Vector vitvir- tse pseudovector, perpendicular to the plane, pobudovanoї on two spіvmultipliers, which is the result of the binary operation "vector multiplication" over vectors in the trivial Euclidean space. The vector TV does not have the power of commutativity and associativity (є anticommutative) i, on the vіdmіnu vіd scalar creation of vectorіv, є vector. Widely victorious in rich technical and physical additions. For example, the moment of impulse and the Lorentz force are mathematically written as a vector. The vector extension of the corisny for "reversing" the perpendicularity of vectors - the module of the vector extension of two vectors to the additional extension of their modules, as they are perpendicular, and change to zero, as vectors are parallel or anti-parallel.

It is possible to calculate the vector addition in a different way, and theoretically, in the space, whether there is a largeness of n, it is possible to calculate the addition of n-1 vectors, taking away a single vector, perpendicular to them we have. But if tvir is surrounded by non-trivial binary creations with vector results, then the traditional vector tvir is assigned only to the trivial and seven-world spaces. The result of a vector creation, like a scalar one, lies in the Euclidean space metric.

On the other hand, in the form of a formula for calculating the coordinates of the vector in a scalar recursive coordinate system in a three-dimensional rectangular coordinate system;

Appointment:
The vector complement of the vector a vector b in the space R 3 is called the vector c , which satisfies the upcoming vimog:
The length of the vector c is extended to the length of the length of the vectors a and b by the sine of the cut between them:
|c|=|a||b|sin φ;
vector c is orthogonal to the skin vector s a and b;
the vector c of rectifications so that the trinity of vectors in abc is right;
the space R7 needs the associativity of the trio of vectors a, b, c.
Designation:
c===a×b

Rice. 1. The area of ​​the parallelogram is equal to the module of the vector creation

Geometric power of vector art:
Necessary that enough mental The collinarity of two non-zero vectors is equal to zero of their vector creation.

Vector creative module dorivnyuє area S parallelogram inspired by vectors reduced to the cob aі b(Div. fig. 1).

Yakscho e- single vector, orthogonal vector aі b and vibranium so that three a,b,e- rights, and S- the area of ​​the parallelogram induced on them (pointing to the cob), then the following formula is valid for the vector creation:
=S e

Fig.2. The volume of the parallelepiped at the substitution of vector and scalar additional vectors; dotted lines show the projections of the vector c on a × b and the vector a on b × c, the first line is the significance of scalar creations

Yakscho c- which vector, π - be-yak flat, scho vengeance tsey vector, e- a single vector that lies near the plane π ta orthogonal to c,g- single vector orthogonal to the plane π and straightening so that three vectors ecgє right, then for someone who lies at the square π vector a the correct formula is:
=Pr e a |c|g
de Pr e a vector projection of e onto a
|c|-modulus of the vector h

When choosing a vector and a scalar creation, you can use the parallelepiped, inspired by the vectors reduced to the cob a, bі c. Such a dobutok of three vectors is called zmishanim.
V=|a (b×c)|
The little one shows that this can be achieved in two ways: the geometric result is saved when replacing the “scalar” and “vector” creations with objects:
V=a×b c=a b×c

The magnitude of the vector creation lies in the sine of the cut between the cob vectors, so the vector tvir can be taken as the steps of perpendicularity of the vector in the same way, like the scalar tvir can be seen as the steps of parallelism. Vectorial addition of two single vectors to 1 (alone vector), as well as vectors and perpendicular, and as 0 (zero vector), as vectors and parallel or anti-parallel.

Viraz for vector creation in Cartesian coordinates
Yakscho two vectors aі b assigned by their rectangular Cartesian coordinates, or rather, represented in an orthonormal basis
a = (a x, a y, a z)
b = (b x, b y, b z)
and the coordinate system is right, then your vector tvir may look
=(a y b z -a z b y ,a z b x -a x b z ,a x b y -a y b x)
For memorization ts_єї formulas:
i = ∑ε ijk a j b k
de ε ijk- a symbol of Levi-Chiviti.

7.1. Designation of vector creation

Three non-coplanar vectors a, b and c, taken in the designated order, satisfy the right triple, as if from the end of the third vector h, the shortest turn from the first vector a to the other vector b is visible, which follows the anti-godinnikov arrow, and lev, as for the godinnikov ( div. Fig. 16).

The vector extension of a vector to the vector b is called the vector h, which is:

1. Perpendicular to vectors a and b, so z ^ a and c ^ b;

2. May dozhina, numerically equal to the area of ​​the parallelogram, based on the vectors a ib yak at the sides (div. fig. 17), tobto.

3. The vectors a, b and h satisfy the right trinity.

Vector tvir is denoted by a x b or [a, b]. From the designation of the vector creation without intermediary squeal such spіvvіdnoshennia between orts i jі k(div. fig. 18):

i x j = k , j x k = i , k x i = j .
We bring, for example, that i xj = k.

1) k ^ i, k ^ j;

2) |k |=1, but | i x j| = | i | |J| sin(90°)=1;

3) vectors i, j i k satisfy the right of the trio (Fig. 16).

7.2. The power of the vector creation

1. When rearranging the sp_multipliers, the vector does not change the sign, that is. and xb \u003d (b xa) (div. Fig. 19).

Vectors a хb і b х are collinear, may have the same modules (the area of ​​the parallelogram is left unchangeable), and also straight lines (triples a, b, а xb і a, b, b x a of the opposite orientation). Became booty axb = -(bxa).

2. The vector tvir can lose the power of a scalar multiplier, so l (a xb) \u003d (l a) x b \u003d a x (l b).

Let l>0. Vector l (a xb) perpendicular to the vectors a and b. Vector ( l a) x b also perpendicular to the vectors a i b(Vector a, l but lie at one flat). Mean, vector l(a xb) that ( l a) x b collinear. Obviously, scho th direct zbіgayutsya. Mayut the same dovzhina:

Tom l(a x b) = l a xb. It is similarly brought with l<0.

3. Two non-zero vectors a i b kolіnearnі tіlki tіlki tіlі, if їхній vector tvіr dоrіvnyuє zero vector, then а ||b<=>and xb = 0.

Zokrema, i * i = j * j = k * k = 0 .

4. Vector tvir may rozpodіlnu power:

(a+b) xs = a xs + b xs.

Acceptable without confirmation.

7.3. Viraz vector creation through coordinates

Mi vikoristovuvatimemo table of the vector creation of the vector i, j and k:

so that the shortest path from the first vector to the other one goes straight with the arrows, tvir reaches the third vector, otherwise it goes away - the third vector is taken with a minus sign.

Give the task two vectors a = a x i + a y j+az k i b = b x i+by j+bz k. We know the vector creation of these vectors, multiplying them as rich terms (apparently up to the power of the vector creation):

Otriman's formula can be written shorter:

shards of the right part of the equality (7.1) show the third-order ranking of the arbitrator behind the elements of the first row. Equity (7.2) is easy to forget.

7.4. Deyakі programs of vector creation

Establishing collinearity of vectors

The value of the area of ​​the parallelogram and trikutnik

Zgіdno z vznachennyam vector creative vectorіv a i b |a xb | =| a | * | b | sin g, i.e. S pairs = | and x b |. І, later, D S = 1/2 | a x b |

Assignment to the moment of force or shodo point

Come on at the point A force is applied F = AB and let me Pro- Dejaka point to space (div. Fig. 20).

From physics it is clear that moment of force F shodo points Pro called vector M, which pass through a point Pro ta:

1) perpendicular to the plane, to pass through points O, A, B;

2) numerically more strength on the shoulder

3) establish the right trio with vectors OA and A .

Otzhe, M \u003d OA x F.

Significance of line swiddenness wrapping

Shvidkist v points M of a solid body w on a non-rotating axis is determined by the Euler formula v \u003d w xr, de r \u003d OM, de O-deac, the point of the axis is non-rotating (div. Fig. 21).

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. At the branch of mathematicians, little firewood was fired, it’s better to hang out on Pinocchio. Really, the material is more wide and simple - hardly more foldable, lower than the same scalar tvir, there will be less typical tasks. Golovne in analytic geometry, like a lot of people who change their minds and already messed up, DON'T HAVE PARTY 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. You can’t get juggling at once, the shards of our eyes can be seen only space vectors, and the flat vectors from two coordinates are left behind. Why? Such data were already born - vector and zmіshany tvіr vektorіv is designated to practice in trivimir space. Already easier!

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

diya herself be appointed let's come in rank: . Іsnuyet Іnshi varianty, but I also use the sound to designate a vector TV 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 NUMBER:

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 non-colinear vectoriv, taken from given order, called VECTOR, dozhina numerically better area of ​​the parallelogram inspired by these vectors; vector orthogonal vector, and directings so that the basis has the right orientation:

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

Also, you can see the following moments:

1) Outside vectors, marked with red arrows, for the designated not collinear. Vipadok kolіnearnyh vektorіv bude rozglyady vіznіshe before the river.

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 VECTOR, which means blue color. As a vector and multiply in reverse order, we take equal for the length and the prolongation for the direct vector (crimson color). Tobto fair jealousy .

3) Now cognizable from the geometric zm_st vector creation. This is an 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 length of the vectorial extension is not the same as 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. Therefore, following from what was said above, the formula for calculating the DOVJINI 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 trickster, inspired by vectors (black shading), can be known by the formula:

4) No less important fact is that the vector is orthogonal to the vectors, that . 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 will marvel 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, they can be divided into 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 . To pay attention to the fact that the vector 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 doboot of a 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 know 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 vectors, such as

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 create). For a specific formula:

Vidpovid:

If you ate about dovzhina, then it’s obvious that you are peaceful - alone.

b) It is necessary for the mind to know area a parallelogram based on vectors. The area of ​​this parallelogram is numerically more advanced than the vector one:

Vidpovid:

To show respect, that there is no way around vector television, we were inquired about square figures vіdpovіdno rozmіrnіst – square units.

Always marveling at what is necessary to know beyond the mind, clear proof. You can get away with letters, but you can turn to the middle of the vistachas, and with good chances to turn to doopratsyuvannya. 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, you need to try on the control, virishuyuchi be-like zavdannya z mathematician and z іnshih subjects tezh.

Where did the large 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 practice, the heads are actually wider, they can be rolled 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 rozіbrana more, іnоdі її name 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:

It's time 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 doboot 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) The first and the last additions are equal to zero (zero vector) points 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 task to make it wider in the control robots, the axis of the butt for an independent variance:

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: the coordinate vectors are written in the upper row of the signifier, the coordinate vectors are “stacked” in the other and third rows, and 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:
a)
b)

Solution: The revision is based on one of the principles of this lesson: since vectors are collinear, then their vector complement 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 production of vectors.

Denmark would not be great, the shards of the head, de victorious zmіshane tvіr vectorіv, not rich. In fact, everything is taken into account by the appointment, geometric change and a couple 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 cuboid volume, 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:

Porinaemo 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: zmіshany dobutok vector_v: . 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) Let's not worry about understanding the orientation of the basis and space again. 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.

Denmark online calculator calculates vector income vector. We hope to report a solution. To calculate the vector addition vector, enter the coordinates of the vector in the box and click on the button "Calculate."

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Instructions for entering data. Numbers are entered as whole numbers (apply: 487, 5, -7623 thin.), tenth numbers (eg. 67., 102.54 thin.) or fractions. The fraction is required to be typed at the sight of a / b, de a і b (b> 0) tsіlі or tens of numbers. Apply 45/5, 6.6/76.4, -7/6.7 thin.

Vector TV vector

First of all, go to the designation of the vector creation of the vector, let's look at it ordered vector trio, left vector vector trio, right vector trio.

Definition 1. Three vectors are called ordered by trinity(abo triykoy), as it is designated, which of these vectors is the first, which is the other and which is the third.

Record cba- meaning - first є vector c, others є vector b and third є vector a.

Designation 2. Trinity of non-coplanar vectors abc it is called right (left), as when the cob is reduced to the cob, the vectors are ruffled so, as the raster is obviously great, unbending, and the middle finger of the right (left) hand.

Destination 2 can be formulated and otherwise.

Designation 2". Trinity of non-coplanar vectors abc called the right (left), as when brought to the cob, vector c roztashovuetsya on that bіk vіd of the area, which is marked by vectors aі b, stars shortest turn a before b vіdbuvaєtsya against the year's arrow (behind the year's arrow).

Three vectors abc, shown in fig. 1 є right, and three abc shown in fig. 2 є left.

Like two triplets of vectors and right or left, it seems that they are the same orientation. Otherwise, it seems that the stench is of an opposite orientation.

Appointment 3. The Cartesian or affine coordinate system is called right (left), as the three base vectors satisfy the right (left) triple.

For the sake of singing, let us look at the right coordinate systems.

Appointment 4. vector creative vector a per vector b called vector h, which is denoted by the symbol c=[ab] (or c=[a,b], or c=a×b) and pleased with the coming three vimog:

• dozhina vector h dobootku dozhin vektoriv aі b on sinus kuta φ between them:

|c|=|[ab]|=|a||b|sinφ;

(1)

• vector h orthogonal to dermal vector aі b;
• vector c straightening so that three abcє right.

Vector dobutok vector in may still power:

• [ab]=−[ba] (antipermutability spіvmulnikiv);
• [(λa)b]=λ [ab] (happiness shodo numerical multiplier);
• [(a+b)c]=[ac]+[bc] (rozpodіlnіst shodo sumi vectoriv);
• [aa]=0 for any vector a.

Geometric power of the vector creation.

Theorem 1. For two vectors to be collinear, it is necessary to achieve zero equality of their vector creation.

Bringing. Necessity. Come on vectori aі b collinear. Todi cut between them 0 or 180° sinφ=sin180=sin 0 = 0. Again, reversed by (1), the vector c equals zero. Todi c null vector.

Availability. Come on vector dobutok vector_v aі b nav to zero: [ ab]=0. Let us know that the vectors aі b collinear. I want one of the vectors aі b zero, then q_ vectors are collinear (because the zero vector can't be ignored directly and it can be considered a collinear vector).

Yakshcho OK offense vectori aі b nonzero, then | a|>0, |b|>0. Todi z [ ab]=0 and (1) sinφ=0. Otzhe vectori aі b collinear.

The theorem has been completed.

Theorem 2. Dovzhin (module) of vector creation ab] more expensive area S a parallelogram inspired by pointing to the cob vectors aі b.

Bringing. As you can see, the area of ​​the parallelogram is more expensive to add the sum of the sides of this parallelogram to the sine of the cut between them. Father:

Todo vector dobutok tsikh vector_v may look:

Rozkrivayuchi vyznachnik behind the elements of the first row, we take into account the layout of the vector. a×b basis i, j, k, which is equivalent to formula (3).

Proof of Theorem 3. We add up all possible bets from the base vectors i, j, k and porahuєmo їhnіy vector TV. It is necessary to ensure that the basis vectors are mutually orthogonal, to satisfy the right of a trio and to have a single douzhina (in other words, one can assume that i={1, 0, 0}, j={0, 1, 0}, k= (0, 0, 1)). Todi maєmo:

For the rest of the equanimity and spіvvіdnoshen (4), we take:

Adding a 3×3 matrix, the first row is the basis vector i, j, k, and other rows filled with vector elements aі b:

In this rank, the result of the vector creation of vectors aі b will vector:

.

Example 2. Find the vectorial addition of vectors [ ab], de vector a representations by two points. Pochatkov point of the vector: , end point vector a: , vector b may look .

Rozvyazannya. Move the first vector to the cob of coordinates. For which one is visible from the most visible coordinates of the endpoint, the coordinates of the cobpoint are:

Let's count the vyznachnik ts_єї matrix, rozklavshi її on the first row. As a result, we calculate the vector additional vector aі b.

ZMISHANIY CREATOR OF THREE VECTORS IN THAT YOGO OF POWER

Zmіshanim creative three vectors name the number that is good. Appointed . Here the first two vectors are multiplied vectorially and then the subtracting vector is scalarly multiplied by the third vector . Obviously, such a TV is a sprat.

Let's look at the power of the mixed creation.

1. Geometric zmist crazy creation. Zmishane tvir 3 vectors with accuracy up to the sign of the old obyagu paralepiped, prompted by these vectors, like at the ribs, tobto. .

   In such a manner, .

   

   Bringing. Vіdklademo vektori vіd zagalnogo cob and pobuduєmo on them paralepiped. Significantly and respectfully, scho. For the purpose of the scalar creation

   Allowing what i know through h the height of the parallelepiped, we know.

   In such a manner, at

   Well, then th. Father, .

   Ob'ednuyuchi insults and vipadki, otrimuєmo either.

   Z confirmation of the quality of the zokrema is clear, that the trinity of vectors is right, then zmishane tvir, and yakshcho - leva, then.

2. For whatever vectors , , equality is fair

   The proof of the power's power is evident from the power 1. It's true, it's easy to show that it is. Until then, the signs "+" and "-" are taken at the same time, because kuti mizh vectors ta і one hour gostrі or stupid.

3. When rearranging, whether there are two spіvmulnіnіv zmіshanі tvіr change the sign.

   It’s true, as if we can look at the confusion of TV, then, for example, or

4. Zmіshany tvіr tіlki tіlki tіlki і, if іz сpіvmіnnіkіv dоrіvnyuє zero аbо vectors аrе coplanar.

   Bringing.

   Including, the necessary and sufficient mental coplanarity of 3 vectors and the equality to zero of their mixed creation. In addition, it is obvious that three vectors establish a basis for space, for example.

   As well as the vectors and tasks in the coordinate form, it is possible to show that these changes are known by the formula:

   .

   Thus, zmіshany dobutok dobutok to the third-order signifier, who in the first row has the coordinates of the first vector, in the other row - the coordinates of another vector i in the third row - the third vector.

   apply.

ANALYTICAL GEOMETRY IN SPACE

Rivnyannia F(x, y, z)= 0 is assigned to the space Oxyz deaku surface, tobto. geometrical place point, coordinates of which x, y, z satisfy whomever is jealous. The line is called equal to the surface, and x, y, z- current coordinates.

However, often the surface is asked not by equals, but as an impersonal point of space, which may have that other power. And here it is necessary to know the equivalence of the surface, from її geometrical powers.

AREA.

NORMAL AREA VECTOR.

LEVELING THE AREA TO PASS THROUGH A GIVEN POINT

Let's look at the expanse of a large area σ. The position is dependent on the given vector perpendicular to the given plane, that fixed point M0(x0, y 0, z0) that lies near the plane σ.

The vector perpendicular to the plane σ is called normal vector qієї area. Let the vector have coordinates.

Let us see the plane σ, which passes through given a point M0 and may be a normal vector. For which we take on the plane σ a sufficient point M(x, y, z) i look at the vector.

For whatever point MÎ σ vector. Therefore, the scalar addition to zero Tsya jealousy is the mind of the point MО σ. It is fair for all points in the plane and breaks down, like only a point M lean back pose with a plane σ.

How to know through the radius-vector of a point M, is the radius vector of the point M0, then th equal can be recorded at a glance

Tse equal is called vector equal to the area. Let's write yoga in the coordinate form. Oscilki, then

Otzhe, we took away the flatness of the area, to pass this point. In this way, in order to fold the flatness of the plane, it is necessary to know the coordinates of the normal vector and the coordinates of the deuce point that lie on the plane.

Respectfully, that the plane is equal to the equal of the 1st stage of the flow coordinates x, yі z.

apply.

ZAHALNE RIVNYANNYA SQUARE

Can you show how equal the first step is to Cartesian coordinates x, y, zє equal to deykoї area. The price is recorded as:

Ax+By+Cz+D=0

and is called wild jealous planes, and the coordinates A, B, C here are the coordinates of the normal vector of the area.

Let's look at the surroundings of the falls infamous jealousy. Of course, as the plane of the coordinate system is expanded, it means that one or more decal coefficients are adjusted to zero.

A - the core of the vіdrіzka, which is seen by the plane on the axis Ox. Similarly, it can be shown that bі c- Dovzhini vіdrіzkіv, scho vіdsіkayutsya flat on the axes, scho to be seen. Ouchі Oz.

Rivnyannyam flats at the windbreaks are handy to scorch for raising the flats.