Porch 

Vector geometric projection. Projections of vectors on coordinate axes. Expand a and b in terms of basis vectors

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 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 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

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 a in question 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 .

Decision

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.

Decision

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:

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 → → .

The 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.

Decision

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.

Decision

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.

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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

(X) I J K
I
J
K

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:

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

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

Coordinate axis denoted by any letter: X, Y, Z, s, t ... Usually, a point (arbitrarily) is selected on the axis, which is called the origin and, as a rule, denoted by the letter O. Distances to other points of interest to us are measured from this point.

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

Point coordinate per axis is a number, the absolute value of which is equal to the length of the segment of the axis (in the selected scale) enclosed between the beginning 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 beginning 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 segment of the axis (in 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 just say - projection of a vector onto an axis, that is, the word scalar lowered. Vector projection 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, say the Y axis, its projection will be denoted as y (Fig. 9).

To calculate vector projection onto the 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.

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 that axis.

Figure 11 shows that a x = a Cos α

That is, the projection of the vector onto the axis is equal to the product of the vector modulus 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.

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), then when calculating projections, the angles can be counted both clockwise and counterclockwise.

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 (similarly for other axes),

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

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

Remember!

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.

Once again - FAST!

Let two vectors and be given in space. Set aside from an arbitrary point O vectors and . corner between the vectors and is called the smallest of the angles. Denoted .

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

Angle between vector and axis l understand the angle between the vectors and .

So let l is some axis and is a vector.

Denote by A 1 and B1 projections on the axis l points A and B. Let's pretend that A 1 has a coordinate x 1, a B1- coordinate x2 on axle l.

Then projection vector per axis l is called difference x 1 – x2 between the coordinates of the projections of the end and beginning of the vector onto this axis.

Projection of a vector onto an axis l we will denote .

It is clear that if the angle between the vector and the axis l sharp then x2> x 1, and the projection x2 – x 1> 0; if this angle is obtuse, then x2< x 1 and projection x2 – x 1< 0. Наконец, если вектор перпендикулярен оси l, then x2= x 1 and x2– 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 a vector onto an axis is a number or a scalar.

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

Let's look at some of the main projection properties.

LINEARLY DEPENDENT AND LINEARLY INDEPENDENT SYSTEMS OF VECTORS

Let's consider several vectors.

Linear combination of these vectors is any vector of the form , where are some numbers. The numbers are called the coefficients of the linear combination. It is also said that in this case is linearly expressed in terms of given vectors , i.e. obtained from them by linear operations.

For example, if three vectors are given, then 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 decomposed along these vectors.

The vectors are called linearly dependent, if there are such numbers, not all equal to zero, that . It is clear that the 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 proved similarly.

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

Proof.

BASIS

Basis is the collection of non-zero linearly independent vectors. The elements of the basis will be denoted by .

In the previous subsection, we saw that two non-collinear vectors in the 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. Therefore, three non-coplanar vectors are called a basis in space.

The following assertion 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. Such a decomposition is unique.

Proof.

Thus, the basis allows you to uniquely associate each vector with a triple of numbers - the coefficients of the expansion of this vector in terms of the vectors of the basis: . The converse is also true, each triple of numbers x, y, z using the basis, you can match the vector if you make a linear combination .

If the basis and , then the numbers x, y, z called coordinates vectors in the given basis. The vector coordinates denote .


CARTESIAN COORDINATE SYSTEM

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

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

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

Consider an arbitrary point in the chosen coordinate system M. Let us introduce the concept of a point coordinate M. The vector that connects the origin to the point M. called radius vector points M.

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

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

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

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

If the vectors taken as a basis in the chosen 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.