projection vector on an axis is called a vector, which is obtained by multiplying the scalar projection of a vector on this axis and the unit vector of this axis. For example, if a x is scalar projection vector a on the x-axis, then a x i- its vector projection on this axis.

Denote vector projection just like 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 denote a x ( oily 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 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.

Vector coordinates are the coefficients of the only possible linear combination of basis vectors in the chosen coordinate system equal to the given vector.

where are the coordinates of the vector.

Dot product of vectors

SCOAL PRODUCT OF VECTORS[- in finite-dimensional vector space is defined as the sum of the products of the same components of the multiplied vectors.

For example, S. p. 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

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 axis. 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

.

Solving problems on the equilibrium of converging forces by constructing closed force polygons is associated with cumbersome constructions. A universal method for solving such problems is the transition to determining the projections of given forces on the coordinate axes and operating with these projections. The axis is called a straight line, which is assigned a certain direction.

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

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

Consider a number of cases of projecting forces onto an axis:

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

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

1. Fx = F cosα

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 cosα = F cos180 0 - φ =- F cos phi.

Force projection F per axle oh in this case is negative.

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

Projection of force F on the axis X zero

F x= F cos 90° = 0.

Force located on a plane howe(Fig. 18), can be projected onto two coordinate axes oh and OU.

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

From Δ OAB: F x= F cosα, F x= F sinα.

From Δ SLA: F x= F cos phi, F x= F sin phi.

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

The projection of the vector sum or the resultant on any axis is equal to the algebraic sum of the projections of the terms of the vectors on the same axis.

Consider 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

Drop from the vertices of the force polygon onto 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 terms of vectors. Their projections enter the above equation with the appropriate sign.

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

The axis is the direction. Hence, the projection onto an axis or onto a directed line is considered the same. Projection can be algebraic or geometric. In geometric terms, the projection of a vector onto an axis is understood as a vector, and in algebraic terms, it is a number. That is, the concepts of the projection of a vector on an axis and the numerical projection of a vector on an axis are used.

If we have an axis L and a non-zero vector A B → , then we can construct a vector A 1 B 1 ⇀ , denoting the projections of its points A 1 and B 1 .

A 1 B → 1 will be the projection of the vector A B → onto L .

Definition 1

The projection of the vector onto the axis a vector is called, the beginning and end of which are projections of the beginning and end of the given vector. n p L A B → → it is customary to denote the projection of A B → onto L . To construct a projection on L, drop the perpendiculars on L.

Example 1

An example of the projection of a vector onto an axis.

On the coordinate plane O x y, a point M 1 (x 1, y 1) is specified. It is necessary to build projections on O x and O y for the image of the radius vector of the point M 1 . Let's get the coordinates of the vectors (x 1 , 0) and (0 , y 1) .

If we are talking about the projection of a → onto a non-zero b → or the projection of a → onto the direction b → , then we mean the projection of a → onto the axis with which the direction b → coincides. The projection a → onto the line defined by b → is denoted n p b → a → → . It is known that when the angle is between a → and b → , we can consider n p b → a → → and b → codirectional. In the case when the angle is obtuse, n p b → a → → and b → are oppositely directed. In the situation of perpendicularity a → and b → , and a → is zero, the projection of a → along the direction b → is a zero vector.

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

Definition 2

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

The numerical projection of A B → onto L is denoted n p L A B → , and a → onto b → - n p b → a → .

Based on the formula, we get n p b → a → = a → · cos a → , b → ^ , whence a → is the length of the vector a → , a ⇀ , b → ^ is the angle between the vectors a → and b → .

We get the formula for calculating the numerical projection: n p b → a → = a → · cos a → , b → ^ . It is applicable for known lengths a → and b → and the angle between them. The formula is applicable for known coordinates a → and b → , but there is a simplified version of it.

Example 2

Find out the numerical projection a → onto a straight line in the direction b → with the length a → equal to 8 and the angle between them is 60 degrees. By condition we have a ⇀ = 8 , a ⇀ , b → ^ = 60 ° . So, we substitute the numerical values ​​into the formula n p b ⇀ a → = a → · cos a → , b → ^ = 8 · cos 60 ° = 8 · 1 2 = 4 .

Answer: 4.

With known cos (a → , b → ^) = a ⇀ , b → a → · b → , we have a → , b → as the scalar product of a → and b → . Following from the formula n p b → a → = a → · cos a ⇀ , b → ^ , we can find the numerical projection a → directed along the vector b → and get n p b → a → = a → , b → b → . The formula is equivalent to the definition given at the beginning of the clause.

Definition 3

The numerical projection of the vector a → on the axis coinciding in direction with b → is the ratio of the scalar product of the vectors a → and b → to the length b → . The formula n p b → a → = a → , b → b → is applicable for finding the numerical projection of a → onto a straight line coinciding in direction with b → , with known a → and b → coordinates.

Example 3

Given b → = (- 3 , 4) . Find the numerical projection a → = (1 , 7) onto L .

Solution

On the coordinate plane n p b → a → = a → , b → b → has the form n p b → a → = a → , b → b → = a x b x + a y b y b x 2 + b y 2 , with a → = (a x , a y ) and b → = b x , b y . To find the numerical projection of the vector a → onto the L axis, you need: n p L a → = n p b → a → = a → , b → b → = a x b x + a y b y b x 2 + b y 2 = 1 (- 3) + 7 4 (- 3) 2 + 4 2 = 5 .

Answer: 5.

Example 4

Find the projection a → onto L , coinciding with the direction b → , where there are a → = - 2 , 3 , 1 and b → = (3 , - 2 , 6) . A three-dimensional space is given.

Solution

Given a → = a x , a y , a z and b → = b x , b y , b z calculate the scalar product: a ⇀ , b → = a x b x + a y b y + a z b z . We find the length b → by the formula b → = b x 2 + b y 2 + b z 2. It follows that the formula for determining the numerical projection a → will be: n p b → a ⇀ = a → , b → b → = a x b x + a y b y + a z b z b x 2 + b y 2 + b z 2 .

We substitute numerical values: n p L a → = n p b → a → = (- 2) 3 + 3 (- 2) + 1 6 3 2 + (- 2) 2 + 6 2 = - 6 49 = - 6 7 .

Answer: - 6 7 .

Let's look at the connection between a → on L and the length of the projection of a → on L . Draw an axis L by adding a → and b → from a point to L , after which we draw a perpendicular line from the end of a → to L and project onto L . There are 5 image variations:

The first the case when a → = n p b → a → → means a → = n p b → a → → , hence n p b → a → = a → cos (a , → b → ^) = a → cos 0 ° = a → = n p b → a → → .

Second case implies the use of n p b → a → ⇀ = a → cos a → , b → , so n p b → a → = a → cos (a → , b →) ^ = n p b → a → → .

Third case explains that when n p b → a → → = 0 → we get n p b ⇀ a → = a → cos (a → , b → ^) = a → cos 90 ° = 0, then n p b → a → → = 0 and n p b → a → = 0 = n p b → a → → .

Fourth case shows n p b → a → → = a → cos (180 ° - a → , b → ^) = - a → cos (a → , b → ^) , follows n p b → a → = a → cos (a → , b → ^) = - n p b → a → → .

Fifth case shows a → = n p b → a → → , which means a → = n p b → a → → , hence we have n p b → a → = a → cos a → , b → ^ = a → cos 180 ° = - a → = - n p b → a → .

Definition 4

The numerical projection of the vector a → on the axis L , which is directed like b → , has the meaning:

• the length of the projection of the vector a → onto L provided that the angle between a → and b → is less than 90 degrees or equal to 0: n p b → a → = n p b → a → → with the condition 0 ≤ (a → , b →) ^< 90 ° ;
• zero under the condition of perpendicularity a → and b → : n p b → a → = 0 when (a → , b → ^) = 90 ° ;
• the length of the projection a → onto L, times -1 when there is an obtuse or flattened angle of the vectors a → and b → : n p b → a → = - n p b → a → → with the 90° condition< a → , b → ^ ≤ 180 ° .

Example 5

Given the length of the projection a → onto L , equal to 2 . Find the numerical projection a → given that the angle is 5 π 6 radians.

Solution

It can be seen from the condition that this angle is obtuse: π 2< 5 π 6 < π . Тогда можем найти числовую проекцию a → на L: n p L a → = - n p L a → → = - 2 .

Answer: - 2.

Example 6

Given a plane O x y z with the length of the vector a → equal to 6 3 , b → (- 2 , 1 , 2) with an angle of 30 degrees. Find the coordinates of the projection a → onto the L axis.

Solution

First, we calculate the numerical projection of the vector a → : n p L a → = n p b → a → = a → cos (a → , b →) ^ = 6 3 cos 30 ° = 6 3 3 2 = 9 .

By condition, the angle is acute, then the numerical projection a → = is the length of the projection of the vector a → : n p L a → = n p L a → → = 9 . This case shows that the vectors n p L a → → and b → are co-directed, which means that there is a number t for which the equality is true: n p L a → → = t · b → . From here we see that n p L a → → = t b → , so we can find the value of the parameter t: t = n p L a → → b → = 9 (- 2) 2 + 1 2 + 2 2 = 9 9 = 3 .

Then n p L a → → = 3 b → with the coordinates of the projection of the vector a → onto the L axis are b → = (- 2 , 1 , 2) , where it is necessary to multiply the values ​​by 3. We have n p L a → → = (- 6 , 3 , 6). Answer: (- 6 , 3 , 6) .

It is necessary to repeat the previously studied information about the condition of vector collinearity.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

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 the 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 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 vector and the axis:

Orth axis. Decomposition of a vector in terms of coordinate vectors. Vector coordinates. Coordinate properties

Answer:

Horts of 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 orts is equal to the dimension of the coordinate system, and they are all perpendicular to each other.

In the three-dimensional case, the orts are usually denoted

AND Symbols with arrows and can also be used.

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

Decomposition of a vector in terms of coordinate vectors.

The orth of the coordinate axis is denoted by , axes - by , axes - by (Fig. 1)

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

If the vector is located in space, then the expansion in terms of 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 end coordinates: (x2 - x1; y2 - y1).

Coordinate properties.

Consider a coordinate line with the origin at the point O and a 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 the axis, their coordinates are added.

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

Scalar product of vectors. Properties.

Answer:

The scalar product of two non-zero vectors is a number,

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

Properties:

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

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

Answer:

Dot product (×) orts

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

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

Vector product of two vectors. Vector product properties.

Answer:

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

Cross product of a vector a per vector b called vector with which:

1. Perpendicular to vectors a and b

2. Has a length numerically equal to the area of ​​the parallelogram formed on a and b vectors

3. Vectors, a,b, and c form the right triple of vectors

Properties:

1.

3.

4.

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

Answer:

Vector product of coordinate vectors.

Determination of the vector product of vectors given 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.

We expand a and b in terms of 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 obtain

[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 . (one)

By the definition of a vector product, we find

= 0, = k, = - j,

= - k, = 0, = i,

= j, = - i. = 0.

Given these equalities, 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 cross product of two vectors given 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 remembering:

Usually formula (3) is written even shorter: