Let two vectors and be given in space. Let's postpone from an arbitrary point O vectors and . Angle between vectors is called the smallest of the angles. Designated .

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

At an angle between the vector and the axis l understand the angle between the vectors and .

So let l is some axis and is a vector.

Let us denote by A 1 And B 1 projections onto the axis l respectively points A And B. Let's assume that A 1 has a coordinate x 1, A B 1– coordinate x 2 on the axis l.

Then projection vector per axis l called difference x 1 – x 2 between the coordinates of the projections of the end and beginning of the vector onto this axis.

Projection of the vector onto the axis l we will denote .

It is clear that if the angle between the vector and the axis l spicy then x 2> x 1, and projection x 2 – x 1> 0; if this angle is obtuse, then x 2< x 1 and projection x 2 – x 1< 0. Наконец, если вектор перпендикулярен оси l, That x 2= x 1 And x 2– 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 the vector onto the axis is a number or a scalar.

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

Let's look at some basic properties of projections.

LINEARLY DEPENDENT AND LINEARLY INDEPENDENT VECTOR SYSTEMS

Let's consider several vectors.

Linear combination of these vectors is any vector of the form , where are some numbers. The numbers are called linear combination coefficients. They also say that in this case it is linearly expressed through these vectors, i.e. is obtained from them using linear actions.

For example, if three vectors are given, then the following 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 laid out along these vectors.

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

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

Proof.

BASIS

Basis is a collection of non-zero linearly independent vectors. We will denote the elements of the basis by .

In the previous paragraph, we saw that two non-collinear vectors on a 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. Consequently, we call three non-coplanar vectors a basis in space.

The following statement 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. This is the only decomposition.

Proof.

Thus, the basis allows each vector to be uniquely associated with a triple of numbers - the coefficients of the expansion of this vector into the basis vectors: . The converse is also true, for every three numbers x, y, z using the basis, you can compare the vector if you make a linear combination .

If the basis and , then the numbers x, y, z are called coordinates vector in a given basis. Vector coordinates are denoted by .

CARTESIAN COORDINATE SYSTEM

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

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

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

Consider an arbitrary point in the selected coordinate system M. Let us introduce the concept of point coordinates M. Vector connecting the origin to a point M. called radius vector points M.

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

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

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

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

If the vectors taken as a basis in the selected 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.

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

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

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

2. How is the displacement vector of a 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 body coordinates.

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 a point decreases over time, then the projection of the displacement vector onto 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 guide of the axis itself.

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

4. If the displacement vector is parallel to the X axis, then the modulus of the vector’s projection onto this axis is equal to the modulus of the vector itself, and its projection onto 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 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 is equal in absolute value to the length of the vector s 1 . With such a movement, 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 magnitude to the length of the vector s 1 . With such a movement, 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 magnitude to the length of the vector s 3 . With such a movement, 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 magnitude to the length of the vector s 4 . With such a movement, 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 on the X axis is negative and equal in magnitude to the length of the vector s 5 . With such a movement, the X coordinate of the body will decrease by the length of the vector s 5.

6. If the distance traveled is large, then can the displacement module 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 as close as desired to the initial position of the body. If the final and initial positions of the body coincide, the displacement module will be equal to zero.

7. Why is the vector of movement 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 vector of movement of the body, we can determine the coordinates of the body, i.e. the position of the body at any moment in time, and knowing only the distance traveled, we cannot determine the coordinates of the body, because we have no information about the direction of movement, but can only judge the length of the path traveled at a given time.

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 to design.

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 directions, their magnitudes 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 plotted, for example, from the point of intersection of the axes. Let its length be equal to one. Then its magnitude

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 a geometric sense, the projection of a vector onto an axis is a vector, and in an algebraic sense 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 we 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 a marked right triangle, from the definition of the 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.

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

According to the definition we have

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

In space, definition 2 of the vector projection onto the axis remains in force, only the projection direction is specified 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 a vector is a projection of this vector onto the axis defined by the corresponding basis vector. In this case, the design direction is specified 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 side

From we find

Substituting AC into equality (2), we obtain

Since the numbers x and the same sign in both cases under consideration ((Fig. 5, a) ; (Fig. 5, b), then from equality (4) it follows

Comment. In what follows, we will consider only the orthogonal projection of the vector onto the axis and therefore the word “ort” (orthogonal) will be omitted from the notation.

Let us present a number of formulas that are used later in solving problems.

a) Projection of the vector onto the 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 is the distance from point M to plane b (Fig. 6).

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

• G) The 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 and be projections of points A and B onto, then

e) Distance from a point to a line.

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

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

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

f) The angle between a straight line and a plane.

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

As is known, the angle μ between a straight line l and its projection onto plane b is called the angle between the line and the plane. We have

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