First, let's remember what is coordinate axis, projection of a point onto an axis and coordinates of a point on the axis.

Coordinate axis is a straight line that is given a direction. You can think of it as a vector with an infinitely large modulus.

Coordinate axis denoted by any letter: X, Y, Z, s, t ... Usually, a point (arbitrarily) is selected on the axis, which is called the origin and, as a rule, denoted by the letter O. Distances to other points of interest to us are measured from this point.

Projection of a point onto an axis- this is the base of the perpendicular dropped from this point to the given axis (Fig. 8). That is, the projection of a point onto the axis is a point.

Point coordinate per axis is a number, the absolute value of which 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.

Scalar projection of a vector onto an axis- This 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. Important! Usually instead of the expression scalar projection of a vector onto an axis they just say - projection of a vector onto an axis, that is, the word scalar lowered. Vector projection 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, say the Y axis, its projection will be denoted as y (Fig. 9).

To calculate vector projection onto the 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.

We must remember: the scalar projection of a vector onto an axis (or, simply, the projection of a vector onto an axis) is a number (not a vector)! Moreover, the projection can be positive if the value x k is greater than the value x n, negative if the value x k is less than the value x n and equal to zero if x k is equal to x n (Fig. 10).

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.

Figure 11 shows that a x = a Cos α

That is, the projection of the vector onto the axis is equal to the product of the vector modulus and the cosine of the angle between axis direction and vector direction. If the angle is acute, then Cos α > 0 and a x > 0, and if it is obtuse, then the cosine of the 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), then when calculating projections, the angles can be counted both clockwise and counterclockwise.

When solving problems, the following properties of projections will often be used: if

a = b + c +…+ d, then a x = b x + c x +…+ d x (similarly for other axes),

a= m b, then a x = mb x (similarly for other axes).

The formula a x = a Cos α will be Often meet when solving problems, so it must be known. You need to know the rule for determining the projection by heart!

Remember!

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.

Once again - FAST!

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

.

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 full description which the knowledge of only a numerical measure turns out to be 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:

In addition, a vector is often denoted by a pair of letters (usually in capital letters), with the first letter indicating the beginning of the vector, and the second letter indicating its end.

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 chosen (arbitrarily) 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

(non-bold letter denoting a vector, but with an arrow at the top (!) and a subscript of the axis name).

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.

a. The projection of point A onto the axis PQ (Fig. 4) is the base a of the perpendicular dropped from a given point to a given axis. The axis on which we project is called the projection axis.

b. Let two axes and a vector A B be given, as shown in Fig. 5.

The vector whose beginning is the projection of the beginning and the end - the projection of the end of this vector, is called the projection of the vector A B on the PQ axis, It is written like this;

Sometimes the PQ indicator is not written at the bottom, this is done in cases where, apart from PQ, there is no other axis that could be projected on.

with. Theorem I. The values ​​of vectors lying on the same axis are related as the values ​​of their projections on any axis.

Let the axes and vectors shown in Fig. 6 be given. From the similarity of triangles, it can be seen that the lengths of the vectors are related as the lengths of their projections, i.e.

Since the vectors in the drawing are directed in different directions, their magnitudes have different values, therefore,

Obviously, the projection values ​​also have a different sign:

substituting (2) into (3) into (1), we obtain

Reversing the signs, we get

If the vectors are equally directed, then there will be one direction and their projections; there will be no minus signs in formulas (2) and (3). Substituting (2) and (3) into equality (1), we immediately obtain equality (4). Thus, the theorem is proved for all cases.

d. Theorem II. The value of the projection of a vector onto any axis is equal to the value of the vector multiplied by the cosine of the angle between the axis of projections and the axis of the vector. Let the vector be given to the axis as shown in Fig. 7. Let's construct a vector equally directed with its axis and postponed, for example, from the point of intersection of the axes. Let its length be equal to one. Then its value

In physics for grade 9 (I.K. Kikoin, A.K. Kikoin, 1999),
task №5
to chapter " CHAPTER 1. GENERAL INFORMATION ABOUT MOVEMENT».

1. What is called the projection of a vector onto the coordinate axis?

1. The projection of the vector a onto coordinate axis call the length of the segment between the projections of the beginning and end of the vector a (perpendiculars lowered from these points onto the axis) onto this coordinate axis.

2. How is the displacement vector of the body related to its coordinates?

2. The projections of the displacement vector s on the coordinate axes are equal to the change in the corresponding coordinates of the body.

3. If the coordinate of a point increases over time, then what sign does the projection of the displacement vector onto the coordinate axis have? What if it decreases?

3. If the coordinate of a point increases over time, then the projection of the displacement vector onto the coordinate axis will be positive, because in this case, we will go from the projection of the beginning to the projection of the end of the vector in the direction of the axis itself.

If the coordinate of the point decreases over time, then the projection of the displacement vector on the coordinate axis will be negative, because in this case, we will go from the projection of the beginning to the projection of the end of the vector against the directing axis itself.

4. If the displacement vector is parallel to the X axis, then what is the module of the projection of the vector onto this axis? What about the projection module of the same vector onto the Y-axis?

4. If the displacement vector is parallel to the X axis, then the module of the vector projection on this axis is equal to the module of the vector itself, and its projection on the Y axis is zero.

5. Determine the signs of the projections onto the X axis of the displacement vectors shown in Figure 22. How do the coordinates of the body change during these displacements?

5. In all of the following cases, the Y coordinate of the body does not change, and the X coordinate of the body will change as follows:

a) s 1 ;

the projection of the vector s 1 onto the X axis is negative and modulo equal to the length of the vector s 1 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 1 .

b) s 2 ;

the projection of the vector s 2 onto the X axis is positive and equal in absolute value to the length of the vector s 1 . With such a displacement, the X coordinate of the body will increase by the length of the vector s 2 .

c) s 3 ;

the projection of the vector s 3 onto the X axis is negative and equal in absolute value to the length of the vector s 3 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 3 .

d) s 4 ;

the projection of the vector s 4 onto the X axis is positive and equal in absolute value to the length of the vector s 4 . With such a displacement, the X coordinate of the body will increase by the length of the vector s 4 .

e) s 5 ;

the projection of the vector s 5 onto the X axis is negative and equal in absolute value to the length of the vector s 5 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 5 .

6. If the distance traveled is large, can the displacement modulus be small?

6. Maybe. This is due to the fact that displacement (displacement vector) is a vector quantity, i.e. is a directed straight line segment connecting the initial position of the body with its subsequent positions. And the final position of the body (regardless of the distance traveled) can be arbitrarily close to the initial position of the body. If the final and initial positions of the body coincide, the displacement modulus will be equal to zero.

7. Why is the displacement vector of a body more important in mechanics than the path it has traveled?

7. The main task of mechanics is to determine the position of the body at any time. Knowing the displacement vector of the body, we can determine the coordinates of the body, i.e. the position of the body at any time, and knowing only the distance traveled, we cannot determine the coordinates of the body, because we do not have information about the direction of movement, but we can only judge the length of the path traveled at a given time.