example,
means that the vector in some chosen basis has a decomposition:
.

From the properties of multiplication of a vector by a number and addition of vectors, an assertion follows regarding linear actions on vectors that are given by coordinates.

In order to find the coordinates of a vector, if the coordinates of its beginning and end are known, it is necessary to subtract the coordinate of the beginning from the corresponding coordinate of its end.

Linear operations on vectors

Linear operations on vectors are the operations of adding (subtracting) vectors and multiplying a vector by a number. Let's consider them.

Definition Vector product per number
is called a vector coinciding in direction with the vector , if
, which has the opposite direction, if
negative. The length of this vector is equal to the product of the length of the vector per modulo number
.

P example . Build Vector
, if
and
(Fig. 2.3).

When a vector is multiplied by a number, its coordinates are multiplied by that number..

Indeed, if , then

Vector product on the
called vector
;
- opposite direction .

Note that a vector whose length is 1 is called single(or ortho).

Using the operation of multiplying a vector by a number, any vector can be expressed in terms of a unit vector of the same direction. Indeed, dividing the vector for its length (i.e. multiplying on the ), we get a unit vector of the same direction as the vector . We will denote it
. Hence it follows that
.

Definition The sum of two vectors and called vector , which comes out of their common origin and is the diagonal of a parallelogram whose sides are vectors and (Fig. 2.4).

.

By definition of equal vectors
that's why
-triangle rule. The triangle rule can be extended to any number of vectors and thus obtain the polygon rule:
is the vector that connects the beginning of the first vector with the end of the last vector (Fig. 2.5).

So, in order to construct the sum vector, it is necessary to attach the beginning of the second to the end of the first vector, to the end of the second to attach the beginning of the third, and so on. Then the sum vector will be the vector that connects the beginning of the first of the vectors with the end of the last.

When vectors are added, their corresponding coordinates are also added

Indeed, if and
,

If the vectors
and are not coplanar, then their sum is a diagonal
a parallelepiped built on these vectors (Fig. 2.6)

,

where

Properties:

- commutativity;

- associativity;

- distributivity with respect to multiplication by a number

.

Those. a vector sum can be transformed according to the same rules as an algebraic one.

DefinitionThe difference of two vectors and is called such a vector , which, when added to the vector gives a vector . Those.
if
. Geometrically represents the second diagonal of the parallelogram built on the vectors and with a common beginning and directed from the end of the vector to the end of the vector (Fig. 2.7).

Projection of a vector onto an axis. Projection Properties

Recall the concept of a number line. A numerical axis is a straight line on which:

direction (→);

reference point (point O);

segment, which is taken as a unit of scale.

Let there be a vector
and axis . From points and let's drop the perpendiculars on the axis . Let's get the points and - point projections and (Fig. 2.8 a).

Definition Vector projection
per axle called the length of the segment
this axis, which is located between the bases of the projections of the beginning and end of the vector
per axle . It is taken with a plus sign if the direction of the segment
coincides with the direction of the projection axis, and with a minus sign if these directions are opposite. Designation:
.

O definition Angle between vector
and axis called the angle , by which it is necessary to turn the axis in the shortest way so that it coincides with the direction of the vector
.

Let's find
:

Figure 2.8 a shows:
.

On fig. 2.8 b): .

The projection of a vector onto an axis is equal to the product of the length of this vector and the cosine of the angle between the vector and the projection axis:
.

Projection Properties:

If a
, then the vectors are called orthogonal

Example . Vectors are given
,
.Then

.

Example. If the beginning of the vector
is at the point
, and end at a point
, then the vector
has coordinates:

O definition Angle between two vectors and called the smallest angle
(Fig. 2.13) between these vectors, reduced to a common beginning .

Angle between vectors and symbolically written like this: .

It follows from the definition that the angle between vectors can vary within
.

If a
, then the vectors are called orthogonal.

.

Definition. The cosines of the angles of a vector with the coordinate axes are called direction cosines of the vector. If the vector
forms angles with the coordinate axes

.