Golovna > Calendar chips > Zmishane TV vector. Vector vitvir vector_v

Zmishane TV vector. Vector vitvir vector_v

Let's look at TV vectors, і , folded in an advancing rank:
. Here the first two vectors are multiplied vectorially, as the result is scalarly multiplied by the third vector. Such a creation is called a vector-scalar or a mixed creation of three vectors. Zmishany tvirє deakim number.

Z'yasuemo geometric sense virazu
.

Theorem . Zmіshany dobutok 3 vectorіv dorіvnyuє obyagu paralepiped, pobudovanogo on tsіh vectors, taken zі sign "plus", yakscho tsі vectori utvoryuyut the right troіyka, і zі sign "minus", yakshcho utavlyayut left troyka.

Proof.. Let's make a parallelepiped, the edges of which are vectors , , ta vector
.

Maemo:
,
, de - the area of the parallelogram based on the vectors і ,
for the right trio of vectors and
for left, de
- The height of the parallelepiped. We take:
, then.
, de - obsyag paralepiped, adorned with vectors , і .

The dominance of the mixed creation

1. Change tvir does not change when cyclical permutations of yoga spіvmulnіnіv, tobto. .

In fact, in times, neither the circumstance of the parallelepiped nor the orientation of the ribs is changed.

2. Zmіshane tvіr not zmіnyuєtsya pіd h zmіni mіstsami signs of vector and scalar multiplication, tobto.
.

True,
і
. The sign at the right part of these equivalences is taken by the same one, that is, the trio of vectors , , і , , - one orientation.

Otzhe,
. Tse allow you to record zmіshane tvіr vectorіv
at the sight
unsigned vector scalar multiplication.

3. Zmіshane tvіr zmіnyuє sign at zmіnі mіsts whether there are two vectors-spіvmultipliers, that is.
,
,
.

Indeed, such a permutation is more common permutation of spivmultipliers in a vector creation, which changes a sign in a creation.

4. Changes in the number of non-zero vectors , і one to zero i todi, if they are coplanar.

2.12. Calculation of mixed creation in coordinate form in orthonormal basis

Give the task vectors
,
,
. Let's know їх zmіshany tvir, vikoristovuyuchi vrazi in coordinates for vector and scalar creations:

. (10)

Otriman's formula can be written shorter:

shards of the right of a part of equality (10) є rozladannya vyznachnik of the third order for the elements of the third row.

Otzhe, zmіshane tvіr vektorі v dorіvnyuє vyznachnik of the third order, folded from the coordinates of the vectorіv, which are multiplied.

2.13. Acts of additions to the mixed creation

Designated mutual orientation of vectors in space

Designated mutual orientation of vectors , і Gruntuyetsya on the upcoming mirkuvannyah. Yakscho
, then , , - Rights three; yakscho
, then , , - Liva three.

Umov’s vector complanarity

Vectors , і complanarnі tіlі і tіlki tіlki іtі, if їхнє змішане tvіr dоrіvnyuє zero (
,
,
):

vectors , , coplanar.

Designed to fit the paralepiped and the tricot pyramid

It does not matter to show that the parallelepiped, based on vectors, , і counted as
and obsyag trikutnoї piramidi, pobudovanoї on tsikh same vectors, dorivnyuє
.

example 1. Bring what vectori
,
,
coplanar.

Solution. We know the change of tvir tsikh vectorіv for the formula:

Tse means that vectors
coplanar.

butt 2. Given the vertices of the tetrahedron: (0, -2, 5), (6, 6, 0), (3, -3, 6),
(2, -1, 3). Know the length of yoga heights lowered from the top .

Solution. We know the backbone of the tetrahedron
. We take into account the formula:

Oskelki vyznachnik is more expensive to a negative number, then you need to take a minus sign in front of the formula. Otzhe,
.

Shukanu size h significant from the formula
, de S - Base area. Significantly flat S:

Oskilki

Submitting a formula
meaning
і
, taken h= 3.

Example 3. Chi appease vectors
the basis of space? Lay out vector
on the basis of vectors.

Solution. Just as vectors establish a basis for space, all stinks lie in one flat, that is. є non-coplanar. We know the sound of tvir vector_v
:
,

Also, the vectors are not coplanar and establish a basis for space. If vectors establish a basis for space, then be it a vector you can look at a linear combination of basic vectors,
,de
vector coordinates at the basis of vectors
. We know qi coordinates, adding and rozvyazshi system equal

Virishyuchi її Gauss method, maybe

Zvіdsi
. Todi .

in such a manner,
.

Example 4. The tops of the pyramid are located at the points:
,
,
,
. Calculate:

a) face area
;

b) obsyag pyramids
;

c) vector projection
off road vector
;

d) cut
;

e) check what vectors
,
,
coplanar.

Solution

a) From the designation of the vector creation, it is clear that:

We know the vector
і
, vicorist formula

,
.

For vectors, given by their projections, the vector TV is known by the formula

, de
.

For our vipadu

The value of the taken vector is known, vicorist formula

,
.

and then
(sq. Od.).

b) The improvement of three vectors by the absolute value of the increase in the volume of the parallelepiped, based on the vectors , , yak on the ribs.

Zmishane tvir are calculated according to the following formula:

We know the vector
,
,
, which run from the ribs of the pyramid, which converge to the top :

Zmіshany tvіr tsikh vektorіv

Oskіlki obsyag pіramіdі dorіvnyuє porіnі obyagu paralepiped, pobudovanogo vektoryov
,
,
, then
(Cub. od.).

c) Vicorist formula
, which means the scalar additional vector , , can be written like this:

de
or
;

or
.

For the projection of the vector
off road vector
we know the coordinates of the vectors
,
, and then, zastosovuchi formula

acceptable

d) For the meaning of kuta
variable vectors
,
, what to wash the scorching cob at the point :

Let's follow the formula of the scalar creation

e) In order to have three vectors

,
,

were coplanar, necessary and sufficient, so that their change would be equal to zero.

Our mind can
.

Again, vectors are coplanar.

On this level, we can look at two more operations with vectors: vector booth vector_vі Zmіshany tvіr vectorіv (Vіdrazu possilannya, who needs the very thing). It's nothing terrible, so sometimes it's just for total happiness, krim scalar creative vector, Need more and more. This is the vector axis of drug addiction. Might add up to the enemy, scho we climb into the net of analytic geometry. Tse not so. For whom the great mathematicians have taken little firewood, it’s better to hang out on Pinocchio. Really, the material is more wide and simple - hardly more foldable, lower than the same scalar doboot, there will be less typical tasks. Golovne in analytic geometry, like a lot of people who change their minds and already having a mess, DO NOT HAVE MERCY IN HIVISLE. Repeat like a spell, and you will be happy.

Like vectors and vibrate here far away, like glitters on the horizon, don’t be, start from the lesson Vectors for teapots, in order to learn or to gain basic knowledge about vectors. Readers can learn more about this information, I have tried to pick the most complete collection of applications, which are often used by practical robots

What will make you happy? If I'm small, then I've learned to juggling two and wrapping three in bags. It was creepy. At the same time, juggling will not happen in a flash, the shards of our eyes can be seen only space vectors, and the flat vectors from two coordinates are left behind. Why? This is how the data were already born - the vector is not the same zmіshane tvіr vektorіv is designated to practice in the trivial space. Already easier!

In this operation, just like in a scalar creation, take part two vectors. Let there be immortal letters.

diya herself be appointed let's come in rank: . Іsnuyut and іnshі options, but I also use the sound to designate a vector tvir vector in the same way, in square arms with a cross.

I immediately food: yakscho in scalar creation of vectors take the fate of two vectors, and here also multiply two vectors, then what difference? Clear difference, first for everything, as a RESULT:

The result of the scalar vector creation is є:

VECTOR: , then the vector is multiplied and the vector is taken again. Closed club. Vlasne, the sound is the name of the operation. In different primary literature, the meaning of the same can be changed, I choose the letter .

Designation of vector creation

I'll be back with a picture, then comments.

Appointment: Vector creative noncollinear vectoriv, taken from given order, called VECTOR, dozhina numerically better area of the parallelogram, based on these vectors; vector orthogonal to vectors, and directings so that the basis has the right orientation:

We choose the appointment by the brushes, there is a lot of cicada here!

Again, you can name the following moments:

1) Outside vectors, marked with red arrows, for the designated not collinear. Vipadok kolіnearnyh vektor_v before the river will look trohi pіznіshe.

2) Take vectors in a strictly defined order: – "a" multiplied by "be", and chi is not "be" to "a". The result of the multiplication of vectorsє A vector with blue color values. If you multiply the vectors y in the reverse order, then we take away the vector equal to the distance and the straight vector (crimson color). Tobto fair jealousy .

3) Now cognizable from the geometric zm_st vector creation. This is an extremely important point! The length of the blue vector (and, also, i of the crimson vector) is numerically greater than the area of the parallelogram, based on the vectors. On the little one is a parallelogram of shading with black color.

Note : armchair є schematic, і, naturally, the nominal value of the vector creation is not equal to the area of the parallelogram.

We guess one of the geometric formulas: the area of the parallelogram is more expensive to add the sum of the sides to the sine of the cut between them. To that, according to the foregoing, the formula for calculating the DOVZHINI of the Vector creation is valid:

I reiterate that the formulas have about the DOWN of the vector, and not about the vector itself. What practical zmist? And the sense is such that in the problems of analytical geometry the area of a parallelogram is often known through the concept of a vector creation:

Let's take a friend an important formula. The diagonal of the parallelogram (black dotted line) divides the yogo into two equal tricots. Later, the area of the tricutnik, inspired by vectors (black shading), can be known by the formula:

4) No less important fact believe that the vector is orthogonal to the vectors , so . Understandably, the straightening vector (crimson arrow) is also orthogonal to the outward vectors.

5) The vector of straightening so that basis may law orientation. On the lesson about go to a new basis I report back about plane orientation and at once we will figure out what kind of orientation to space. I will explain on your fingers right hand. Think about it eye-catching finger with vector i middle finger with a vector. Ring finger and little finger press down to the valley. As a result thumb- Vector tvir is uphill. Price and є right-orientation basis (on a small scale itself). Now remember the vectors ( expressive and middle fingers) by the hands, as a result, the thumb will flare up, and the vector tvir will already move down. This is also a right-orientation basis. Possibly, you have a winklo of food: what kind of basis can I have a left orientation? "Invite" the same fingers left hand vectors , and take away the left basis and the left orientation of the space (in my case, the great finger is spread out at the straight line of the lower vector). Figuratively, apparently, the bases “twist” or orient the space at the different sides. And if we don’t understand it, let’s think about it abstractly - so, for example, the orientation of the space changes the size of the mirror, and it’s like “strike the object out of the mirror”, then you can’t get into the “original” in the wild. Before speech, put three fingers to the mirror and analyze the impression;-)

... it’s still good, what do you now know about right and left orientation bases, more scary talk of such lecturers about changing orientation =)

Vector tvir collinear vectors

The appointment was reportedly disassembled, there was no more clarification, what is needed, if the vectors are collinear. As vectors are collinear, then they can be expanded on one straight line and our parallelogram can also be folded into one straight line. Such an area, as it seems to be mathematicians, virogenous The parallelogram is equal to zero. Tse w vyplivaє i z formulas - the sine of zero or 180 degrees to zero, and therefore, the square of zero

In such a rank, yakscho, then і . It is important to pay attention to the fact that the vector dobutok itself is equal to the zero vector, but in practice it is often difficult to write that the vector is also equal to zero.

Okremy vipadok - vector tvir of the vector on itself:

For the help of the vector creation, the collenarity of the trivi- mer vectors can be reversed;

For the perfection of practical applications, you may need trigonometric table, to find the meaning of sinuses.

Well, let's fire the fire:

butt 1

a) Know the value of the vector creation of vectors, so

b) Find the area of the parallelogram based on the vectors

Solution: Hі, tse not a drukarska pardon, vihіdnі danі in the points of the mind, I navmisno zrobiv the same. That's why the design decision is taken care of!

a) It is necessary for the mind to know dozhina vector (vector creation). For a specific formula:

Vidpovid:

If you ate about dovzhina, then it seems that you are showing peace - loneliness.

b) It is necessary for the mind to know area a parallelogram based on vectors. The area of this parallelogram is numerically superior to the vector creation:

Vidpovid:

To give respect to the fact that there are no warnings about vector witting, we were inquired about square figures vіdpovіdno rozіrnіst - kvadnі odinіtsі.

Always marvel at what it is necessary to know beyond the mind clear proof. You can get away with letters, ale letters in the middle of vikladachiv vistacha, and with good chances to turn around for additional treatment. Although the reasoning is not particularly strained - if it is not correct, then there is a reaction that the person does not understand in simple speeches and / or does not delve into the essence of the task. At this moment, it’s necessary to try on control, violating whether it’s a task of higher mathematics of that and of other subjects tezh.

Where did the great letter "en" go? In principle, її it was possible to stick to the decision, but with the method of speeding up the recording, I didn’t kill it. I spodіvayus, all zrozumіlo, scho and tse signification of one and the same.

A popular butt for independent vision:

butt 2

Know the area of trikutnik, inspired by vectors, yakscho

The formula for the area of the tricot through the vector dobutok is given in the comments before the appointment. The solution is to follow the example of the lesson.

In fact, the dressing room is really wide, they can roll it up with tricots.

For the accomplishment of other tasks, we need:

The power of the vector creative vector

We already looked at the leaders of the authority of the vector creation, I will include them in the list.

For more vectors and a greater number, the following powers are valid:

1) In other sources of information, this item is not heard by authorities, but it is still important in practical terms. So let it be.

2) - Power tezh rozіbrano more, іnоdі yogo call anticommutative. Otherwise, apparently, the order of the vector may be significant.

3) - happy or associative laws of vector practice. Konstanty seamlessly blame for intervector creativity. Really, what do they need to do?

4) - rozpodіlnі abo distributive laws of vector practice. There are also no problems for opening the shackle.

As a demonstration, a short butt is looked at:

butt 3

Know yakscho

Solution: For the mind, it is necessary to know the realm of the vector creation. Let's write our miniature:

(1) Zgіdno z associative laws, we blame the constant for intervector creation.

(2) We blame the inter-module constant, its own module has a “minus” sign. Dovzhina can be negative.

(3) I understood further.

Vidpovid:

The hour has come to add firewood to the fire:

butt 4

Calculate the area of the trickster, inspired by vectors, as

Solution: The area of \u200b\u200bthe trikutnik is known by the formula . The catch is that the vectors "ce" and "de" themselves are represented as a sum of vectors. The algorithm here is standard and guess what, apply No. 3 and 4 to lesson Scalar tvir vector_v. For clarity, the solution is divided into three stages:

1) On the first crochet, we can see the vector tvir through the vector tvir, in fact, virazimo vector through vector. About dozhini still no words!

(1) Represented by a number of vectors.

(2) Vikoristovuyuchi distributive laws, opening the arches for the rule of multiplication of rich terms.

(3) Vikoristovuyuchi associative law, we blame all the constants for intervector creations. With a small dosvіdі dії 2 і 3 it is possible to beat one hour.

(4) First and foremost, the rest of the additions to zero (zero vector) are the rewards of receiving power. Another addendum has the power of anticommutativity of the vector creation:

(5) Suggest similar dodanki.

As a result, the vector appeared through the vector, which is necessary to achieve:

2) At another stage, we will know the length of the vector creation we need. Tsya deya guessing Butt 3:

3) We know the area of the shukan tricoutnik:

Stages 2-3 solutions can be completed in one row.

Vidpovid:

Take a look at the tasks to finish wider in control robots, Axis butt for independent vision:

butt 5

Know yakscho

Briefly, the solution is to illustrate the lesson. Surprisingly, how much you were respectful of the front butts ;-)

Vector tvіr vectorіv y coordinates

, given in the orthonormal basis , expressed by the formula:

The formula is really simple: at the top row of the signifier, coordinate vectors are written, at the other and third rows, the coordinates of the vectors are “stacked”, moreover, it is in strict order- First coordinates of the "ve" vector, then coordinates of the "double-ve" vector. If vectors need to be multiplied in a different order, then the rows should be remembered as spaces:

butt 10

Verify, what are the next vectors and space:
but)
b)

Solution: Verification is based on one of the solids given lesson: as vectors and collinear, їх vector tvir is equal to zero (zero vector): .

a) We know the vector TV:

In this manner, the vectors are not collinear.

b) We know the vector TV:

Vidpovid: a) not collinear; b)

Axis, maybe, and all the main information about vector creation of vectors.

Tsej rasdіl bude small, oskolki zavdan, de vikoristovuetsya zmіshane tvіr vektorіv, not rich. Practically everything will fit into the design, geometrical change and sprat of working formulas.

Zmishany TV vector:

The axis stinks so much like a train and check, do not check, if they are charged.

On the back of my head, I’ll rediscover that picture:

Appointment: Created with creativity non-coplanar vectoriv, taken from given order, called obsyag paralepiped, based on these vectors, with the “+” sign, so the basis is right, and the “–” sign, so the basis is left.

We see the little ones. The lines invisible to us are crossed with a dotted line:

Zanuryuёmosya at the appointment:

2) Take vectors in song order, so the permutation of vectors in creation, as you guess, does not pass without traces.

3) Before that, as a commentary on a geometrical change, I will state an obvious fact: zm_shany tv_r vectorіv є NUMBER: . In the initial literature, the design can be somehow different, I mean the sound is zmishane tvir through, and the result is calculated with the letter “ne”.

For appointment zmіshany tvіr - tse obsyag paralelepiped, based on vectors (the figure is crossed with red vectors and black color lines). That is the number of the old obyagu of this parallelepiped.

Note : chairs are sketchy.

4) Don't try again to understand the orientation of the basis and space. The sense of the final part of the one who can take the obligatory sign is minus. In simple words, Zmishane tvir can be negative: .

The following is a formula for calculating the volume of a parallelepiped based on vectors.

For vectors , i , given by their coordinates , the difference tvir is calculated according to the following formula: .

Zmіshany tvіr zastosovuyut: 1) for calculating the obsyagіv tetrahedron and parallelepiped, on the vectors , i , like on the edges, according to the formula: ; 2) yak umova complanarity of vectors , i : i - complanarnі.

Topic 5. Straight lines and flats.

The normal vector of lines , any non-zero vector of perpendiculars to a given line is called. Direct vector direct , any non-zero vector parallel to a straight line is called.

Straight on the flat

1) - wildly equal straight line, de normal vector straight line;

2) - Alignment of a straight line to pass through a point perpendicular to a given vector;

3) canonically equal );

5) - alignment of straight lines with cut coefficient , de - The point through the yaku is straight to pass; () - Kut, which is a direct warehouse of weight; - Dovzhina vіdrіzka (zі zі sign) scho vіdsіkaєtsya direct on the axis (the sign "", i.e. vіdrіzok vіdsіkaєtsya on the positive part of the axis i "", i.e. on the negative).

6) - alignment of straight lines at the windbreaks, de i - dozhini vіdrіzkіv (zі sign), which are seen straight on the coordinate axes i (the sign "", i.e., vіdrіzok vіdsіkaєtsya on the positive part of the axis "", i.e. on the negative).

Walk from the point to the straight line , Assigned to the deep levels on the flat, to be known for the formula:

Kut, ( )between straight lines i , by setting the highest equals or the equals with the top coefficient, to be known for one of the following formulas:

Yakshcho abo.

Yakshcho abo

Coordinate points of the cross line and how to solve the system of linear lines: or .

Normal plane vector , any non-zero vector of perpendiculars to a given plane is called.

flat the coordinate system can be set equal to one of the advancing views:

1) - wildly equal area, de normal area vector;

2) - leveling of the plane to pass through the point perpendicular to the given vector;

3) - Rivnyannya flat, scho to pass through three points i ;

4) - leveling of the area at the windbreaks, de , i - Dini vіdrіzkіv (zі sign ), which are seen by the plane on the coordinate axes , i (sign "", i.e., vіdrіzok vіdsіkaєtsya on the positive part of the axis i "", i.e., on the negative).

V_dstan v_d points to the plane , Assigned to the ardent equals, to know the formula:

Kut,( )between flats і , given by galvanized equals, be found behind the formula:

Straight in space the coordinate system can be set equal to one of the advancing views:

1) - wildly equal straight line, like a line between two planes, de - normal vectors of planes i ;

2) - Alignment of a straight line to pass through a point parallel to a given vector ( canonically equal );

3) - Alignment of a straight line to pass through two given points;

4) - alignment of a straight line that passes through a point parallel to a given vector , ( parametric alignment );

Kut, ( ) between straight lines і in space , Given the canonical equivalences, follow the formula:

Line point coordinates , given by parametric equalities that flat , assigned to the main lines, they are rebuying as a decoupling of the system of linear lines: .

Kut, ( ) between a straight line , set by canonical equals that flat , Assigned to the ardent equals to know the formula:

Topic 6. Curves in a different order.

Algebraic curves of a different order the curve is called in the coordinate system, wildly equal How can I look:

de numbers - do not reach zero overnight. Now comes the classification of curves in a different order: 1) yakshcho , then the fiercely equal shows the curve elliptical type (circumference (at ), elіps (at ), empty multiplier, dot); 2) yakscho, then - curve hyperbolic type (hyperbole, a couple of straight lines that are tinted); 3) yakscho, then - curve parabolic type(parabola, empty faceless, straight line, a pair of parallel lines). Circle, elips, hyperbola and parabola are called non-virgin curves of a different order.

Zagalne rivnyannya , de , What signifies a non-virogenous curve (colo, ellipse, hyperbola, parabola), zavzhdi (by the method of seeing the outer squares) can be brought to align one of the advancing views:

1a) - alignment of the stake with the center at the point and radius (Fig. 5).

1b)- alignment of the ellipse with the center at the point and the axes of symmetry, parallel to the coordinate axes. The numbers are called ellipse seeds the main rectangle of the elips; ellipse tops .

To create an elіps in the coordinate system: 1) the center of the ellipse is visible; 2) carried out through the center of the dotted line of the axis of symmetry of the ellipse; 3) the main rectangle of the ellipse will be a dotted line with the center and sides parallel to the axes of symmetry; 4) depicting the suctile line of the ellipse, fitting them into the main rectangle of the ellipse, standing on the other side at the top of the ellipse (Fig. 6).

Similarly, there will be a colo, the main rectangle of whichever side (Fig. 5).

Fig.5 Fig.6

2) - equalization of hyperbole (titles pov'yazanimi) with the center at the point and the axes of symmetry parallel to the coordinate axes. The numbers are called with hyperbole ; a rectangle with sides parallel to the axes of symmetry and a center at a point - the main rectangle hyperbola; crosspoints of the main rectangle with axes of symmetry - vertices of hyperbolas; straight lines, which pass through the proliferating vertices of the main rectangle - asymptotes of hyperbolas .

To induce hyperbole in the coordinate system: 1) the center of hyperbole is evident; 2) carried out through the center of the dotted line of the axis of symmetry of hyperbole; 3) the main rectangle of hyperbola with the center and sides and parallel to the axes of symmetry will be a dotted line; 4) drawn through the vertices of the main rectangle with a dotted straight line, which are hyperbole asymptotes, which are not close, with an indistant view of the cob of coordinates, hyperbole lines are approached, not overlapping; 5) depicted with a suctile line of a hyperbola (Fig. 7) or a hyperbola (Fig. 8).

small 7 small 8

3a)- alignment of the parabola with the vertex at the point of symmetry, parallel to the coordinate axis (Fig. 9).

3b)- alignment of the parabola with the vertex at the point of symmetry, parallel to the coordinate axis (Fig. 10).

To create a parabola in a coordinate system: 1) show the top of the parabola; 2) drawn through the vertex by a dotted line of all symmetry of the parabola; 3) depicting a suctile line of a parabola, with a straight line of a needle, with a fixed sign of the parabola parameter: at - in the negative direction of the coordinate axis (Fig. 9b and 10b).

Rice. 9a Mal. 9b

Rice. 10a Small. 10b

Topic 7. Anonymous. Numerical multipliers. function.

Pid impersonal to understand for a day the succession of objects of be-like nature, remembered by one another and I think as a single whole. Objects that become impersonal, call yoga elements . Bagato can be inexhaustible (accumulates from an inexhaustible number of elements), end (accumulates from an innumerable number of elements), empty (do not avenge the same element). Impersonal means:, like elements:. An empty multiplicity means.

Nameless name multiplied multiply, so that all the elements of the multiplier lie in the multiplicity and write. Nameless and name equal , as if the stench is formed from the quiet elements themselves and write. Two multiplies will be equal to the same and only a little to the same, if i .

Nameless name universal (Within the framework of this mathematical theory) , yakscho yogo elements є all objects that are seen in this theory.

Bezlich you can ask: 1) pererakhuvannyam all yogo elements, for example: (less than kіntsevih multiplies); 2) zavdannya rules for the assignment of belonging to the element of the universal multiplier, given multiplier: .

United

Peretin many and are called impersonal

Retail many and are called impersonal

Additions multiplier (up to the universal multiplier) is called impersonal.

Two multiplications are called equivalent and write ~ as between the elements of these multiples can be set mutually unambiguously. The impersonal is called rakhunkovim , which is equivalent to the impersonal natural numbers : ~ . An empty lot for the appointments to lie down to the hospital.

Understanding the tightness of the multiplier is blamed when the multiplicity is equal for the number of elements that are hidden in them. The pressure of the multiplier means. The intensity of the final multiplier is the quantity of yogo elements.

Equivalent multipliers may equal tension. The impersonal is called indistinguishable , which is why the tightness is greater for the tightness of the multiplier.

Diysnim (speech) number called innumerable dozens of drib, taken with the sign "+" or "". The correct numbers are drawn from the points of the number line. module (absolute value) of a decimal number is called an unknown number:

The impersonal is called numeric , yakscho yogo elements є dіysnі numbers. intermissions multiplies of numbers are called: , , , , , , , , , .

The absence of all points on the number line, which pleases the mind, de - skilki is always a small number, called -outskirts (or just outskirts) dots i are indicated. The impersonality of all the points of the mind, de - skilki is always a great number, it is called - outskirts (or just outskirts) inconsistencies and signify.

The value that takes one and the same numerical value is called fast. The value that takes different numerical values is called zminnoy. function the rule is called, as a skin number should be put in one whole number, i write. The impersonal is called the area of appointment functions, - impersonal ( or the region ) value functions, - argument , - function values . The most extensive way to define a function is an analytical method, in which a function is defined by a formula. Designated natural area The function is called the impersonal value of the argument, for which the formula is given ma sens. Schedule function , In a rectangular coordinate system , an impersonal point of the plane with coordinates , is called.

The function is called steam room on a multiplier, symmetrical to a point, as for all the minds win: i unpaired how to win the mind. In a different way - the function of a foreign mind or neither pair nor unpair .

The function is called periodic on a plurality, which is the main number ( function period ), such that everyone wins the mind: . The smallest number is called the main period.

The function is called monotonously growing (subsiding ) by a multiplier, as the larger value of the argument gives a larger (less) value of the function.

The function is called obmezhenoyu on a plurality, which is a number, such that everyone wins minds:. In another way, the function - not bordered .

Zvorotny to function , such a function is called, as it is designated on impersonal and skin

Set at vіdpovіdnіst tak, scho. For znahodzhennya funktії, zvorotnoї to funktії , it is necessary to virishity equal shodo. What is the function , є strictly monotonous on, there is a turn around, at your own, as the function grows (changes), then the turn function also grows (decreases).

The function, which is seen, de, is the function of such a function, that the area assigned to the function is to replace the impersonal value of the function, is called folding function independent argument. Change is called by its intermediate argument. A collapsible function is also called a composition of functions, and write: .

Basic elementary functions are important: static function , showing function ( , ), logarithmic function ( , ), trigonometric functions , , , , turning trigonometric functions , , , . Elementary a function is called, separated from the basic elementary functions by a finite number of arithmetic operations and compositions.

As for the tasks of the schedule of the function, then the schedule of the function will be brought up to a number of transformations (zsuv, squeezing or stretching, exaggeration) graphics:

1) 2) the transformation is symmetrically showing the schedule of the axis; 3) the transformation breaks the graph along the axis by one (- to the right, - to the left); 4) turning the graph along the axis to one (- uphill, - down); 5) reworking the schedule of the axis of the axis stretching at times, like squeezing at times, like; 6) the reworking of the schedule is squeezing the axis at times, it’s like stretching it at times, it’s like.

The sequence of transformations in case of a prompt schedule of a function can be seen symbolically:

Note. In case of a victorious reworking, there is a trace on the uvaz, that the value of the zsuv vzdovzh axis is signified by that constant, as it is added without intermediary to the argument, and not to the argument.

The graph of the function is a parabola with a vertex at a point, the needles are straight uphill, or down, like. A graph of a fractional-linear function is a hyperbola with a center at a point, asymptotes that pass through the center, parallel to the coordinate axes. that satisfy the minds. called.

For vectors , i , given coordinates, , zmishane tvir are calculated according to the following formula: .

Topic 5. Lines on the flat.

The normal vector of lines , any non-zero vector of perpendiculars to a given line is called. Direct vector direct , any non-zero vector parallel to a straight line is called.

Straight on the flat the coordinate system can be set equal to one of the advancing views:

1) - wildly equal straight line, de normal vector straight line;

2) - Alignment of a straight line to pass through a point perpendicular to a given vector;

3) - Alignment of a straight line to pass through a point parallel to a given vector ( canonically equal );

4) - Alignment of a straight line to pass through two given points;

Walk from the point to the straight line , Assigned to the deep levels on the flat, to be known for the formula:

Kut, ( )between straight lines i , by setting the highest equals or the equals with the top coefficient, to be known for one of the following formulas:

Yakshcho abo.

Yakshcho abo

Coordinate points of the cross line and how to solve the system of linear lines: or .

Theme 10 Anonymous. Numerical multipliers. Options.

Nameless name multiplied multiply, so that all the elements of the multiplier lie in the multiplicity and write.

Nameless and name equal , as if the stench is formed from the quiet elements themselves and write. Two multiplies will be equal to the same and only a little to the same, if i .

Nameless name universal (Within the framework of this mathematical theory) , yakscho yogo elements є all objects that are seen in this theory.

United

Peretin many and are called impersonal

Retail many and are called impersonal

Additions multiplier (up to the universal multiplier) is called impersonal.

Diysnim (speech) number called innumerable dozens of drib, taken with the sign "+" or "". The correct numbers are drawn from the points of the number line.

module (absolute value) of a decimal number is called an unknown number:

The impersonal is called numeric yakscho yogo elements є dіysnі numbers. Numerical intermissions are called multiples

numbers: , , , , , , , , , .

The function is called periodic on a plurality, which is the main number ( function period ), such that everyone wins the mind: . The smallest number is called the main period.

The function is called monotonously growing (subsiding ) by a multiplier, as the larger value of the argument gives a larger (less) value of the function.

The function is called obmezhenoyu on a plurality, which is a number, such that everyone wins minds:. In another way, the function - not bordered .

Zvorotny to function , such a function is called, as it is designated on an impersonal and dermal basis to put it in such a way that. For znahodzhennya funktії, zvorotnoї to funktії , it is necessary to virishity equal shodo. What is the function , є strictly monotonous on, there is a turn around, at your own, as the function grows (changes), then the turn function also grows (decreases).

The graph of the function is a parabola with a vertex at a point, the needles are straight uphill, or down, like.

In some cases, when the schedule of the function is gradually divided, the area is designated for a sprat of spaces, which do not overlap, and subsequently the schedule will be on the skin of them.

The skin of orderings typing from the real numbers is called dot -peaceful arithmetic (coordinate) space and are designated either, in their number they are called її coordinates .

Let і - deyaki multiply points і. If the skin points are put at the same time as the rule of thumb for one whole number, then it seems that a numerical function is set on the multiplier in the form of changes and write or write briefly i, with which it is called the area of appointment , - meaningless meaning , - arguments (independent change) functions.

The function of two variables is often used, the function of three variables -. The scope of the function is a sprat of a point of the plane, the function is a sprat of a point of space.

Topic 7. Numerical sequences and rows. Between sequences. Between the functions and the continuity.

As a skin natural number, according to the deaky rule, it is set to one whole number, then it seems that it is given numerical sequence . Mean briefly. The number is called sleepy member of the sequence . Sequence is also called a function of a natural argument. Consequences must always avenge the impersonal elements, among which they can be equal.

The number is called boundary sequence , and write, as if there were some date, there is a number such that there will be an imbalance.

Consequence, which may be the end of the border, is called similar , in a different way - disperse .

: 1) subsiding , yakscho; 2) growing , yakscho; 3) costly , yakscho; 4) non-growth yakscho. All reincarnations and more sequences are called monotone .

The sequence is called obmezhenoyu , which is the number of such that everyone wins the mind: . In a different way, the sequence - not bordered .

Be it monotonous, the sequence may be between ( Weierstrass theorem).

The sequence is called infinitely small yakscho. The sequence is called infinitely great (what to go to neskіchennostі), yakshcho.

number called between sequences, de

Postiyna is called a non-peer number. The logarithm of a number on the basis is called the natural logarithm of the number and is assigned.

Viraz mind, de - sequence of numbers, called numerical near i be appointed. The sum of the first terms in the series is called -oh with a private sum row.

The row is called similar yakscho rozbіzhnym as if there is no boundary. The number is called sumo row what to go , when to write.

If the series converge, then (a necessary sign of comfort in a row ) . Zvorotne firmness is wrong.

Yakshcho, then the row diverge ( enough sign of diversity in a row ).

Enlightened by the harmonies nearby name a series that converges for and diverges for.

Geometric next name a series that converges at , at its own sum is more expensive and diverges at . there is a number chi symbol. (left flare, right flare) and

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