Dovzhina vector, cut between vectors - tsі understand є natural and zastosovnymi and іntuїtivno zrozumіlimi schodo vector yak vіdrіzka sing straight. Below, we learn how to distinguish between vectors in a trivial space, yogo cosine and we can look at the theory on butts.

For a better understanding of the kuta between vectors, we turn to a graphic illustration: let's put two vectors a → and b →, which are non-zero, on the plane or in the trivi- mer space. We also set a sufficient point O and add it to the vector O A → = b → and O B → = b →

Appointment 1

Kutom between vectors a → і b → is called a cut between the exchanges PRO and PRO.

The subtraction of kut is signified by such a rank: a → , b → ^

Obviously, it is possible to get a value from 0 to π or from 0 to 180 degrees.

a → , b → ^ = 0 if the vectors are codirectional and a → , b → ^ = π if the vectors are oppositely directed.

Appointment 2

The vectors are called perpendicular yakscho cut between them is 90 degrees or π 2 radians.

If we want one of the vectors to be null, then a → , b → ^ is not assigned.

The cosine of the kuta between two vectors, and also, і well kut, can be used either for the help of the scalar creation of the vectors, or for the help of the cosine theorem for the tricot, based on the two given vectors.

Vіdpovіdno scalar TVіr є a → , b → = a → b → cos a → , b → ^ .

If the given vectors a → and b → non-zero, then we can divide the right and the left part of the equality into additional two vectors, omitting in this way the formula for the value of the cosine of kuta between non-zero vectors:

cos a → , b → ^ = a → , b → a → b →

Tsya formula vikoristovuetsya, if mid-weekend danikh є dozhini vectorіv yogo scalar tver.

butt 1

External data: vectors a → and b → . Dovzhini їх equals 3 and 6 are clear, like a scalar twіr dorіvnyuє - 9. It is necessary to calculate the cosine of the cut between vectors and to know the cut itself.

Solution

There are enough data in the past to complete the formula, then cos a → , b → ^ = - 9 3 6 = - 1 2 ,

Now it is significant between vectors: a → , b → ^ = a r c cos (- 1 2) = 3 π 4

Suggestion: cos a → , b → ^ = - 1 2 , a → , b → ^ = 3 π 4

Most often, the tasks are fixed, the vectors are given by the coordinates of a rectangular coordinate system. For such variations, it is necessary to enter the same formula, but in coordinate form.

The length of the vector is defined as the square root of the sum of the squares of its coordinates, and the scalar addition of the vector is the sum of the sum of the corresponding coordinates. Then the formula for the value of the cosine of kuta between vectors on the plane a → = (a x , a y) , b → = (b x , b y) looks like this:

cos a → , b → ^ = a x b x + a y b y a x 2 + a y 2 b x 2 + b y 2

And the formula for the value of the cosine kuta between vectors in the trivial space a → = (ax , ay , az) , b → = (bx , by , bz) looks like: cos a → , b → ^ = ax bx + ay by + az bzax 2 + ay 2 + az 2 bx 2 + by 2 + bz 2

butt 2

External data: vectors a → = (2, 0, - 1), b → = (1, 2, 3) in a rectangular coordinate system. It is necessary to designate a cut between them.

Solution

1. To complete the task, we can immediately put the formula:

cos a → , b → ^ = 2 1 + 0 2 + (- 1) 3 2 2 + 0 2 + (- 1) 2 1 2 + 2 2 + 3 2 = - 1 70 ⇒ a → , b → ^ = arc cos (-170) = - arc cos 170

1. You can also assign a kut to the formula:

cos a → , b → ^ = (a → , b →) a → b → ,

and then expand the forward vector_v and scalar line behind the coordinates: a → = 2 2 + 0 2 + (- 1) 2 = 5 b → = 1 2 + 2 2 + 3 2 = 14 a → , b → ^ = 2 1 + 0 2 + (- 1) 3 = - 1 cos a → , b → ^ = a → , b → ^ a → b → = - 1 5 14 = - 1 70 ⇒ a → , b → ^ = - arc cos 1 70

Suggestion: a → , b → ^ = - a r c cos 1 70

Also, the extension of the task, if the coordinates of three points in a rectangular coordinate system are given, and it is necessary to specify the same cut. Also, in order to assign points between vectors and given coordinates, it is necessary to calculate the coordinates of the vectors at the different points of the cob and end of the vector.

butt 3

Outside data: on the plane in a rectangular coordinate system given points A (2, - 1), B (3, 2), C (7, - 2). It is necessary to find the cosine of the coota between the vectors A C → B C → .

Solution

We know the coordinates of the vectors behind the coordinates of the given points A C → = (7 - 2 , - 2 - (- 1)) = (5 , - 1) B C → = (7 - 3 , - 2 - 2) = (4 , - 4)

Now we can find a formula for defining the cosine of kuta between vectors on the plane in coordinates: cos AC → BC → ^ = (AC → BC →) AC → BC → = 5 4 + (- 1) (- 4) 5 + (-1 ) 2 4 2 + (- 4) 2 = 24 26 32 = 3 13

Value: cos A C → , B C → ^ = 3 13

Kut mіzh vectors can be calculated by the theorem of cosinus. We add to the point O the vector O A → = a → і O B → = b → then, zgіdno with the cosine theorem for the tricutnik OAB, it will be true:

A B 2 \u003d O A 2 + O B 2 - 2 O A O B cos (∠ A O B) ,

what is equal:

b → - a → 2 = a → + b → - 2 a → b → cos (a → , b →) ^

and we will show the formula for the cosine of kuta:

cos (a → , b →) ^ = 1 2 a → 2 + b → 2 - b → - a → 2 a → b →

For zastosuvannya otrimanoї formulas, we need two vectors, yakі clumsily assigned to their coordinates.

If you want to assign a method, it’s possible, nevertheless, it’s more common to put the formula:

cos (a → , b →) ^ = a → , b → a → b →

How did you remember the pardon in the text, be kind, see it and press Ctrl + Enter

ωn = υ 2

Substituting in the whole virase υ z (10.9), it is known that

ωn = ω2 R

Modulus of tangential acceleration is fine up to (9.8) better

I renew my equals (10.9), we take:

(ωR)

t → 0

t → 0

t → 0

t → 0

ωτ = βR

(10.10) d dt? huddled

Rβ,

Also, as normal, and tangentially accelerated linearly from R - the point in the direction of the wrapping axis.

§eleven. Link between vectors v and ω

Krіm looked at earlier the operations of folding and multiplying vectors, as well as multiplying a vector by a scalar (div. §2), and also the operation of multiplying vectors. Two vectors can be multiplied one by one in two ways: the first way results in a new vector, the other is reduced to a scalar value. It is significant that there is no operation to subdivide a vector onto a vector.

Let's take a look at sectoral vitvir vectors at once. Scalar dobutok vector_v we will introduce later, if you need wine.

The vector creation of two vectors A and B is called the vector Z, which volody such powers:

1) the module of the vector Z is a good addition to the modules of the vectors that are multiplied by the sine of the cut α between them (Fig. 35):

2) the vector C perpendicular to the plane, in which vectors A and B lie, moreover, the straight lines of the lines with the straight lines A and B according to the rule of the right screw: to marvel at the vector C, the turn along the shortest path from the first sp_multiplier to the other zdіysnyuєtsya for the year's arrow.

Symbolically vector TV can be written in two ways: | AB | or A×B.

We will use the first of these methods, and sometimes, for easier reading of formulas, we will put someone between the multipliers. It is not necessary to block the oblique cross and square arches at the same time: [А×В], Inadmissible record of this form: [AB]=ABsinα. Zliva is a vector here, right-handed is the modulus of the vector, which is a scalar. Righteous jealousy is coming:

| [AB] |= ABsinα.

Shards of a vector creation are directly related to the wrapper from the first multiplier to another, the result of vector multiplication of two vectors lies in the order of the multipliers. Changing the order of the multiplicands of the call, changing the direction of the resulting vector on the length (Fig. 35)

= −

B×A = − (A×B).

In such a rank, vector tvir cannot have the power of commutativity. You can tell that vector tvir is distributive, that

[A, (B1 + B2 + ... + BN)] = [AB1] + [AB2] + ... + [ABN].

The vector of the robot has two polar and two axial vectors and an axial vector. The vectorial addition of the axial vector to the polar (or otherwise) will, however, be a polar vector. Change the sign, which signifies directly the axial vectors, on the reverse, bring it into the opposite direction to change the sign in front of the vector boost and immediately to change the sign in front of one of the sp_multipliers, As a result, the value that is shown by the vector boost is lost without change.

The module of the vector creation can be given a simple geometric interpretation: ABsinα is numerically more parallel to the plane of the parallelogram created on the vectors A and B (Fig. 36; the vector C=[AB] of straight lines at this slope is perpendicular to the plane of the chair, behind the chair).

Let the vectors A and B be mutually perpendicular (Fig. 37).

1) , i approve s

Utavimo podviyne vektorne tvir tsikh vektoriv:

D = A, [BA],

so we multiply the vector by A, and then we multiply the vector A by the vector, which is the result of the first multiplication. The vector [VA] is the maximum modulus that is good BA(sin α = sin π 2

vectors A and B cuti, equal π/2. Also, the modulus of the vector D is more |A|*||=A*BA=A2 B. Direction of the vector D, as it is easy to see from fig. 37, zbіgaєtsya from the vector V. Tse give us the opportunity to write such a rіvnіst:

			A2B.

By formula (11.3) we have given the corystuvatimos non-one-time. Substantiate that it is only fair in that case, if the vectors A and B are mutually perpendicular.

Alignment (10.9) establishes a link between modules of vectors v and ω. For the help of the vector creation, it can be written viraz, which gives support between the vectors themselves. Let the body wrap around the axis z from the apex swidkistyu ω (Fig. 38). It’s easy to know that the vectorial addition ω to the radius-vector of a point, swidkity v, as we want to know, is a vector that runs directly with the vector v and may be the modulus, equal ωr sinα=ωR, tobto. v [div. formula (10.9)]. In this way, the vector complement [ωR] i after the direct i modulus is complementary to the vector v.

Come on V – n-peaceful vector space, in which two bases are given: e 1 , e 2 , …, e n- old basis, e" 1 , e" 2 , …, e"n- New basis. At a sufficient vector aє coordinates at the skin of them:

a= a 1 e 1 + a2 e 2 + … + a n e n;

a= a" 1 e"1+a" 2 e"2 + … + a" ne"n.

To insert a link between the coordinates of the vector a in the old and new bases, it is necessary to lay out the vectors of the new basis for the vectors of the old basis:

e 1 = a11 e 1 + a 21 e 2 + … + a n 1 e n,

e 2 = a 12 e 1 + a 22 e 2 + … + a n 2 e n,

………………………………..

e"n= a 1 n e 1 + a2 n e 2 + … + a nn e n.

Appointment 8.14. Matrix transition from the old basis to the new basis The matrix is ​​called, it is composed of the coordinates of the vectors in the new basis according to the old basis, writing down the columns, tobto.

Matrix columns T- all the coordinates of the basic, otzhe, linearly independent, vectors, otzhe, tsі stovptsі linearly independent. Matrix with linearly independent columns є non-virgin, її signifier not closer to zero і for the matrix T basic reversal matrix T –1 .

Significantly the coordinates of the vector a in old and new bases, obviously, like a] and [ a]". Behind the additional matrix for the transition, a link is established between [ a] and [ a]".

Theorem 8.10. Set vector coordinates a the old basis has a more advanced matrix transition to the coordinates of the vector a in a new basis, then [ a] = T[a]".

Consequence. Set vector coordinates a the new basis has a more advanced matrix, the return matrix transition, to the vector coordinates a at the old basis, then [ a]" = T –1 [a].

Example 8.8. Fold the transition matrix to the basis e 1 , e 2 , to basis e" 1 , e 2, de e" 1 = 3e 1 + e 2 , e" 2 = 5e 1 + 2e 2 i know the coordinates of the vector a = 2e" 1 – 4e 2 at the old basis.

Solution. The coordinates of the new basis vectors along the old basis are the rows (3, 1) and (5, 2) or the matrix T I'll look. So yak [ a]" = , then [ a] = × = .

Example 8.9. Given two bases e 1 , e 2 - old basis, e" 1 , e 2 - new basis, moreover e" 1 = 3e 1 + e 2 , e" 2 = 5e 1 + 2e 2. Know the coordinates of the vector a = 2e 1 – e 2 for the new basis.

Solution. 1 way. Behind the minds of the given coordinates of the vector but at the old basis: [ a]=. We know the transition matrix to the old basis e 1 , e 2 to new basis e" 1 , e 2. Take away the matrix T= for it we know the inversion matrix T-1 = . Similar to the corollary of Theorem 8.10, it is possible [ a]" = T –1 [a] = × = .

2 way so yak e" 1 , e 2 basis then vector but spread out behind the base vectors with an offensive rank a = k 1 e" 1 – k 2 e 2. We know the numbers k 1 ta k 2 - ce i will be the coordinates of the vector but at the new basis.

a = k 1 e" 1 – k 2 e" 2 = k 1 (3e 1 + e 2) – k 2 (5e 1 + 2e 2) =

= e 1 (3k 1 + 5k 2) + e 2 (k 1 + 2k 2) = 2e 1 – e 2 .

The coordinates of one and the same vector in a given basis are uniquely displayed, maybe the system: Virishyuchi tsyu system, otrimaemo k 1 = 9 that k 2 = -5, i.e. [ a]" = .

In this article, we have discussed with you one of the little sticks-viruchalochki, so as to allow you to take a lot of tasks from geometry to simple arithmetic. Tsya "stick" can actually make your life easier, especially for that kind of person, if you feel unstoppably at the promptings of the wide-ranging articles, revisiting the skinny. bud. Use it all to remember the songs, and show that practical beginners. The method, which we can see here in detail, is to allow you to practically completely abstract from various geometrical motives and mirroring. Ringing method "coordinate method". At this article, we can look at the following food with you:

1. Coordinate plane
2. Points and vectors on the plane
3. Pobudov vectors behind two points
4. Dovzhina vector (stand between two points)
5. Coordinates of the midpoint of the vіdrіzka
6. Scalar doboot vector_v
7. Kut mizh two vectors

I'm guessing, you already guessed why the coordinate method is called that way? That's right, having omitted such a name, the fact that VIN operates not with geometric objects, but with their numerical characteristics (coordinates). And the transformation itself, which allows you to go from geometry to algebra, is based on the advanced coordinate system. If the outward figure was flat, the coordinates are two-world, and if the figure is 3D, then the coordinates are three-dimensional. In these stats, we can see more than a two-dimensional vipadok. And the main meta stats - teach you how to use some basic methods of the method of coordinates (stinks sometimes appear to be the same as the hour of the day when the order is taken from the plan of the measurements in part B of the ED). Discussed methods of completion of the C2 task (the task of stereometry) were assigned to the attack by two divisions according to the topics.

Why would it be logical to talk about the coordinate method? Literally, from the understanding of the coordinate system. Guess, if you are stuck with her first. I’m wondering if I’m in the 7th class, if you know about the basis of the linear function, for example. I'll guess, you'll be її behind the points. Do you remember? You choose a sufficient number, substituting її at the formula and counting in such a rank. For example, yakscho, then, yaksho, those, etc. What are you taking away from the results? And otrimuvav ty specks with coordinates: i. Dali ty drawing a “cross” (coordinate system), choosing the scale (the number of cells you will have is a single cross) and assigning points to the niy you have taken, as if by going down a straight line, the line and the graph of the function have been taken off.

Here are a few moments, like a varto explain to you report:

1. A solitary wreath you choose for the mirroring of clarity, so that everything is beautifully and compactly placed on the little one

2. It is accepted that everything goes to the right, and everything goes uphill

3. The stench is tucked under a straight hem, and the point of the tread is called the cob of coordinates. Vaughn is signified by a letter.

4. In the record of the coordinate of the point, for example, left-handed, the shackles have the coordinate of the point along the axis, and right-handed, along the axis. Zokrema, simply means that at the point

5. In order to set a point on the coordinate axis, you need to specify її coordinates (2 numbers)

6. For any point that lies on the axis,

7. For any point that lies on the axis,

8. All is called all abscissa

9. All are called all ordinates

Now let's go with you zrobimo offensive krok: meaningfully two points. Z'єdnaєmo tsі two points vіdrіzkom. And let's put an arrow like this, we're going to do it from point to point: so we'll straighten our line!

Guess, what is the name of the straighteners? Maybe, wine is called a vector!

In such a rank, as if we were hitting point by point, moreover, we will have point A on the cob, and point B on the end, we take a vector. Qiu pobudovu tezh robiv at the 8th grade, do you remember?

It appears that vectors, like points, can be designated by two digits: qi digits are called the coordinates of the vector. Nutrition: how do you think, what is enough for us to know the coordinates of the cob and the end of the vector, to know the coordinates? It appears that it is so! And it’s even easier to fight:

In this order, since the point of the vector is the cob, and the point is the end, the vector may have advancing coordinates:

For example, yakscho, then the coordinates of the vector

Now let's go ahead and start, we know the coordinates of the vector. What do we need to change for what? So, it’s necessary to remember the cob and the end with the mists: now the vector’s cob will be at the point, and the end – at the point. Todi:

Marvel respectfully, what do vectors look like? Single їhnya vіdminnіst - tse signs in coordinates. The stench is proliferant. This fact is accepted to be written down as follows:

Sometimes, as it is not discussed specifically, like a dot is an ear of a vector, and a yak is a kіntsem, then vectors are not denoted by two great letters, but by one row, for example:, і etc.

Now trochs exercise yourself and find the coordinates of the upcoming vectors:

Revision:

And now rozvyazhi zavdannya troch folded:

Vector with the cob at the point maє co-or-de-na-ti. Find the abs-cis-su points.

All the same, dosit prosaic: Come on - coordinate points. Todi

I have created a system for the purpose of what is the coordinate of the vector. The same point can be coordinated. Us tsіkavit abscissa. Todi

Suggestion:

What else can you work with vectors? That may be all the same, scho і zі zvichaynymi numbers

1. Vectors can be folded one by one
2. Vectors can be seen one from one
3. Vectors can be multiplied (or multiplied) by quite a non-zero number
4. Vectors can be multiplied one by one

All these operations may be entirely geometrically manifested. For example, the tricot rule (or parallelogram) for folding and seeing:

The vector expands or shrinks or changes directly when multiplying or expanding by a number:

However, here we need food, what should we look for with the coordinates.

1. When folding (adding) two vectors, we add (read) element by element their coordinates. Tobto:

2. When multiplying (dividing) the vector by the number of all coordinates, multiply (divide) by the whole number:

For example:

· Find the sum of co-or-di-nat vіk-to-ra.

Let's start by knowing the coordinates of the skin vector. Having offended the stench, you can make the same cob - a point on the cob of coordinates. They have different kints. Todi, . Now we can calculate the coordinates of the vector. Then the sum of the coordinates of the extracted vector is more.

Suggestion:

Now untie yourself on the offensive:

Know the sum of the coordinates of the vector

Verify:

Let's look at the problem now: we have two points on the coordinate plane. How to know how to get between them? Let the first point be, but a friend. Significantly stand between them through. Let's zrobimo for the sake of accuracy, the chair is coming:

What am I doing? First, I connected the points i, and also the points of the line, parallel to the axis, and the points of the line, parallel to the axis. The stench twitched to the point, having made a miraculous figure with whom? Why is she a miracle? You and I may know everything about the straight-cut tricutnik. Well, the Pythagorean theorem, for sure. Shukany vіdrіzok - tse hypotenuse of this tricot, and vіrіzki - catheti. Why are points coordinates equal? So, it’s not easy to know them behind the picture:

Now we are speeding up with the Pythagorean theorem. Dovzhini cathetiv we know, we know the hypotenuse:

In this order, between two points - the root of the sum of the squares of the difference from the coordinates. Abo well - stand between two dots - the price of a dozhina vіdrіzka, what happened to them. It is easy to remember that in the midst of speckles you can’t lay flat in a straight line. Todi:

Zvіdsi robimo three visnovki:

Let's get better at the number of points between two points:

For example, yakscho, then stand between and one

Abo pіdemo іnakshe: we know the coordinates of the vector

І we know the length of the vector:

Yak bachish, one and the same!

Now troch work out yourself:

Task: know the distance between the indicated points:

Verify:

There are a couple of tasks for the same formula, but it really does sound like a stink of a trifle else:

1. Know-dі-those square dovzhini vik-to-ra.

2. Know-dі-te square dovzhini vik-to-ra

I think you can deal with them easily? Verify:

1. And the cost of piling) We already knew the coordinates of the vectors and earlier: . Then the vector can have coordinates. Yogo Square

2. We know the coordinates of the vector

Todi square yogo dozhini dorіvnyuє

Nothing fancy, right? Zvichayna arithmetic, no more.

The coming task cannot be unambiguously classified, the stench is quicker than wild erudition and in the meantime draw simple pictures.

1. Find the sine of the kuta on-clo-on the vіd-rіz-ka, z-є-nya-y-th-th point, z vіsyu abscissa.

і

How can we fix it here? It is necessary to know the sine of the kuta mіzh i vіssyu. And de mi vmієmo shukati sinus? That's right, with a straight-cut tricoutnik. What do we need to grow? Indulge your trickster!

Oskіlki coordinates of the point and then vіdrіzok dorіvnyuє, but vіdrіzok. We need to know the sine of kuta. I’ll tell you that the sinus is the extension of the protilegus leg to the hypotenuse, then

What we have lost zrobiti? Know the hypotenuse. You can work in two ways: by the Pythagorean theorem (katety vіdomі!) or by the formula between two points (indeed, one and the same, which is the first way!). I'm going another way:

Suggestion:

The next day it will be easier for you. Vaughn - on coordinate points.

Task 2. 3 points of descent per-pen-di-cular on the entire abs-cis. Find abs-cis-su os-no-va-nya per-pen-di-ku-la-ra.

Let's crush the little ones:

The substava of the perpendicular is the center point, in the yakіy vіn the entire abscissa (vіs) is changed, at the lower point. On the little one you can see that the coordinates are: . The abscissa is to call us - tobto "iksova" warehouse. She's good.

Suggestion: .

Task 3. At the time of the front task, know the sum of the distances from the points to the coordinate axes.

The head of the fire was elementary, as you know, what is the way to get from the point to the axes. Do you know? I spodіvayus, but all the same I tell you:

Otzhe, on my little one, the troch troch is bigger, have I already painted such a perpendicular? Until what wine axis? Up to the axis. And why is Yogo Dozhina worthy? She's good. Now draw a perpendicular to the axis yourself and find out yoga dozhina. Won dorivnyuvatime, right? Todi їkhnya sum dorivnyuє.

Suggestion: .

Task 4. In the minds of task 2, find the ordinate of a point that is symmetrical to the point along the x-axis.

I think you intuitively understood what symmetry is? Even rich objects can be made with it: rich boudinkov, tables, litakiv, rich geometric shapes: kulu, cylinder, square, rhombus and so on. . Such symmetry is called axial. And what else is it all about? Why is that line, behind which figure you can, mentally seeming, “cut” on the same halves (in this picture, all symmetry is straight):

Now let's turn around to our leader. We see that we are looking for a point that is symmetrical to some axis. Todі tsya all - all symmetry. Otzhe, we need to designate such a point, so that all cut the vines on equal parts. Try it yourself to identify such a point. And now compare with my decisions:

Did you feel like that? Dobre! At the found point, we have to click the ordinate. Won dorivnyuє

Suggestion:

And now tell me, after thinking for a second, why do I need the abscissa of the point, the symmetrical point A, how about the y-axis? What is your opinion? The correct answer is: .

For a zagal vipad, the rule can be written as follows:

Krapka, symmetrical to the point along the abscissa axis, may coordinate:

Krapka, symmetrical to the point along the axis of ordinates, may coordinate:

Well, now it's scary manager: know the coordinates of a point that is symmetrical to a point along the cob of coordinates. Think about it for yourself, and then look at my little one!

Suggestion:

Now Task on the parallelogram:

Task 5: Krapki yav-la-yut-sya ver-shi-na-mi paral-le-lo-lo-gram-ma. Find the op-di-na-tu point.

You can solve problems in two ways: logic and the method of coordinates. I'll start the method of coordinates on the back, and then we'll write it down, as if it's different.

It is quite clear that the abscissa of the point is correct. (Won to lie on the perpendicular, drawn from the point to the abscissa axis). We should know the ordinate. Let's speed up, because our figure is a parallelogram, tse means that. Let's know the double wedge, vicorist formula between two points:

We lower the perpendicular, so that we get a fleck from the veil. I will mark the breakpoint with a letter.

Dovzhina vіdrіzka dorіvnyuє. (Find the problem itself, demi discussed this moment), then we know the difference between the two according to the Pythagorean theorem:

Dovzhina vіdrіzka - exactly zbіgaєtsya z yoga ordinate.

Suggestion: .

Other decision (I'll just bring the little ones, what to illustrate)

Hіd vyshennya:

1. Spend

2. Know the coordinates of the point and the distance

3. Bring what.

Another one Order for a dozhina vіdrіzka:

Krapki are-la-huddle tops-shi-on-mi trikutniks. Find the length of the middle line, parallel.

Do you remember, what is the middle line of the tricutnik? Just the same task is elementary. If you don’t remember, then I’ll guess: the middle line of the knitwear is the whole line, as it happens in the middle of the opposite sides. Vaughn is parallel to the core and the most important half of it.

Pidstava - tse vіdrіzok. Yogo dozhina we had a chance to shukati earlier, even more so. The same is true of the middle line of the second line, which is smaller and older.

Suggestion: .

Commentary: tse zavdannya can be done and in a different way, to the extent that we are able to bear the last three.

In the meantime, the axis of you is a spiel, work out on them, the stench is even simpler, but help to “stuff your hand”, using the best way of coordinates!

1. Krapki yav-la-yut-sya tops-shi-on-mi tra-pe-tsії. Find the length of the middle line.

2. Krapki and yav-la-yut-sya tops-shi-na-mi paral-le-lo-gram-ma. Find the op-di-na-tu point.

3. Know-di-those dovzhina vіd-rіz-ka, z-e-nya-th-th-th point i

4. Know-dі-those area for-beautiful fі-gu-ri on co-or-di-nat-noї flat-to-stі.

5. Surroundings with the center in na-cha-le ko-or-di-nat to pass through the point. Know-dіt її ra-dі-vus.

6. Find-di-te ra-dі-us colo-no-stі, describe-san-noї bіla straight-mo-kut-nі-ka, ver-shi-no-something-ro-go-ko-or -dі-na-ti zі-vіd-vіt-stven-but

Solution:

1. It seems that the middle line of the trapezium is more beautiful than the sum of the basics. The basis is good, but the base. Todi

Suggestion:

2. The simplest way to do this is to remember what (the rule of a parallelogram). Calculate the coordinates of vectors i easily: . When vectors are folded, the coordinates are added. Todi maє coordinates. Qi coordinates maє і point, oskіlki cob vector - tse point with coordinates. Ordinate to squawk us. She's good.

Suggestion:

3. Diemo next to the formula between two points:

Suggestion:

4. Look at the picture and tell me, is the shaded area squeezed between two figures? Vaughn is squeezed between two squares. These are the squares of the shukano figure and the equal squares of the great square, minus the square of the small one. The side of the small square is the tse vіdrіzok, scho z'ednuє points and yogo dozhina dorіvnyuє

Even the area of ​​a small square is more expensive

So it’s done by itself and with a great square: the yogo side is tse vіdrіzok, scho joining the points and yogo dozhina is more expensive

Todi area of ​​the great square is more expensive

The area of ​​the shukano figure is known by the formula:

Suggestion:

5. As soon as the center of the cob of coordinates and pass through the point, then the radius will be exactly the same as the old one of the vіdrіzka (throw the little ones and you understand, why it's obvious). Let's know the length of this wind:

Suggestion:

6. It seems that the radius of the described square of the rectangle of the stake is more than half of the diagonal. We know the dozhina, whether it be from two diagonals (even if the straight-cut stink is equal!)

Suggestion:

Well, did you manage to do it? Bulo is not too easy to grow up, right? There is only one rule here - remember to look at the picture and just “rahuvat” all the data from it.

We have lost our good fortune. There are literally two more points that I would like to discuss.

Let's try to solve the axis of such a simple task. Give the given two points in. Find the coordinates of the middle of the vіdrіzka. The solution of this task is as follows: let the point - the middle is shukana, then the same coordinates:

Tobto: coordinates of the middle of the vіdrіzka = arithmetic mean of the coordinates of the ends of the vіdrіzka.

It’s even simpler and don’t call out the difficulties of the students. Let's marvel at some zavdannya and how victorious it is:

I

2. Krapki yav-la-yut-sya ver-shi-na-mi che-ti-reh-vogі-no-ka. Know-dі-te op-dі-on-that point pe-re-si-che-nya yogo dia-go-on-lei.

3. Know-dі-te abs-cis-su of the center of the circle, describe-san-noї bіla straight-mo-kut-nі-ka, ver-shi-no-something-ro-go-to or-dі-na-ti zі-vіd-vіt-but.

Solution:

1. The first task is just a classic. Dіёmo vіdrazu for the designation of the middle of the vіdrіzka. Vaughn can coordinate. Ordinate is good.

Suggestion:

2. It is easy to bachiti, that this chotirikutnik is a parallelogram (navit a rhombus!). You yourself can bring it, virahuvavshi dozhina sides and equalize them between yourself. What do I know about a parallelogram? Yogo diagonally with a dot of the peretina navpil! Aha! Mean the point of crossing the diagonals - what? Tse middle be like a diagonal! Viberu, zokrema, diagonal. Then the point can be coordinated The ordinate of the point, which is more expensive.

Suggestion:

3. Why is the center of the described square of the stake squared? Vіn zbіgaєtsya with a point of crossing of the yogo diagonals. What do you know about the diagonals of a rectangle? The stench is equal and the point of the cross is navpil. The manager was ringing up to the front. Take, for example, the diagonal. Todi yakshcho is the center of the described stake, then it is the middle. Shukayu coordinates: Abscissa rіvna.

Suggestion:

Now, work out a little bit on your own, I’ll just lead you to the skin care, so that you can’t believe yourself for a moment.

1. Know-di-te ra-di-us of the circumference, describe the san-no ї bele of trikutnik, top-shi-no-so-ro-go may ko-or-de-on ti

2. Know-dі-te or-dі-on-that center of the circle, describe-san-noї bіla trikutnik, the tops of someone may ko-or-dі-on-ti

3. What r-dі-u-su can be buti colo z the center at the point where the axis abs-cis was sticking out?

4. Find-dі-te op-dі-on-that point pe-re-se-che-nya osі і vіd-rіz-ka, z-e-nya-yu-th-th point i

Suggestions:

Is everything gone? I’m already rooting for you! Now - the rest of the row. Be especially respectful now. That material, which I will explain at once, can be applied not only to simple tasks on the method of coordinates of part B, but it is also used everywhere in problem C2.

Yaku zі svoїh obіtsyanok I have not yet finished trimming? Guess what kind of operations on vectors I announced to be carried out and how allowed forever? I didn't forget anything? Forget it! Forgetting to explain what the multiplicity of vectors means.

There are two ways to multiply a vector by a vector. In the opposite way, we will have objects of a different nature:

Vector tvіr vykonuetsya dosit cunningly. How yoga work and now it is necessary, we will discuss with you in the next article. And in tsіy mi zupinimsya on the scalar creation.

There are already two ways that allow us to calculate yoga:

As soon as you guessed, the result can be the same! Otzhe, let's take a look at the first way:

Scalar twir via coordinates

To know: - greedily accept the meaning of the scalar creation

The calculation formula is:

Tobto scalar witwir = sum of creative coordinates of vectors!

Butt:

Find out

Solution:

We know the coordinates of the skin from vectors:

The scalar twir is calculated using the following formula:

Suggestion:

Bachish, nothing complicated!

Anu, now try it yourself:

Know-di-te ska-lyar-not pro-z-ve-de-nie v_k-to-r_v i

Rushed? Perhaps, th approach is a small reminder? Let's revise:

Coordinates of vectors, as in the past! Suggestion: .

Krіm coordinate, є th Іnshy way to calculate the scalar tvіr, and itself through the two vectors and cosine kuta between them:

Designates kut between vectors ta.

That is why the scalar supplementation is more efficient than the augmentation of the vectors by the cosine of the cut between them.

Well, we need a different formula, because we have a first, like a richly simple one, we don’t have any common cosinuses. And you will need it for the fact that with the first and other formulas you can show how to know between vectors!

Come on, guess the formula for the next vector!

Just as I am substituting the qi of the data before the formula of the scalar creation, then I subtract:

Ale from the other side:

What did we take away from you? We now have a formula, so I can calculate between two vectors! Other words for style are written as follows:

This is the algorithm for calculating kuta between attack vectors:

1. Calculable scalar TV in terms of coordinates
2. We know dozhini vector_v and multiply їх
3. We divide the result of point 1 by the result of point 2

Let's practice on the butts:

1. Know-dі-te kut mіzh vіk-to-ra-mi i. Give proof to the gra-du-sah.

2. In the minds of the forward task, find the cosine between vectors

Let's do it this way: first of all, I will help you to do it yourself, and to the other, try to do it yourself! Good? Let's fix it!

1. Qi vectors - our old know. We already respected their scalar tver and vіn equal. The coordinates are as follows: , . Todі we know їх dozhini:

Then there is the cosine between vectors:

The cosine of which kuta is more expensive? Tse cut.

Suggestion:

Well, now I’ll tell my friend the manager himself, and then we’ll fight! I'll give you a little more short solution:

2. can coordinate, can coordinate.

Come on - kut mizh vectors i todi

Suggestion:

Slid assign, scho zavdannya directly on the vector i method of coordinates in the part B of the examination work to complete the examination. However, the more important task C2 can be easily changed by going to the introduction of the coordinate system. So, you can use this article as a foundation, on the basis of such a peaceful time, you can achieve cunning prompting, like we need to complete complex assignments.

COORDINATES AND VECTORS. MIDDLE AT RIVEN

We continue to use the coordinate method. For the past few years, we have developed a number of important formulas that allow:

1. Know the coordinates of the vector
2. Find the length of the vector (alternatively: move between two points)
3. Fold, view vectors. Multiply їх on speech number
4. Know the middle of the wind
5. Calculate the scalar gain of vector_v
6. Know the cut between vectors

Obviously, 6 points do not include the entire coordinate method. Vіn lie in the basis of such a science, like analytical geometry, which you should learn from VNZ. I want to build a foundation, which will allow you to take orders from a single state. exams. Іz zavdannymi part B mi rozіbralis in the hour has come to move on like a new rіven! This article will be devoted to the method of completion task C2, in which case it would be reasonable to go to the method of coordinates. Tsya razumnіstnost vznachaetsya tim, scho zavdannya it is necessary to know, and how to post is given. So, I started to set up the coordinate method, which is how to set the power:

1. Know kut between two flats
2. Know the cut between the straight line and the flat
3. Know the cut between two straight
4. Know the distance from a point to a plane
5. Know the distance from a point to a straight line
6. Know the distance from the straight line to the square
7. Know the difference between two straight

Yakshcho given for the mind of the head of the figure є body wrapping (sack, cylinder, cone ...)

Attached figures for the method of coordinates є:

1. Rectangular parallelepiped
2. Pyramida (trikutna, chotirikutna, six-kutna)

So with my knowledge underestimates the method of coordinates for:

1. Significance of area pereriziv
2. Calculation of obsyagіv tіl

Prote next designate that three "invisible" for the method of coordinates of the situation it is practical to complete the calculations. For the greater, the leader of the wines can become your ryativnik, especially as you are not so strong among the trivi- mer ones (as they often do with the cunning).

What are all the other posts listed by me? The stench is no longer flat, like, for example, a square, a tricutnik, a colo, but a volume! Obviously, we need to consider not a two-world, but a three-world coordinate system. It will be easy to finish it: just crim the axis of the abscissa and the ordinates, we will introduce one more, all the app. On the little one is schematically depicted their mutual roztashuvannya:

All stinks are mutually perpendicular, overlapping in one point, which is what we call the cob of coordinates. All abscissa, like before, meaningfully, all ordinates - , and all the applique - .

Whereas earlier a skin point on a plane was characterized by two numbers - an abscissa and an ordinate, then a skin point in space is already described by three numbers - an abscissa, an ordinate, an applique. For example:

The abscissa of the dot is clearly correct, the ordinate is , and the applique is .

Sometimes the abscissa of the point is also called the projection of the point on the entire abscissa, the ordinate - the projection of the point on the entire ordinate, and the applique - the projection of the point on the entire applicator. Obviously, if a point is given, a point with coordinates:

call the projection of a point onto a plane

call the projection of a point onto a plane

Stay natural nourishment: what are all the formulas that are justified, for a two-world vipadka, in space? The sound is firm, the stench is fair and may be the very sight. For a small detail. I think you already figured it out yourself, after yourself. In all the formulas of guilt, we will add one more member, which is valid for the entire application. And to herself.

1. How to set two points: , then:

• Vector coordinates:
• Move between two points (or two vectors)
• The middle of the vіdrіzka maє coordinates

2. If two vectors are given: i, then:

• Їх scalar tvіr dorіvnyuє:
• Cosine of kuta between vectors do_vnyuє:

However, not everything is so simple in space. How do you understand, adding one more coordinate to introduce a sense of diversity in the spectrum of figures that “live” in this space. And for further rozpovidi me it will be necessary to send a deak, rudely seeming, "zagalnennya" straight. Tsim zagalnennyam will be flat. What do you know about flatness? Try s vіdpoviddu, but what is the flat? It's important to say. Prote mi everything is intuitively revealed, as if looking out:

Roughly kazhuchi, tse yakys neskіchenny "arkush", tucked into the expanse. "Inconsistency" is a trace of understanding that the area expands on all sides, so it's a square of more inconsistency. However, this explanation "on the fingers" does not give the least information about the structure of the plane. And we are out of the way.

Let's guess one of the main axioms of geometry:

• through two different points on the plane there is a straight line, before that there is only one:

Abo її analogue of space:

Obviously, you remember, as for two given points to lead straight lines, it doesn’t matter: if the first point has coordinates: but the other, then straight lines will be attacked:

Tse ty passing at the 7th class. In the expanse of straight lines, the axis looks like this: let's have two points with coordinates:

For example, through the points, go straight:

How can you understand? Tse next to understand the axis yak: the point lies on a straight line, so that the coordinates satisfy such a system:

There is no other way for us to appreciate the direct vector of the straight line, but we need to give respect to the important understanding of the direct vector of the straight line. - be a non-zero vector that lies on the line or is parallel to it.

For example, offensive vectors and є direct vectors of the straight line. Come on - a point that lies on a straight line, and - a direct vector. You can write the same straight lines in such a way:

Once again, I repeat, I won’t be more direct than a straight line, but it’s necessary for me to remember that such a direct vector! One more time: tse BE-YAKIYA non-zero vector that lies on a straight line, or parallel ї th.

Vivesti leveling of the area beyond three given points it's not so obvious anymore, and the sound of food is not seen in the course of middle school. And darma! Tsej priyom zhittєvo nebhіdny, if we go to the method of coordinates on the top of folding tasks. However, I admit, what did you learn from the bajannya about something new? Moreover, you can impress your vikladach at the VNZ, if you know that you are already familiar with the methodology, as you sound at the course of analytical geometry. Otzhe, let's do it.

The flatness of the flat is not overthrown by the flatness of the straight line on the flat, but it can look out of itself:

decimal numbers (usі equal to zero), and zminnі, for example: thinly. As a matter of fact, the flatness of the plane does not even cross over into the straight line (linear function). Prote, guess what we have hardened with you? We said that since we have three points, if they do not lie on one straight line, then the flatness of the plane is uniquely inspired by them. Hello yak? I'll try to explain to you.

Shards of flatness of the area can be seen:

And the points lie on this plane, then when setting the coordinates of the skin point on the plane of the plane, we are responsible for taking the correct identity:

In this rank, it is necessary to make three equals already out of the unknown! Dilemma! However, you can always admit that (for which it is necessary to add). In this rank, we take three equals from the trio of indispensable:

However, we do not violate such a system, but write down a mysterious expression, as if we were screaming from a new one:

Flatness of the plane that passes through three given points

\[\left| (\begin(array)(*(20)(c))(x - (x_0))&((x_1) - (x_0))&((x_2) - (x_0))\\(y - (y_0) )&((y_1) - (y_0))&((y_2) - (y_0))\\(z - (z_0))&((z_1) - (z_0))&((z_2) - (z_0)) \end(array)) \right| = 0\]

Stop! What else is it? What an invisible module! However, the object, like you are walking in front of you, does not have anything to do with the module. This object is called the third-order primate. Vіdteper i nadalі, if you matimesh on the right with the method of coordinates on the plane, then you will often see the signs. What is a third-order vyznachnik? It’s not surprising, it’s more than just a number. I lost my mind, like the very number we set as a signifier.

Let's write down the head of the third order for a savage looker:

De - deakі numbers. Moreover, under the first index we understand the row number, and under the index - the number of the column. For example, it means that the number is on the peretina of another row and the third row. Let's put the foot on the food: what kind of rank do we count such a vyznachnik? So, how will we give you the number itself? For the vyznachnik itself of the third order, heuristically (on the face of it), the trikutnik’s rule looks like this:

1. Additional elements in the head diagonal (from the upper left kuta to the lower right)
2. Extraction of elements in the side diagonal (from the upper right hem to the lower left)
3. Todi vyznachnik more expensive retail value, otrimanih on crocita

To write everything down in numbers, we take the following viraz:

Tim is not less, remember the way of counting in such a look is not necessary, it’s enough in your head to simply keep the tricks and the idea itself, what is going on and what is being seen later).

Let's illustrate the method of tricks on the butt:

1. Calculate the winner:

Let's figure out what we store, and what we see:

Dodanki, how to go from the "plus":

The main diagonal: additional elements of the door

First trikutnik, “perpendicular to the head diagonal: additional elements

Another tricutnik, “perpendicular to the head diagonal: additional elements of wood

We add three numbers:

Dodanki, yakі go with a "minus"

Tse side diagonal: additional elements

The first tricoutnik, “perpendicular to the side diagonal: additional elements

Another tricutnik, “perpendicular to the side diagonal: additional elements

We add three numbers:

Everything that has been left out of work - you can see it with the sum of donations “with plus” the sum of dodankiv with “minus”:

in such a manner,

Yak bachish, there is nothing coherent and supernatural among the counted vyznachniki in the third order. It's just important to remember the tricksters and not allow arithmetic pardons. Now try to independently virahuvati:

Verify:

1. First tricout, perpendicular to the head diagonal:
2. Another tricot, perpendicular to the main diagonal:
3. Amount of dodankіv іz plus:
4. The first tricot, perpendicular to the side diagonal:
5. Another tricot, perpendicular to the side diagonal:
6. Amount of dodankiv with a minus:
7. Amount of dodankiv іz plus minus amount of dodankіv іz minus:

The axis is also a couple of vyznachnikiv, counted their values ​​independently and equalized from the vіdpovіdyami:

Suggestions:

Well, did everything go wrong? Good, then you can collapse far away! Even though it’s difficult, then my pleasure is this: on the Internet, there is a bunch of programs for calculating the manager on-line. All that is necessary for you is to come up with your own leader, calculate it independently, and then we will compensate for it, that the program is important. І so doti, doki results do not start spіvpadati. Upevneniy, tsey moment not zmusit dovgo chekati!

Now let's turn to that signpost, which I wrote down, if I was talking about leveling the plane through three given points:

All that is necessary for you is to calculate the value without a middle (using the trikutnik method) and equate the result to zero. Zvichayno, the shards are change, then you take away a deaky viraz, which should be deposited in them. The very viraz and will be equal to the plane, which will pass through three given points, which will not lie on one straight line!

Let's illustrate what was said in a simple example:

1. Encourage the plane to pass through the points

We add for these three points of the signpost:

Let's just say:

Now yoga is counted without intermediary following the rule of tricks:

\[(\left| right| = \left((x + 3) \right) \cdot 0 \cdot 0 + 2 \cdot 1 \cdot \left((z + 1) \right) + \left((y - 2) \right) \cdot 5 \cdot 6 - )\]

In this order, equal to the plane, which can pass through the points, you can look:

Now try to sing one day on your own, and then let's talk about it:

2. Know the alignment of the plane to pass through the points

Well, now let's discuss the decision:

Let's make a sign:

І calculable value:

Todi leveling of the area may look:

But well, short on, take it away:

Now two tasks for self-control:

1. Encourage the plane to pass through three points:

Suggestions:

Has everything gone wrong? Well, even though it’s a difficult song, then my pleasure is this: you take three points from your head (with a great step of imovirnosti they will not lie on one straight line), you will be flat behind them. And then we will check it online for ourselves. For example, on the website:

However, for the help of the clergymen, we will not only be equal to the area. Guess, I'll show you what is assigned for vectors not only scalar twir. More vector, as well as zmіshany tvіr. If the scalar creation of two vectors i will be a number, then the vector creation of two vectors i will be a vector, moreover, the vector of perpendiculars to the tasks:

Moreover, the yogo module is the same as the area of ​​the parallelogram built on the vectors i. This vector is needed to calculate the number of points from a point to a straight line. How do we get the vector TV of the vectors i, like their coordinates of the task? For help, the third-order vyznachnik comes again. However, first of all, I will move on to the algorithm for calculating the vector creation, I will try to make a small lyrical entry.

Tsey access to the basic vectors.

Schematically, the stink of the image of the little one:

How do you think, why are stinks called basic? On the right in that:

Abo on the image:

The validity of this formula is obvious, even:

Vector vitvir

Now I can start introducing vector art:

The vector creation of two vectors is the vector that is calculated according to the following rule:

Now we are going to add some examples of calculation of vector creation:

Example 1: Know the vector boost of vectors:

Solution: I'm putting together a sign:

I love yoga:

Now looking at the base vector notation, I'll turn to the base vector notation:

In this manner:

Now try.

Ready? Verify:

I traditionally two tasks for control:

1. Find the vector TV of upcoming vectors:
2. Find the vector TV of upcoming vectors:

Suggestions:

Zmishany tvir three vectors

The rest of the construction, as I need it, is the result of the confusion of three vectors. Vono, yak i scalar, є number. There are two ways of calculating. - through vyznachnik, - through zmishane tvir.

And for yourself, come on, we are given three vectors:

Then three vectors, which are indicated through, can be calculated as:

1. - tobto shift tvir - all scalar tvir of the vector on the vector tvir of two other vectors

For example:

Independently try to calculate yoga through the vector twir and change your mind, the results will fall!

I again - two butts for an independent vision:

Suggestions:

Choice of coordinate system

Well, now we have the axis of the entire necessary foundation of knowledge, so that we can create folding stereometric tasks from geometry. However, the first thing to do is to proceed without middle ground to applying that algorithm to their versatility, I respect that it will be corny-colored on any food: as the very choose a coordinate system for those other shapes. Even if you choose the mutual expansion of the coordinate system and the figures in the space, it’s possible to designate, the bulk of the bulk will be calculated.

I'll guess what we see in such a post:

1. Rectangular parallelepiped
2. Straight prism (trikutna, six-kutna ...)
3. Pyramida (trikutna, chotirikutna)
4. Tetrahedron (one and the same as trikutna pyramid)

For a rectangular parallelepiped or a cube, I recommend the following approach:

Tobto figure I will place "in kut". The cube and the parallelepiped are good figures. For them, you can easily know the coordinates of your vertices. For example, yakscho (as shown on the little one)

then the coordinates of the vertices are:

Remembering, zvichayno, not necessary, prote memory, as a better mother cube or a straight-cut paralepiped - bazhano.

Straight prism

Prism - more shkidliva post. Roztashovuvati її in space can be done in a different way. However, the most acceptable option seems to be:

Tricut prism:

Tobto one of the sides of the tricutnik we put it on the whole, moreover, one of the vertices goes with the cob of coordinates.

Six-point prism:

That is why one of the vertices zbіgaєtsya with the cob of coordinates and one of the zі storіn lies on the axis.

Chotirikutna that six-kutna pyramid:

The situation is similar to a cube: two sides of the base are one by one with the coordinate axes, one of the vertices is one by one with the cob of coordinates. A single small folding to unravel the coordinates of the point.

For a six-fold pyramid - similarly, as for a six-fold prism. The main task will be to find out the coordinates of the vertex.

Tetrahedron (tricutna pyramid)

The situation is similar to tієї, as I grafted for a triangular prism: one vertex goes along the cob of coordinates, one side lies on the coordinate axis.

Well, now we are close to you, so that we can proceed to the cherry day. After what I said on the very cob of the article, the same moment the axis of a kind of vysnovok was created: more tasks C2 are divided into 2 categories: tasks on the cut and tasks on the vіdstan. At the back of my head, we’ll look at the well-known kuta with you. The stench of their line is subdivided into the following categories (the world has more folding):

Asking for a search for kutiv

1. Znakhodzhennya kuta mizh two straight lines
2. Znakhodzhennya kuta between two flats

Let's take a look at these tasks one by one: let's look at the knowledge of the kuta between two straight lines. Well, guess what, why didn’t you try similar ones with you earlier? Guess, aje mi already small schos like this ... Mi shukali kut mizh two vectors. I’ll guess you, as two vectors are given: i, then how can they be known from the spіvvіdnosheniya:

Now, however, we may have a meta - a sign of a kuta between two straight lines. Let's go wild to the "flat picture":

Skіlki we have wiyshlo kutіv when crossing two straight lines? Already things. True, only two of them are not equal, others are vertical to them (and they avoid them). Then what kind of kut to us vvazhat kutom between two straight lines: chi? Here the rule is: cut between two straight lines no more than lower degrees. Tobto from two kutiv we will always choose a kut from the smallest degree world. Tobto on this picture cut between two straight lines. In order not to fool around with the joke of the smallest of two kutivs, cunning mathematicians propagated the victorious module. In this order, the cut between the two is directly dependent on the formula:

You, like a respected reader, don’t have enough food: and the stars, well, we take those numbers themselves, as we need for the calculation of the cosine of the kuta? Note: we are brothers of direct vectors of straight lines! In this rank, the algorithm for knowing the kuta between two straight lines looks like this:

1. Zastosovuєmo formula 1.

Abo reporter:

1. Shukaєmo coordinates of the direct vector of the first line
2. Shukaєmo coordinates of the direct vector of the other straight line
3. Calculating the modulus of the new scalar creation
4. Shukaemo dozhina first vector
5. Shukaёmo dovzhina another vector
6. We multiply the results of point 4 by the results of point 5
7. We divide the result of point 3 with the result of point 6. We take the cosine of kuta between the straight lines
8. Even if the result allows exactly virahuvati kut, joking yoga
9. Otherwise, we write through the arc cosine

Well, now it’s time to move on to the day: I will demonstrate the solution of the first two in a report, I will present the solution of the second one in a short look, and before the remaining two days I will give no more than suggestions, all the calculations before them it’s your fault to carry out yourself.

Manager:

1. The right tete-ra-ed-re knows-di-te kut mi-zh vy-so-that tete-ra-ed-ra and me-di-a-noi bo-koi faces.

2. At the right-wild shost-vugilny pi-ra-mi-de hundred-ro-no OS-no-va-nya-to-roї equal, and more-to-vі ribs equal, know the cut between the straight lines i.

3. Keep all the ribs of the correct four-ti-rekh-vugіlnoi pі-ra-mі-di equal among themselves. Know-dі-te kut m_zh straight-mi-mi and yakscho vіd-rіzok - vy-so-ta dan-noї pі-ra-mі-di, dot - se-re-di-on її bo-ko-vo- th edge

4. On the edge of the cube, there is a point so that Nai-di-te cut between straight lines

5. Point - se-re-dі-on the edges of the cube

I'm unfailingly putting the tasks in that order. I still haven’t managed to orient myself in the method of coordinates, I myself will sort out the most problematic figures, but I’ll let you figure out the simplest cube! Step by step, you should learn how to practice with us in figures, I’ll change the order of the day from one to the other.

Let's get to the cherry-pick:

1. Small tetrahedron, move yogo to the coordinate system just like I did earlier. Oskіlki tetraeds are correct - all yogo faces (including the base) - are correct trikutniks. Oskіlki we are not given dovzhina side, then I can accept її equal. I think, you understand, what is really not stale, considering how much our tetrahedron will be “stretched”? I will also draw height and median in tetrahedron. Preferably, I will paint yoga support (we will need it).

It is necessary for me to know kut mizh i. What do we see? We don't have the coordinate of the point. Otzhe, you need to know the coordinate point. Now we think: the point is the whole point of the line of heights (either bisectrix or median) of the tricutnik. A dot is a chained dot. The point w is the middle of the vіdrіzka. Then it is enough to know: coordinate point: .

Let's start from the simplest: point coordinates. Look at the little ones: It is clear that the applicator of the point is equal to zero (the speck lies on the flat). Її ordinate dorіvnyuє (oskіlki - median). It is more convenient to know the її abscissa. However, it is easy to fight on the basis of the Pythagorean theorem: Look at the trickster. Yogo hypotenuse is good, and one of the catheters is good:

Remaining maєmo: .

Now we know the coordinates of the point. It is clear that її the applicate is new to zero, and the її ordinate is the same, like at a point, tobto. We know її abscissa. It’s trivial to try to finish it, as if to remember that heights of an equal-sided knitted fabric with a cross-point to be divided by a proportion top view. Oskіlki: then shukana abscissa of a point In this order, the coordinates of the points are updated:

We know the coordinates of the point. It is clear that її the abscissa and the ordinate go beyond the abscissa and the ordinate of the point. And the applique is a good old one. - this is one of the catheters of trikutnik. The hypotenuse of tricot - ce vіdrіzok - leg. Vіn shukaє z mirkuvan, yaky I saw in bold type:

Krapka is the middle of the vіdrіzka. Then we need to guess the formula for the coordinates of the middle of the vіdrіzka:

That's all, now we can shukati coordinates of direct vectors:

Well, everything is ready: we submit all the data to the formula:

in such a manner,

Suggestion:

It’s not your fault that you’re lying like this “zhahlivy” vіdpovіdі: for problems C2, it’s a great practice. I would sooner zdivuvavsya b "beautiful" vіdpovіdі in this part. So, as a reminder, I practically did not go into anything, except for the Pythagorean theorem and the height of the heights of an equal-sided tricutnik. Therefore, for the completion of a stereometric task, I have chosen a minimum of stereometry. Vigrash at tsyomu is often “extinguished” by bulky charges. Then stink to dosit algorithmic!

2. Imagine the correct six-sided pyramid at once from the coordinate system, as well as the base:

We need to know the cut between straight lines. Otzhe, our zavdannya zavdannya to search for the coordinates of the point: . The coordinates of the remaining three are known by the small little one, and the coordinate of the vertex is known through the coordinate of the point. Robots in bulk, but you need to get to her!

a) Coordinate: it is clear that this ordinate equals zero. We know the abscissa. For whom we can look at a straight-cut tricutnik. It’s a pity that we have less hypotenuse in our house, as it is more beautiful. The leg mi namagatimosya vіdshukati (because it is clear that the bottom of the leg will give us the abscissa of the specks). How can we її shukati? Guess what for posting we have to lie in the basis of the pyramid? Tse is the correct six-piece. And what does it mean? Tse means that the new one has all the sides and all the kuti are equal. It is required to know one such kut. Any ideas? Ideas masa, ale є formula:

The sum of cutiv of the correct n-kutnik is more expensive .

Otzhe, the sum of kutiv of the correct six-kutnik is more degrees. Todi leather from kutіv dorіvnyuє:

Let's look at the picture again. I realized that the windpipe is the bisectrix of the kuta. Todі kut dovnyuє degrees. Todi:

Same zvіdki.

In this rank, maє coordinates

b) Now we can easily know the coordinate of the point: .

c) We know the coordinates of the point. Oskіlki її abscissa zbіgaєtsya z dovzhina vіdrіzka outwardly. Knowing the ordinate is also not too difficult: for example, we get points and a point on a straight line is significant, for example. (Zrobi himself awkwardly pobudova). In this order, the ordinate of the point B is equal to the sum of the dozhins of the vіdrіzkіv. Znovu zvernemosya to trikutnik. Todi

Same as the point can coordinate

d) The coordinates of the point are now visible. Take a look at the rectangle and bring it to such a rank of coordinate points:

e) Lost to know the coordinates of the vertex. It is clear that the abscissa and the ordinate go beyond the abscissa and the ordinate of the point. We know the application. Because. Let's look at a straight-cut tricoutnik. Behind the brain is a bichne rib. Tse hypotenuse of my tricouter. Then the height of the pyramid is the leg.

The same point may have coordinates:

Well, that's it, I have the coordinates of all the points to click on me. I joke the coordinates of direct vectors in straight lines:

Shukaєmo kut mizh tsimi vectors:

Suggestion:

Well, I know, with the completion of this task, I didn’t beat the annual windings, the formulas for the sum of the cuts of the correct n-cut, as well as the designation of the cosine and sine of the straight-cut tricut.

3. Oskіlki we are again not given the rest of the ribs at the pyramid, then I will honor them with equal loneliness. In this order, oskіlki all ribs, and not only bіchnі, equal between themselves, then the basis of the pyramid and less is a square, and the bіchnі faces are the correct trikutniki. Imagine such a pyramid, as well as the basis on the plane, indicating all the data, put in the text of the task:

Shukaemo kut mizh i. I'll work on even short tabs, if I'm looking for point coordinates. You will need to “decipher” them:

b) - the middle of the vіrіzka. Її coordinates:

c) I know Dovzhina vіdrіzka for the theorems of Pіthagoras in trikutnik. I will know for the Pythagorean theorem in trikutnik.

Coordinates:

d) - the middle of the vіrіzka. Її coordinates equal

e) Vector coordinates

f) Vector coordinates

g) Shukaemo cut:

The cube is the simplest figure. I'm sorry that you will figure it out on your own. Vidpovіdі until zavdan 4 and 5 coming:

Znahodzhennya kuta mizh straight and flat

Well, the hour of the simplest tasks is over! Now the butts will be even more foldable. For vіdshukannya kuta mіzh straight and flat, we will fix it like this:

1. Behind three dots there will be equal planes
   ,
   vikoristovuyuchi vyznachnik of the third order.
2. For two points, we can find the coordinates of the direct vector of the straight line:
3. Zastosovuєmo formula for calculating the kuta between a straight line and a plane:

Yak bachish, this formula is already similar to the one, yaku mi zastosovuvali for a joke kutiv between two straight lines. The structure of the right part is simply the same, but now we are talking about the sine, but not the cosine, as before. Well, I got one unacceptable diya - a search for the flatness of the square.

Not applicable to the old screen perfection of applications:

1. Os-no-va-nі-єm direct-my prize-mi yav-la-et-sya rіv-but-poor-ren-ny trikutnik Vi-so-ta prize-mi dorivnyu. Find a cut between my straight and flat brush

2. At a straight-mo-vug_lny pa-ral-le-le-pі-pe-de z-west-ni Nai-di-te cut between my straight and flat brush

3. The correct six-round prism has all the ribs equal. Find a cut between my straight and flat brush.

4. At the right-vіlnіy trikutnіy pi-ra-mi-de z os-no-va-nі-єm іz-west-ni ribs -but-va-nya and straight, passing through se-re-dі-ni ribs i

5. Keep all the edges of the right chotiricut pyramid with the top equal between each other. Know-dі-te kut between a straight line and a flat brush, like a point - se-re-di-on the bo-ko-in-th edge of the p-ra-mi-di.

I am writing the first two tasks in a report, the third - briefly, and the remaining two I leave you for an independent verse. Before that, you have already had a mother on the right with trikutnoy and chotirikutnoy pyramids, and the axis of prisms - still not.

Solution:

1. Imagine a prism, and navit її basis. Sumy її iz the coordinate system that is significant all the data, as given for the mind of the task:

I swear for a day of underestimation of proportions, but for a change, the task is, in fact, not so important. The flat is just the “back wall” of my prism. Just to finish it, guess what kind of flatness you can look at:

However, it is possible to show without intermediary:

Choose enough three points on this plane: for example, .

We store flatness of the area:

It’s right for you: independently virahuvat tsey vyznachnik. Do you have wow? Todi leveling of the area may look:

Abo just

in such a manner,

For example, I need to know the coordinates of the direct vector of the straight line. If the point is scaled with the cob of coordinates, then the coordinates of the vector are simply scaled with the coordinates of the point. For which we know the column of coordinates of the point.

For whom we can look at a trikutnik. Let's draw a height (won - median and bisector) from the top. Oskіlki ordinate of the point is dorivnyuє. In order to know the abscissa of a point, we need to calculate the length of the vdrіzka. Behind the Pythagorean theorem, we can:

The same point may have coordinates:

Krapka - tse “lifted” to the krapka:

Same vector coordinates:

Suggestion:

Yak bachish, there is nothing in principle foldable for the hour of such tasks. In fact, the process will say a little “directness” of such a figure, like a prism. Now let's move on to this butt:

2. Small paralepiped, drawn in a new plane and straight, and also around the lower base:

On the back we know the level of the plane: The coordinates of the three points that it has:

(the first two coordinates are taken away in an obvious way, and you can easily find the remaining coordinate in the picture from the points). Todi warehouse equal area:

We count:

Shukaєmo coordinates of a direct vector: It is clear that the yogo coordinates are shifted from the coordinates of a point, why not? How to know the coordinates? Tse coordinates of the point, move along the axis of the application per unit! . Todi Shukaemo shukanovy kut:

Suggestion:

3. A six-sided pyramid is slightly correct, and then it is carried out straight in the plane.

Here it’s problematic to paint a plane, it doesn’t seem like it’s about the development of this task, the method of coordinates is all the same! Itself in yoga universality and yoga is the main thing!

The plane passes through three points: . Shukaєmo їх coordinates:

one). Find the coordinates for the remaining two points yourself. You need to solve the problem from the six-fold pyramid!

2) The area will be equal:

Shukaєmo coordinates of the vector: . (Again marvel at the worker with a tricot pyramid!)

3) Shukaemo cut:

Suggestion:

Yak bachish, there is nothing supernaturally foldable in these factories. It’s better to be more respectful of the roots. Until the last two days, I will give only a hint:

Like the moment of reconciliation, the technique of solving the task is the same: the main task is to know the coordinates of the vertices and put them in formulas. We were left with one more class to look at the number of kutivs, but for ourselves:

Calculation of kutiv between two flats

The solution algorithm will be like this:

1. Behind three points, we can see the equalness of the first plane:
2. Behind the other three points, we can see the level of another plane:
3. Zastosovuєmo formula:

Yak bachish, the formula is already similar to the two in front, for the help of some of them they shuffled kuti between straight lines and straight lines and flats. So zam'yatati tsyu tobі not warehouse osoblivih trudnoshchiv. Let's move on to the analysis of the task:

1. One hundred-ro-on the basis of the correct tri-cut prize is more expensive, and the diagonal of the side face is more beautiful. Know-dі-te kut mіzh flat-brush and flat-brush OS-no-va-nya prizes.

2. At the right che-ti-rekh-vugіl-noї pі-ra-mі-de, all the ribs are somehow equal, you know the sine of the kuta between the flat brush and the flat brush, scho to pass through the point per-di-ku-lyar-but straight.

3. The correct four-rekh-vugіlnіy prism has a hundred-ro-no OS-but-va-nya equals, and more-to-vі ribs are equal. On the edge of the vіd-me-che-on the point so, scho. Know the cut between the planes

4. At the right-vil-noy chotiricutnoy prize-mі side of the os-no-va-nya is equal, and the ribs are equal. On the edge of the vіd-mі-che-on the point so that Nai-di-te kut mіzh plane-ko-stya-mi i.

5. At the cube, find-de-te co-si-nus kuta m_zh flat-to-stya-mi i

Decommissioning tasks:

1. A small regular (basically - equilateral tricot) tricot prism that is naked on the flat, like a figure for the mind of the head:

We need to know the alignment of two planes: The alignment of the foundations is trivial to enter: you can put the top line behind three points, I’ll put the alignment in a row:

Now we know the level The point is the coordinate point The point - Oskilki is the median and the height of the tricot, then it is easy to know the Pythagorean theorem in the tricot. The same point can coordinate: We know the applicative of the point

Then we need the following coordinates: Folding the plane.

Calculate cut between flats:

Suggestion:

2. Robimo little ones:

Nayskladnіshe - tse zozumіti, scho tse such a taєmnicha flat, yak to pass through a point perpendicularly. What's up, smut, what's up? Golovne - tse respect! Indeed, a straight line is perpendicular. The line is also perpendicular. Then the plane, which will pass through the two straight lines, will be perpendicular to the straight line, i, to the speech, pass through the point. Tsya surface also pass through the top of the pyramid. Todi needs a flat area - And the flat area has already been given to us. Shukaєmo coordinate point.

The coordinate of a point is known through the point. From a little baby it is easy to know that the coordinates of the point will be like this: What is now left to know, to know the coordinates of the top of the pyramid? Still need to virahuvati її visotu. Tse rush for help ієї zh Pіthagoras's theorems: bring the cob, scho (it's trivial from small trikutniki, scho to make a square on a pedestal). Shards for the mind, then maybe:

Now everything is ready: vertex coordinates:

We fold the flatness of the area:

You already fahіvets at the number of vyznachnіv. Without practice you take away:

Abo іnakshe (how to multiply the insults of the parts on the root of the two)

Now we know the level of the area:

(You don’t forget, how we take the flatness of the surface, right? If you don’t understand, the stars took minus one, then turn around to the designated flatness of the flat!

We calculate the signifier:

(You can remember that the flatness of the plane fell on the straight lines that go through the points i! Think why!)

Now we calculate cut:

We need to know the sine:

Suggestion:

3. Tricky food: what is a rectangular prism, how do you think? Why is it all the better to see you paralelepiped! Odrazu OK robimo kreslennya! You can navitt okremo not imagine, but there is not much to look at here:

The flat, as we have already mentioned, is recorded at the sight of the equal:

Now we fold the area

Vіdrazu skladєmo equalization of the area:

Shukaemo cut:

Now we have to wait till the last two days:

Well, now it’s time to reread the troch, and we’ve done well with you and have done a great job!

Vector coordinates. Sticking rіven

In these articles, we will discuss one more class of task, which can be used for the additional method of coordinates: task on the calculation of data. And for yourself, we’ll look at you like this:

1. Calculation between straight lines to cross.

I am ordering the data of the order to the greatest extent of their folding. The simplest thing is to know move from point to plane, and the best way is to know stand between crossed straight lines. I want, well, there is nothing impossible! Let's not put it in the old box and immediately proceed to look at the first class task:

Calculation from point to plane

What do we need to complete this task?

1. Coordinate points

Since then, as soon as we take away all the necessary data, we put the formula:

As I will be equal to the flat, you can already be seen from the front buildings, as I sorted out from the past part. Let's get down to business before tomorrow. The scheme is offensive: 1, 2 - I help you to prove it, moreover, to report it, 3, 4 - only an opinion, you make decisions yourself and correct. Start!

Manager:

1. Danium cube. Dovzhina ribs of the cube are old. Find-dі-te roses-hundred-i-nya in the se-re-dі-ni vіd-rіz-ka to flat-to-stі

2. Dana is a great-vіlna che-ti-rekh-vugіl-on pi-ra-mi-yes. Know-dі-those roses-hundred-I vіd specks to the flat-bone de - se-re-dі-on the ribs.

3. The right-vil-noi trikutnoy pi-ra-mi-de z os-no-va-nі-єm has one rib, and one hundred-ro-on the os-no-vanya is dorіvnyuє. Know-dі-those roses-hundred-I-nya from the top to the flat.

4. The correct six-cornered prize is equal to all ribs. Know-dі-te vіdstan vіd points to the plane.

Solution:

1. Small cube with single ribs, it will be a cross that plane, the middle of the rail is meaningful with a letter

.

Let's take a look at the legend: we know the coordinates of the point. Bo (guess the coordinates of the middle of the wind!)

Now we add up the alignment of the area for three points

\[\left| (\begin(array)(*(20)(c))x&0&1\y&1&0\z&1&1\end(array)) \right| = 0\]

Now I can proceed to the search for the answer:

2. We start anew from the chair, on which all the gifts are given!

For the piramidi, it would have been beautifully painted the base.

Bring on the fact that I'm drawing like a trigger with my paw, it's not easy for us to break the task!

Now it's easy to know the coordinates of the points

Oskіlki coordinate points, then

2. Oskіlki coordinates of the point a - the middle of the vіdrіzka, then

Without problems we know the coordinates of two points on the plane. We add up the flatness of the area and, quite simply, yoga:

\[\left| (\left| (\begin(array)(*(20)(c))x&1&(\frac(3)(2))\\y&0&(\frac(3)(2))\z&0&(\frac( ( \sqrt 3 ))(2))\end(array)) \right|) \right| = 0\]

Oskіlki point may coordinate: , then we can calculate the number:

Vidpovid (duzhe rіdkіsna!):

Well, what, did you get it? I’m guessing that it’s so technical here, as if in quiet butts, which we saw with you in the front part. So I’m sorry, because you’ve been slandered by this material, then it’s not important for you to write down two tasks that you have lost. I’ll give you more hints:

Calculation in the line in a straight line to the flat

There is really nothing new here. How can you straighten up that flat one at a time? They have all the possibilities: to turn over, otherwise it is straight parallel to the plane. How do you think, what is the best way to go straight up to the flat, with which straight line to cross? I’m guessing that it’s clear here that such a thing is worth zero. Nesіkaviy drop.

Another twist is tricky: there is already a non-zero one. However, the shards are straight parallel to the plane, then the skin point of the straight line is equally distant from the plane:

In this manner:

And it means that my task was going to the front: we are looking for coordinates, whether there are points on a straight line, we are looking for a flat surface, we can calculate from a point to a flat. Indeed, such tasks in EDI are rarely heard in the region. I was far away to know only one task, and then the data in the new were such that the method of coordinates before it was no more stagnant!

Now let's move on to another, richly important class of tasks:

Calculation of points to a straight line

What do we need?

1. Coordinates of the point, as we can see:

2. Coordinates of any point that lies on a straight line

3. Coordinates of the direct vector of the straight line

How to fix the formula?

What does the banner of this shot mean, and so it can be clear: the head of the direct vector of the straight line. There's a tricky number here! Viraz means the module (dovzhina) of the vector creation of the vector, and How to calculate the vector vitver, we spun with you at the front part of the robot. Refresh your knowledge, we need to stink at once!

In this rank, the algorithm for decoupling tasks will come:

1. Shukaєmo coordinates of the point, as for which we are joking:

2. Shukaєmo coordinate any point on the straight line, to which we shukâєmo go:

3. Be a vector

4. It will be a direct vector

5. Calculating vector TV

6. Shukaєmo dozhina otrimenny vector:

7. Calculating the number of:

We have a lot of robots, but the butts will be quite foldable! So now take all the respect!

1. Dana pra-vіlna trikutna pі-ra-mi-yes with ver-shi-noy. Hundred-ro-on the os-no-va-nya pi-ra-mi-di dorіvnyuє, vi-so-ta dorіvnyuє. Know-dі-those roses-hundred-i-nya in the se-re-dі-nee-bo-ko-go-th rib to the straight line, de points i - se-re-dі-no ribs and zі-vіd-vіd- stven-but.

2. Dovzhini ribs and straight-vugіl-no-go parale-le-le-pі-pe-da equal co-vіd-vet-stvo-but і Nay-dі-te ras-st-i-ny vіd ver-shi -none to straight

3. At the right-wild sixth-vuhilny prize, all the edges are equal, find-di-those roses-standing from the point to the straight line

Solution:

1. Robimo is more accurate chair, on which all data is assigned:

Robots with us are impersonal! I want to briefly describe in words what we can say in order:

1. Coordinate point

2. Coordinate points

3. Coordinate point

4. Coordinates of vectors and

5. Your vector TV

6. Dovzhina vector

7. Dovzhina vector creation

8. Wait until

Well, well, robots mi maєmo chimalo! We take over her, rolled up our sleeves!

1. To know the coordinates of the height of the pyramid, we need to know the coordinates of the point Її applique to zero, and the ordinate to the abscissa її to the top of the vіdrіzka. Remaining, took away the coordinates:

Point coordinates

2. - the middle of the cut

3. - the middle of the cut

The middle of the vіdrіzka

4.Coordinates

Vector coordinates

5. Calculating vector TV:

6. Dovzhina of the vector: the simplest thing is to replace, which is the middle line of the tricot, also, in the middle half of the base. So what.

7. Dear vector creation:

8. Nareshti, we know you should:

Wow, that's all! I’ll honestly say to you: the execution of which task was carried out by traditional methods (through prompting) would have been richer. Natomist here I am calling everything to the finished algorithm! I think so, what is the algorithm of your wisdom? Therefore, I will ask you to write two tasks on your own. Porivniaemo vіdpovidі?

Well, I’ll repeat again: it’s easier (sweeter) to see it through prompting, and not going into the coordinate method. I have demonstrated such a way of doing nothing more than to show you a universal method that allows you to "get nothing".

Nareshti, let's look at the rest of the class head:

Calculation of the number of times between straight lines to cross

Here the algorithm for solving problems will be similar to the previous one. What do we have:

3. Whether there is a vector that connects the points of the first line and the other straight line:

How do we joke about standing between straight lines?

The formula is:

Chiselnik - the whole module of the mixed creation (my yogo was introduced in the front part), and the banner - like i in the front formula (the module of the vector creation of direct straight vectors, stand between them with you).

I will tell you what

also the formula for the view can be rewritten in the view:

Such sobі vyznachnik diliti on vyznachnik! I want to, honestly, I’m not up to jarring here! Tsya formula is really rather cumbersome and to bring it to a foldable calculation. In your place, I would go into an extreme depression before her!

Let's try vyrishiti kіlka zavdan, vikoristovuyuchi and more method:

1. At the right-vil-noy trikutnoy prize-mі, all the ribs are like something equal, you know the distance between the straight lines.

2. Dana is a right-wild trikutna priz-ma all ribs os-no-va-something equal Se-che-nie, passing through the side rib and se-re-di- well, the ribs are quad-ra-tom. Find-di-those roses-hundred-I-nya mіzh straight-mi-mi i

I'm cheating on it, and spiraling on it, you're cheating on a friend!

1. I mean a small prism straight

Point C coordinates: todi

Point coordinates

Vector coordinates

Point coordinates

Vector coordinates

Vector coordinates

\[\left((B,\overrightarrow (A(A_1))) \overrightarrow (B(C_1)) ) \right) = \left| (\begin(array)(*(20)(l))(\begin(array)(*(20)(c))0&1&0\end(array))\\(\begin(array)(*(20) (c))0&0&1\end(array))\(\begin(array)(*(20)(c))(\frac((\sqrt 3 ))(2))&( - \frac(1) ( 2))&1\end(array))\end(array)) \right| = \frac((\sqrt 3 ))(2)\]

Please, vector TV between vectors that

\[\overrightarrow (A(A_1)) \cdot \overrightarrow (B(C_1)) = \left| \begin(array)(l)\begin(array)(*(20)(c))(\overrightarrow i )&(\overrightarrow j )&(\overrightarrow k )\end(array)\\\begin(array )(*(20)(c))0&0&1\end(array)\\\begin(array)(*(20)(c))(\frac((\sqrt 3 ))(2))&( - \ frac(1)(2))&1\end(array)\end(array) \right| - \frac((\sqrt 3 ))(2)\overrightarrow k + \frac(1)(2)\overrightarrow i \]

Now rahuemo yogo dozhina:

Suggestion:

Now try to carefully vikonati friend zavdannya. Vіdpoviddu neї: .

Vector coordinates. Brief description of the main formulas

Vector - straight lines. - The cob of the vector, - the end of the vector.
The vector is designated as either.

Absolute value vector - dozhina v_drіzka, which depicts the vector. It is signified as.

Vector coordinates:

,
de end vector \displaystyle a .

Sum of vectors: .

Twіr vectorіv:

Scalar tvir vectorіv:

Scalar increment of the vector in the increment of the increment of their absolute values ​​by the cosine of the cut between them:

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