Definition of projection on coordinate axes. Projection of force onto the axis. Projection of the vector sum of forces onto the axis. Classification of vector projections

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

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.

Scalar 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

Answer:

Projection properties:

Vector Projection Properties

Property 1.

The projection of the sum of two vectors onto an axis is equal to the sum of the projections of vectors onto the same axis:

This property allows you to replace the projection of a sum of vectors with the sum of their projections and vice versa.

Property 2. If a vector is multiplied by the number λ, then its projection onto the axis is also multiplied by this number:

Property 3.

The projection of the vector onto the l axis is equal to the product of the modulus of the vector and the cosine of the angle between the vector and the axis:

Orth axis. Decomposition of a vector in coordinate unit vectors. Vector coordinates. Coordinate Properties

Answer:

Unit vectors of the axes.

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

In the three-dimensional case, the unit vectors are usually denoted

And Arrow symbols and may also be used.

Moreover, in the case of a right-handed coordinate system, following formulas with vector products of vectors:

Decomposition of a vector in coordinate unit vectors.

The unit vector of the coordinate axis is denoted by , axes by , axes by (Fig. 1)

For any vector that lies in the plane, the following expansion takes place:

If the vector located in space, then the expansion in unit vectors of the coordinate axes has the form:

Vector coordinates:

To calculate the coordinates of a vector, knowing the coordinates (x1; y1) of its beginning A and the coordinates (x2; y2) of its end B, you need to subtract the coordinates of the beginning from the coordinates of the end: (x2 – x1; y2 – y1).

Properties of coordinates.

Consider a coordinate line with the origin at point O and the unit vector i. Then for any vector a on this line: a = axi.

The number ax is called the coordinate of the vector a on the coordinate axis.

Property 1. When adding vectors on an axis, their coordinates are added.

Property 2. When a vector is multiplied by a number, its coordinate is multiplied by that number.

Dot product of vectors. Properties.

Answer:

The scalar product of two non-zero vectors is the number

equal to the product of these vectors and the cosine of the angle between them.

Properties:

1. The scalar product has the commutative property: ab=ba

Scalar product coordinate unit vectors. Determination of the scalar product of vectors specified by their coordinates.

Answer:

Dot product (×) of unit vectors

(X)	I	J	K
I
J
K

Determination of the scalar product of vectors specified by their coordinates.

The scalar product of two vectors and given by their coordinates can be calculated using the formula

The cross product of two vectors. Properties of a vector product.

Answer:

Three non-coplanar vectors form a right-handed triple if, from the end of the third, the rotation from the first vector to the second is made counterclockwise. If clockwise, then left. If not, then in the opposite direction ( show how he showed with “handles”)

Cross product of a vector A to vector b called a vector from which:

1. Perpendicular to vectors A And b

2. Has length, numerically equal to the area parallelogram formed on a And b vectors

3. Vectors, a ,b, And c form a right-hand triplet of vectors

Properties:

Vector product of coordinate unit vectors. Determination of the vector product of vectors specified by their coordinates.

Answer:

Vector product of coordinate unit vectors.

Determination of the vector product of vectors specified by their coordinates.

Let the vectors a = (x1; y1; z1) and b = (x2; y2; z2) be given by their coordinates in the rectangular Cartesian coordinate system O, i, j, k, and the triple i, j, k is right-handed.

Let's expand a and b into basis vectors:

a = x 1 i + y 1 j + z 1 k, b = x 2 i + y 2 j + z 2 k.

Using the properties of the vector product, we get

[A; b] = =

= x 1 x 2 + x 1 y 2 + x 1 z 2 +

+ y 1 x 2 + y 1 y 2 + y 1 z 2 +

+ z 1 x 2 + z 1 y 2 + z 1 z 2 . (1)

By the definition of a vector product we find

= 0, = k, = - j,

= - k, = 0, = i,

= j, = - i. = 0.

Taking these equalities into account, formula (1) can be written as follows:

[A; b] = x 1 y 2 k - x 1 z 2 j - y 1 x 2 k + y 1 z 2 i + z 1 x 2 j - z 1 y 2 i

[A; b] = (y 1 z 2 - z 1 y 2) i + (z 1 x 2 - x 1 z 2) j + (x 1 y 2 - y 1 x 2) k. (2)

Formula (2) gives an expression for the vector product of two vectors specified by their coordinates.

The resulting formula is cumbersome. Using the notation of determinants, you can write it in another form that is more convenient for memorization:

Usually formula (3) is written even shorter:

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

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

Coordinate axis denoted 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, denoted by the letter O. From this point the distances to other points of interest to us are measured.

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

Point coordinate on axis- this is a number whose absolute value is equal to the length of the axis segment (on the selected scale) contained 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.

Scalar projection of a vector onto an axis- This 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. Important! Usually instead of the expression scalar projection of a vector onto an axis they simply say - projection of the vector onto the axis, that is, the word scalar lowered. Vector projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a lower (as a rule) index 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, say, the Y axis, its projection will be denoted a y (Fig. 9).

To calculate projection of the vector onto the 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.

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 this axis.

From Figure 11 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 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.

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 (similar to other axes),

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

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

Remember!

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.

Once again - by heart!

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 is zero, which means the projection is 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 module 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.