Calculation of the projection of a vector onto an axis. Projection of force onto the axis. Projection of the vector sum of forces onto the axis. Scalar and vector quantities

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

1. Value of vector and scalar…………………………………….4

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

3. Projection of the vector onto the axis………………………………………………………...6

4. Basic formula vector algebra……………………………..8

5. Calculation of the modulus of a vector from its projections…………………...9

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

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

Introduction:

Physics is inextricably linked with mathematics. Mathematics gives physics the means and techniques for a general and precise expression of the relationship between physical quantities, which are discovered as a result of experiment or theoretical research. After all, the main method of research in physics is experimental. This means that a scientist reveals calculations using measurements. Denotes the relationship between various physical quantities. Then, everything is translated into the language of mathematics. Formed mathematical model. Physics is a science that studies the simplest and at the same time the most general laws. The task of physics is to create such a picture in our minds physical world, which 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 - to choose the main thing from all the diversity.

Physical models are mathematical, but mathematics is not their basis. Quantitative relationships between physical quantities are determined 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. Meaning of vector and 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, for example, force, position, speed, acceleration, torque, momentum, electric and magnetic field strength. They can be contrasted with other quantities such as mass, volume, pressure, temperature and density, which can be described by an ordinary number, and are called " scalars".

They are written either in regular font letters or in numbers (a, b, t, G, 5, −7....). Scalar quantities can be positive or negative. At the same time, some objects of study may have such properties that full description For which knowledge of only a numerical measure turns out to be insufficient, it is also necessary to characterize these properties by 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 letter in regular (non-bold) font, but with an arrow above it:

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

The modulus of a vector, that is, the length of a directed straight line segment, is denoted by the same letters as the vector itself, but in normal (not bold) writing and without an arrow above them, or in exactly the same way as a 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 simultaneously characterized by both magnitude and direction.

There are also no positive and negative vectors. But 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 notation is true a= b. It should also be borne in mind that the vector symbol may be preceded by a minus sign, for example - c, however, this sign symbolically indicates that the vector -c has the same module as the vector c, but is directed in the opposite direction.

Vector -c is called the opposite (or inverse) of vector c.
In physics, 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 significant.

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

Axis- This is a straight line that is given some direction.
An axis is designated by some letter: X, Y, Z, s, t... Usually a point is selected (arbitrarily) on the axis, which is called the origin and, as a rule, is designated by the letter O. From this point the distances to other points of interest to us are measured.

Projection of a point on an axis is the base of a perpendicular drawn from this point onto a given axis. That is, the projection of a point onto the axis is a point.

Point coordinate on this axis is called a number, absolute value which is equal to the length of the axis segment (on the selected scale) enclosed between the origin 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 origin and with a minus sign if in the opposite direction.

3. Projection of the vector onto the 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 vector a onto the X axis, then a x ·i is its vector projection onto this axis.

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

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

Scalar projection vector per axis is called number, the absolute value of which is equal to the length of the axis segment (on 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 they simply say - projection. The projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a lower index (as a rule) 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 by a x. When projecting the same vector onto another axis, if the axis is Y, its projection will be denoted as y.

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

a x = x k − x n.

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

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

and equal to zero if x k equals 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 this axis.

From the figure it is clear 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 the obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

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

To find the projection of a vector onto an axis, the modulus 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.

Let's project vector a on the X and Y axes of the rectangular coordinate system. Let's find the vector projections of vector a on these axes:

a x = a x ·i, and y = a y ·j.

But in accordance with the rule of vector addition

a = a x + a y.

a = a x i + a y j.

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

Vector projections a x and a y are called components or components of the vector a. The operation we performed is called the decomposition of a 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 be written like this.

In this article we will understand the projection of a vector onto an axis and learn how to find the numerical projection of a vector. First, we will give a definition of the projection of a vector onto an axis, introduce notation, and also provide a graphic illustration. After this, we will voice the definition of the numerical projection of a vector onto an axis, consider methods for finding it, and show solutions to several examples in which it is necessary to find the numerical projection of a vector on an axis.

Page navigation.

Projection of a vector onto an axis – definition, designation, illustrations, example.

Let's start with some general information.

An axis is a straight line for which a direction is indicated. Thus, the projection of a vector onto an axis and the projection of a vector onto a directed line are one and the same.

The projection of a vector onto an axis can be considered in two senses: geometric and algebraic. IN geometric sense the projection of a vector onto an axis is a vector, and in algebraic it is a number. Often this distinction is not stated explicitly but is understood from the context. We will not ignore this distinction: we will use the term “” when we are talking about the projection of a vector in a geometric sense, and the term “” when we are talking about the projection of a vector in an algebraic sense (the next paragraph of this article is devoted to the numerical projection of a vector onto an axis) .

Now we move on to determining the projection of the vector onto the axis. To do this, it won't hurt to repeat.

Let us be given an L axis and a nonzero vector on a plane or in three-dimensional space. Let us denote the projections of points A and B onto line L, respectively, as A 1 and B 1 and construct a vector. Looking ahead, let's say that a vector is a projection of a vector onto the L axis.

Definition.

Projection of a vector onto an axis is a vector whose beginning and end are, respectively, the projections of the beginning and end of a given vector.

The projection of the vector onto the L axis is denoted as .

To construct a projection of a vector onto the L axis, you need to lower perpendiculars from points A and B onto the directed straight line L - the bases of these perpendiculars will give the beginning and end of the desired projection.

Let's give an example of a vector projection onto an axis.

Let a rectangular coordinate system Oxy be introduced on the plane and a certain point be specified. Let us depict the radius vector of point M 1 and construct its projections onto the coordinate axes Ox and Oy. Obviously, they are vectors with coordinates and, respectively.

You can often hear about the projection of one vector onto another non-zero vector, or the projection of a vector onto the direction of a vector. In this case, we mean a projection of the vector onto a certain axis, the direction of which coincides with the direction of the vector (in general, there are infinitely many axes whose directions coincide with the direction of the vector). The projection of a vector onto a straight line, the direction of which is determined by the vector, is denoted as .

Note that if the angle between the vectors and is acute, then the vectors and are codirectional. If the angle between the vectors and is obtuse, then the vectors and are oppositely directed. If the vector is zero or perpendicular to the vector, then the projection of the vector onto the straight line, the direction of which is specified by the vector, is the zero vector.

Numerical projection of a vector onto an axis - definition, designation, examples of location.

The numerical characteristic of the projection of a vector onto an axis is the numerical projection of this vector onto a given axis.

Definition.

Numerical projection of a vector onto an axis is a number that is equal to the product of the length of a given vector and the cosine of the angle between this vector and the vector that determines the direction of the axis.

The numerical projection of the vector onto the L axis is denoted as (without the arrow on top), and the numerical projection of the vector onto the axis defined by the vector is denoted as .

In this notation, the definition of the numerical projection of a vector onto a line directed as a vector will take the form , where is the length of the vector, is the angle between the vectors and .

So we have the first formula for calculating the numerical projection of a vector: . This formula is applied when the length of the vector and the angle between the vectors and are known. Undoubtedly, this formula can be applied when the coordinates of the vectors and relative to a given rectangular coordinate system are known, but in this case it is more convenient to use another formula, which we will obtain below.

Example.

Calculate the numerical projection of a vector onto a line directed as a vector if the length of the vector is 8 and the angle between the vectors and is equal to .

Solution.

From the problem conditions we have . All that remains is to apply the formula to determine the required numerical projection of the vector:

Answer:

We know that , where is the scalar product of vectors and . Then the formula , which allows us to find the numerical projection of a vector onto a line directed like a vector, will take the form . That is, we can formulate another definition of the numerical projection of a vector onto an axis, which is equivalent to the definition given at the beginning of this paragraph.

Definition.

Numerical projection of a vector onto an axis, the direction of which coincides with the direction of the vector, is the ratio of the scalar product of the vectors and to the length of the vector.

It is convenient to use the resulting formula of the form to find the numerical projection of a vector onto a straight line, the direction of which coincides with the direction of the vector, when the coordinates of the vectors and are known. We will show this when solving examples.

Example.

It is known that the vector specifies the direction of the L axis. Find the numerical projection of the vector onto the L axis.

Solution.

The formula in coordinate form is , where and . We use it to find the required numerical projection of the vector onto the L axis:

Answer:

Example.

With respect to the rectangular coordinate system Oxyz, two vectors are given in three-dimensional space And . Find the numerical projection of the vector onto the L axis, the direction of which coincides with the direction of the vector.

Solution.

By vector coordinates And you can calculate the scalar product of these vectors: . The length of a vector from its coordinates is calculated using the following formula . Then the formula for determining the numerical projection of the vector onto the L axis in coordinates has the form .

Let's apply it:

Answer:

Now let's get the connection between the numerical projection of the vector onto the L axis, the direction of which is determined by the vector, and the length of the vector's projection onto the L axis. To do this, we depict the L axis, plot the vectors and from a point lying on L, lower a perpendicular from the end of the vector to the straight line L and construct a projection of the vector onto the L axis. Depending on the measure of the angle between the vectors and the following five options are possible:

In the first case, it is obvious that , therefore, then .

In the second case, in the marked right triangle from the definition of cosine of an angle we have , hence, .

In the third case, it is obvious that, and , therefore, and .

In the fourth case, from the definition of the cosine of an angle it follows that , where .

In the latter case, therefore, then
.

The following definition of the numerical projection of a vector onto an axis combines the results obtained.

Definition.

Numerical projection of the vector onto the L axis, directed as a vector, this is

Example.

The length of the projection of the vector onto the L axis, the direction of which is specified by the vector, is equal to . What is the numerical projection of the vector onto the L axis if the angle between the vectors and is equal to radians.

A. The projection of point A onto the PQ axis (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, shown in Fig. 5.

A vector whose beginning is the projection of the beginning and whose end is the projection of the end of this vector is called the projection of vector A B onto the PQ axis. It is written like this;

Sometimes the PQ indicator is not written at the bottom; this is done in cases where, besides PQ, there is no other OS on which it could be designed.

With. Theorem I. The magnitudes of vectors lying on one axis are related as the magnitudes of their projections onto any axis.

Let the axes and vectors indicated in Fig. 6 be given. From the similarity of the triangles it is clear 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 sides, then their values have different signs, therefore,

Obviously, the magnitudes of the projections also have different signs:

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

Reversing the signs, we get

If the vectors are equally directed, then their projections will also be of the same direction; there will be no minus signs in formulas (2) and (3). Substituting (2) and (3) into equality (1), we immediately obtain equality (4). So, the theorem has been proven for all cases.

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

Projection 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 – scalar projection vector A to the X axis, then a x i- its vector projection onto this axis.

Let's denote vector projection the same 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 we denote A x( fat a letter denoting a vector and a subscript of the axis name) or (a non-bold letter denoting a vector, but with an arrow at the top (!) and a subscript of the axis name).

To calculate the projection vector on an axis (for example, the X axis), it is necessary to subtract the coordinate of the starting point from the coordinate of its end point, that is
a x = x k − x n.
The projection of a vector onto an axis is a number. 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 equals 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 this axis.

From the figure it is clear 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 the obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

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

Vector coordinates are the coefficients of the only possible linear combination of basis vectors in the selected coordinate system, equal to this vector.

where are the coordinates of the vector.

Dot product vectors

Scalar product of vectors[- in finite-dimensional vector space is defined as the sum of the products of identical components being multiplied vectors.

For example, S.p.v. a = (a 1 , ..., a n) And b = (b 1 , ..., b n):

(a , b ) = a 1 b 1 + a 2 b 2 + ... + a n b n

Solving problems on the equilibrium of converging forces by constructing closed force polygons involves cumbersome constructions. A universal method for solving such problems is to move on to determining the projections of given forces onto the coordinate axes and operating with these projections. An axis is a straight line that is assigned a specific direction.

The projection of a vector onto an axis is a scalar quantity, 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.

A vector projection is considered positive if the direction from the beginning of the projection to its end coincides with the positive direction of the axis. A vector projection 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 onto the coordinate axis is equal to the product of the force modulus and the cosine of the angle between the force vector and the positive direction of the axis.

Let's consider a number of cases of projecting forces onto an axis:

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

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

1. F x = F cos α

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 cos α = F cos180 0 - φ =- F cos φ.

Projection of force F per axis oh in this case it is negative.

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

Projection of force F onto the axis X equal to zero

F x = F cos 90° = 0.

Force located on the plane howe(Fig. 18), can be projected on two coordinate axes Oh And oh.

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

From Δ OAV: F x = F cos α, F x = F sin α.

From Δ OAS: F x = F cos φ, F x = F sin φ.

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

The projection of a vector sum or resultant onto any axis is equal to the algebraic sum of the projections of the summands of the vectors onto the same axis.

Consider the 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

Let us drop from the vertices of the force polygon to 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 vector terms. Their projections enter the above equation with the corresponding sign.

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