Projecting vectors onto axes. Projection (geometric, algebraic) of a vector onto an axis. Properties of projections. Types of projections by definition vector projection

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).

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 a 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 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.

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.

Vector coordinates— coefficients of the only possible linear combination of basis vectors in the selected coordinate system, equal to the given 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

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 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

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 specifying 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.

We 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 right triangle MNA, 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.

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 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 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. The coordinates of the vector 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 unique 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.

A vector description of movement is useful, since in one drawing you can always depict many different vectors and get a visual “picture” of movement before your eyes. However, using a ruler and a protractor every time to perform operations with vectors is very labor-intensive. Therefore, these actions reduce to actions with positive and negative numbers– projections of vectors.

Projection of the vector onto the axis called a scalar quantity equal to the product of the modulus of the projected vector and the cosine of the angle between the directions of the vector and the selected coordinate axis.

The left drawing shows a displacement vector, the module of which is 50 km, and its direction forms obtuse angle 150° with the direction of the X axis. Using the definition, we find the projection of the displacement on the X axis:

sx = s cos(α) = 50 km cos(150°) = –43 km

Since the angle between the axes is 90°, it is easy to calculate that the direction of movement forms an acute angle of 60° with the direction of the Y axis. Using the definition, we find the projection of displacement on the Y axis:

sy = s cos(β) = 50 km cos(60°) = +25 km

As you can see, if the direction of the vector forms an acute angle with the direction of the axis, the projection is positive; if the direction of the vector forms an obtuse angle with the direction of the axis, the projection is negative.

The right drawing shows a velocity vector, the module of which is 5 m/s, and the direction forms an angle of 30° with the direction of the X axis. Let's find the projections:

υx = υ cos(α) = 5 m/s cos( 30°) = +4.3 m/s
υy = υ · cos(β) = 5 m/s · cos( 120°) = –2.5 m/s

It is much easier to find projections of vectors on axes if the projected vectors are parallel or perpendicular to the selected axes. Please note that for the case of parallelism, two options are possible: the vector is co-directional to the axis and the vector is opposite to the axis, and for the case of perpendicularity there is only one option.

The projection of a vector perpendicular to the axis is always zero (see sy and ay in the left drawing, and sx and υx in the right drawing). Indeed, for a vector perpendicular to the axis, the angle between it and the axis is 90°, so the cosine equal to zero, which means the projection is equal to zero.

The projection of a vector codirectional with the axis is positive and equal to its absolute value, for example, sx = +s (see left drawing). Indeed, for a vector codirectional with the axis, the angle between it and the axis is zero, and its cosine is “+1”, that is, the projection is equal to the length of the vector: sx = x – xo = +s .

The projection of the vector opposite to the axis is negative and equal to its absolute value, taken with a minus sign, for example, sy = –s (see the right drawing). Indeed, for a vector opposite to the axis, the angle between it and the axis is 180°, and its cosine is “–1”, that is, the projection is equal to the length of the vector taken with a negative sign: sy = y – yo = –s .

The right-hand sides of both drawings show other cases where the vectors are parallel to one of the coordinate axes and perpendicular to the other. We invite you to make sure for yourself that in these cases, too, the rules formulated in the previous paragraphs are followed.