Vector. Basic power. Coordinates and vectors. Vicerniy guid (2020) Zvorotny vector
VECTORS... DIABOVEVector. scalar,
Vektornі, zmіshana TVIR VEKTORІV.
1. VECTORS, DIV OVER vector.
Basic values.
Value 1. The value, more often characterized by its numerical values in the selected system of one, is called scalar abo scalar .
(Masa tila, obsyag, hour, etc.)
Value 2. A quantity that is characterized by numerical values and directly, is called vector abo vector .
(Change, strength, speed, etc.)
Designation:, abo ,.
Geometric vector - the chain of conversions of the form.
For a vector - a point A- ear, point V- the end of the vector.
Value 3.module vectors - the price of the food from the AB.
Value 4. A vector, the modulus of which is equal to zero, be called nullable , to be identified.
Business value 5. Vektori, roztasovani on parallel straight lines or on one straight line are called collinear ... If two collinear vectors are of the same kind, then the stench is called co-directional .
Business value 6. Two vectors vvazayutsya rivnim , how stink co-direction and є equal modulo.
Diy over vectors.
1) Suma vectors.
Def. 6.bag two vectors and є diagonal of a parallelogram, prompted on cich vectors, to go from the backward point (Rule of parallelogram).
Fig. 1.
Def. 7. The sum of three vectors, is called the diagonal of a parallelepiped, stimulated by qi vectors (Rule of parallelepipeda).
Def. eight. yaksho A, V, Z - certain points, then + = (Tricutnik rule).
fig. 2
The power of giving.
1 about . + = + (Traveling law).
2 about . + (+) = (+) + = (+) + (Received law).
3 about . + (– ) + .
2) Introduce vectors.
Def. nine. pid growth vector = - such, scho + = .
Have a parallelogram - tse іnsha diagonal SD (see pic. 1).
3) Multiplying a vector by a number.
Def. ten. cheese vectors per scalar k be called vector
= k = k ,
maє dovzhinu ka , and directly, such as:
1.stabit directly from the vector, which k > 0;
2.protelezhno straight vector, where k < 0;
3.Dovіlno, yaksho k = 0.
The power of multiplying a vector by a number.
1 about . (k + l ) = k + l .
k ( + ) = k + k .
2 o . k (l ) = (kl ) .
3 o . 1 = , (–1) = – , 0 = .
The power of vectors.
Def. eleven. Two vectors i are called collinear , Yaksho stench roztashovani on parallel straight for on one straight.
A null vector is collinear to any vector.
Theorem 1. Two nonzero vectors i collinear, if the stench is proportional tobto
= k , k is a scalar.
Def. 12. Three vectors are called coplanar , Like the stench is parallel to the deyak_y area or to lie in it.
Theorem 2. Three nonzero vectors ,, compliance, if one of them is a linear combination of two, tobto
= k + l , k , l - scalars.
Vector projection to the sky.
Theorem 3. Vector projection on vertical axis (straight line) l dorіvnyu dobutku add vector to cosine kuta mіzh straight vector і straight axis, tobto = a c os , = ( , l).
2. VECTOR COORDINATES
Def. 13. Vector projections on coordinate axes Oh, OU, Оz call myself vector coordinates. Designation: a x , a y , a z .
Dovzhin vector: 
butt: Enumerate the amount of the vector.
Decision:
Show me by points 
і 
to be calculated by the formula: .
butt: Know where the points are M (2,3, -1) and K (4,5,2).
Diy over vectors in coordinate form.
Given a vector = a x , a y , a z і = b x , b y , b z .
1. ( )= a x b x , a y b y , a z b z .
2. = a x , a y , a z, de is a scalar.
Scalar add-on vectors.
value: With scalar cream two vectors
There is a number that will add a number of vectors to the cosine of a circle between them, tobto = , - kut mіzh vectors i.
The power of scalar creation:
1. =
2. ( + ) =
3.
4.
5. , De - scalars.
6.two vectors perpendicular (orthogonal), .
7.tody and only todi, if 
.
Scalar tvir in coordinate form mє viglyad: 
,
de i .
butt: Know the scalar add-on vectors
Decision:
Vector not conducting vectors.
value: For vector creative two vectors and for a vector, for which:
The module of the path of the parallelogram, stimulated on the given vectors, tobto 
, De cut mіzh vectors i
Tsey vector of perpendiculars multiplied by vectors, tobto
If the vector is not collinear, then the stench is right to the three vectors.
The power of the vector create:
1.When changing the order of the multiplier, the vector add-on, change the sign to the back-to-back, save the module, tobto 
2 .Vector square to the zero vector, tobto
3
The scalar multiplier can be blamed for the vector create sign, tobto 
4
.For any three vectors, parity is fair 
5 It is not necessary and sufficient mind of collinearity of two vectors:
Vector dobutok in coordinate form.
Where we see the coordinates of the vectors , then їх vector addition is behind the formula:
.
The value of the vector create the following, but the area of the parallelogram, prompted on the vectors i, can be calculated using the formula:
butt: Count the area of the tricycle with the vertices (1; -1; 2), (5; -6; 2), (1; 3; -1).
Decision: 
.
The area of the tricycle ABC will be calculated as follows:
,
Zmіshane tvіr vectors.
value: A number is called a (vector-scalar) creation of vectors, as it starts with the formula: 
.
The power of the wicked create:
1.
The change of TV does not change with the cyclical rearrangement of the multiplier, tobto 
.
2. When rearranging the two suspicious cofactors of the minus, add a change of svіy sign to the opposite one, tobto.
3 There is no need and sufficient understanding of the coplanarity of three vectors : =0.
4 .Mixed TV three vectors in the door of a parallelepiped, prompted on the same vectors, taken with a plus sign, if the vectors are right, and with a minus sign, as if the stink is done with your three, .
when you see coordinates vector ,
then it’s worthwhile to follow the formula: 
butt: Calculate the number of questions to add vectors.
Decision: 
3. Basis of systems of vectors.
Viznachennya. Before the system of vectors, there are a number of vectors, so that they should lie in one and the same space R.
Respect. If the system is stored in the end number of vectors, then it means one and the same letter with different indices.
Butt. 
Viznachennya. Be-vector mind = 
to be called a linear combination of vectors. Numbers - by the parameters of the line combination.
Butt. 
.
value... Yakscho vector є linear combination of vectors , then it seems that the vector is spinning linearly through the vector .
Viznachennya. The system of vectors is called line-independent However, the vector of the system cannot be used as a linear combination of vectors. In general, the system is called line-fallow.
butt... vector system 
line-fallow, because the vector 
.
Viznachennya to the basis. The system of vectors will establish a basis, when:
1) there is a line-square,
2) be a vector of space through it linearly turn.
Butt 1. Basis for spaciousness :.
2.
In vector systems 
basis є vector: 
Linearly twist through the vectors.
Respect. To know the basis of a given system and vectors, it is necessary:
1) write the coordinates of the vectors into the matrix,
2) for the help of elementary conversions, bring the matrix to a tricut viewer,
3) nonzero rows of matrices will be the basis of the system,
4) number of vectors in the basis of one rank of the matrix.
Storinka 1 z 2
Food 1. Is it a vector? Yak poznachayutsya vector?
View. The vector we will call the alignments of the drives (Fig. 211). The vector straightens start with the meanings of the ear and the end. On the chair, the direction of the vector is denoted by the arrow. For the designation of vectors we will use small Latin letters a, b, c, .... You can also designate a vector for the meaning of the ear and the end. At the same time, an ear of vector is placed on the first m_sci. Replace the word "vector" over the letters of the meaning of the vector, and put a arrow about rice. The vector for little 211 can be defined as follows:
\ (\ Overline (a) \), \ (\ overrightarrow (a) \) or \ (\ overline (AB) \), \ (\ overrightarrow (AB) \).
Food 2. Are the vectors called the same straight (proto-straight)?
View. The vectors \ (\ overline (AB) \) і \ (\ overline (CD) \) are called the same straight, as the AB and CD swaps are the same.
The vectors \ (\ overline (AB) \) і \ (\ overline (CD) \) are called prototypically straight, as the AB and CD interchanges are protractedly straight.
On little 212, the vectors \ (\ overline (a) \) і \ (\ overline (b) \) are nevertheless straight, and the vectors \ (\ overline (a) \) і \ (\ overline (c) \) are curtailed.
Food 3. So is the absolute value of the vector?
View. The absolute value (or modulus) of the vector is called the image of the vector. The absolute value of the vector \ (\ overline (a) \) is denoted | \ (\ overline (a) \) |.
Food 4. So also a zero vector?
View. An ear of vector can be scored from the end. Such a vector will be called a zero vector. A zero vector is denoted by a zero in the line (\ (\ overline (0) \)). Do not talk about the direction of the zero vector. The absolute value of the zero vector is equal to zero.
Food 5. Are the vectors called rivnim?
View. The two vectors are called r_vny, as the stench is summed up in parallel to the transference. Tse means, that іsnuє parallel transfer, which translates the ear and end of one vector apparently on the ear and end of the vector.
Food 6. Make sure that the equal vectors are nevertheless corrected and equal to the absolute value. I navpaki: however, the vectors are straight, equal in absolute value, equal.
View. With a parallel transferred, the vector zberіgaє svіy straight, as well as its absolute value. This means that the equal vectors are straightened, however, and equal to the absolute value.
Nekhai \ (\ overline (AB) \) і \ (\ overline (CD) \) - the same vector is straight, equal in absolute value (Fig. 213). Parallel transfer, which translates point C to point A, along the half-line CD from exchange AB, so as the stench is equally straight. And so as AB and CD pivni, then at the same point D goes to point B, so that the parallel transfer translates the vector \ (\ overline (CD) \) into the vector \ (\ overline (AB) \). This means that the vector \ (\ overline (AB) \) і \ (\ overline (CD) \) рівні, which needs to be completed.
Food 7. To tell that from any point it is possible to view a vector equal to a given vector, and only one.
View. Let the CD be straight, and the vector \ (\ overline (CD) \) is a part of the straight CD. Let AB be a straight line, the straight line CD should be transformed into yak with a parallel transfer, \ (\ overline (AB) \) is a vector, into which, with a parallel transfer, the vector \ (\ overline (CD) \) should be passed, and hence the vector \ (\ overline (AB) \) і \ (\ overline (CD) \) рівні, and straight AB and CD parallel (div. fig. 213). As it is known, through the point you do not lie on the straight line, you can spend on the area not more than one straight, parallel line (axiom of parallel straight lines). This means that one straight line can be drawn through point A, parallel to the straight line CD. Since the vector \ (\ overline (AB) \) is a part of the straight line AB, then one vector \ (\ overline (AB) \) can be drawn through the point A, equal to the vector \ (\ overline (CD) \).
Food 8. So also the coordinates of the vector? Why is the absolute value of the vector with coordinates a 1, a 2?
View. Move the vector \ (\ overline (a) \) by hand to point A 1 (x 1; y 1), and end point A 2 (x 2; y 2). The coordinates of the vector \ (\ overline (a) \) will be called the numbers a 1 = x 2 - x 1, a 2 = y 2 - y 1. The coordinates of the vector will be assigned to the intermediate values of the vector, in the given vypad \ (\ overline ( a) \) (a 1; a 2) or just \ ((\ overline (a 1; a 2)) \). The coordinates of the zero vector go to zero.
From the formulas that turn around two points through the coordinates, next, the absolute value of the vector with coordinates a 1, a 2 road \ (\ sqrt (a ^ 2 1 + a ^ 2 2) \).
Food 9. Bring the vector to the correct coordinates, and the vector from the equal to the equal coordinates.
View. Leave A 1 (x 1; y 1) і A 2 (x 2; y 2) - ear and end of vector \ (\ overline (a) \). So if the vector \ (\ overline (a ") \) go from the vector \ (\ overline (a) \) to parallel transfers, then it will be like A" 1 (x 1 + c; y 1 + d ), A "2 (x 2 + c; y 2 + d). It can be seen that the vector is offended \ (\ overline (a) \) і \ (\ overline (a") \) coordinates: x 2 - x 1, y 2 - y 1.
Bringing it now is a ringing firmness. Don't know the coordinates of the vectors \ (\ overline (A 1 A 2) \) і \ (\ overline (A "1 A" 2) \) рівні. Brought to you, scho vector rіvnі.
Lead x "1 і y" 1 - coordinates of point A "1, and x" 2, y "2 - coordinates of point A" 2. For the wash theorem x 2 - x 1 = x "2 - x" 1, y 2 - y 1 = y "2 - y" 1. Stars x "2 = x 2 + x" 1 - x 1, y "2 = y 2 + y" 1 - y 1. Parallel transfer, assignments by formulas
x "= x + x" 1 - x 1, y "= y + y" 1 - y 1,
translate point A 1 to point A "1, and point A 2 to point A" 2, so vectors \ (\ overline (A 1 A 2) \) і \ (\ overline (A "1 A" 2) \) рівні , which will bring the demand.
Food 10. Give the value of the sumi vectors.
View. Sum vectorів \ (\ overline (a) \) і \ (\ overline (b) \) with coordinates a 1, a 2 і b 1, b 2 is called vector \ (\ overline (c) \) with coordinates a 1 + b 1, a 2 + ba 2, tobto
\ (\ Overline (a) (a 1; a 2) + \ overline (b) (b 1; b 2) = \ overline (c) (a 1 + b 1; a 2 + b 2) \).
At the end of the statistics with you, there is almost no reason to agree on the same "charming sticks", as to allow you to bring a lot of geometries to simple arithmetic. Tsia "stick" can easily fall to life especially in that kind of fall, if you do not see yourself in the prompting of spacious figurines, repetitive and so on. The method, which, for some reason, can be seen here, is to allow you to practically increase abstraction from all kinds of geometric motives and worlds. I will name the method "Coordinate method"... At the danija statti mi with you the onset of food is discernible:
- coordinate area
- Points and vectors on the area
- Pobudova vectors by two points
- Dovzhina vector (shown between two dots)
- Coordinates of the middle of the vidrizka
- Scalar add-on vectors
- Kut mіzh dvoma vectors
I think, having already wondered, why is the coordinate method so called? Well, if I call it this, I’m not working with geometric objects, but with numerical characteristics (coordinates). And the very re-creation, which allows you to go from geometry to algebra, polarity in the introduced coordinate system. If the figure is flat, then the coordinates are two-dimensional, and if the figure is large, then the coordinates are trivial. In this statute we will only look at two-world views. And the main meta statty is to tell you how to use the basic methods of coordinates (the stench in which you appear to be corny in the first tasks on the planimetry in part B). The discussion of the methods for the revision of the plant C2 (stereometry control) assigning the beginning two sections on the basis of the theme.
Why would it be logical to consider the coordinate method? Melodiously, I understand the coordinate systems. Guess if you are stuck with her. I will be built in the 7th grade, if I know about the introduction of the line function, for example. I'm guessing, I'm going to be on the points. Pam'yataєsh? Tie vibrated a sufficient number, put it in the formula and numbered in such a rank. For example, if, then, like, then and so on. So what if you cut it off as a result? And having trimmed the points with the coordinates: i. Distant scribbled the "cross" (coordinate system), vibrating to a new scale (the thumbnails of the clips will be the same one for you) and by means of which you rejected the points, like by taking a straight line, you will get it out of the line.
Here, a few moments, like a varto, explain the three of the lectures:
1. A single vіdrіzok ti viberaєsh from the world of lightness, so, everything is beautiful and compactly placed on the little one
2. It is accepted that going down the hill to the right, and going down the hill
3. The stench is overturned by a straight cut, and the point is called a cob of coordinates. Vaughn is designated by a letter.
4. At the record of the coordinates of a point, for example, the angle in the arches is the coordinate of the point along the axis, and the right-hander, along the axis. Zokrema, simply means, at the point
5. In order to set a point on the coordinate axis, you need to set the coordinate (2 numbers)
6. For be-like a point to lie on an axis,
7. For be-like a point to lie on an axis,
8. Vіs be called vіssu abscis
9. Vіs be called vіssu ordinate
Now let’s come with you a brutally advancing croc: two points are evident. Z'єdnaєmo tsі two points vіdrіzkom. I can put a line in such a way that it can be carried out from point to point: so that we can quickly straighten our way!
Guess how to be called the conjugation of the forms? Well, we call ourselves a vector!
In such a rank, where there is a point with a point, moreover, on the cob we will have point A, and the end will be point B, then we are a vector. Tse motivate ty tezh robiv in the 8th grade, memory?
To appear, vectors, like і points, can be designated by two numbers: і numbers are called coordinates of a vector. Pitannya: yak ti think, why is it enough for us nobility to coordinate the cob and the end of the vector, why should we know its coordinates? Appear, so well! And to be afraid at the same time is simpler:
In such a rank, so yak in the vector point is the ear, and the end, then the vector of the beginning coordinates:
For example, yaksho, then the coordinates of the vector
Now let's zrobimo navpaki, we know the coordinates of the vector. What do we need to remember? So, it is necessary to remember with small pieces the ear and the end: now the ear of the vector will be in points, and the end - in points. Todi:
Submitting respectfully, what are the vectors i? Unity їх visibility - all signs in coordinates. The stench of protylezhny. Tsey fact is accepted to write down the axis as follows:
Inodi, if it is not special, like a point є with a cob of a vector, and a yak with a knot, then the vector is not meant by two great letters, but by one row, for example :, etc.
now trohi tinker yourself and know the coordinates of the advance vectors:
Perevirka:
And now the verse of the trocha is more foldable:
Vek-tor z na-cha-lom in point maє ko-or-di-na-ti. Nay-di-ti abs-cis-su points.
All the same to finish prosaistically: Nekhai - the coordinates of the point. Todi
The system of sklavs depends on the value of the vector coordinates. Todi point maє coordinates. The abscissa clicks us. Todi
as follows:
How can you rob with vectors? So, all of the same ones, like, are more common, with extraordinary numbers (either way, delay is not possible, but multiplication can be done in two ways, one of which is discussed here in three ways)
- The vector can be folded one by one
- Vector can be seen one by one
- The vector can be multiplied (or delay) by a pretty non-zero number
- Vector can be multiplied one by one
All operations may be completely geometrically manifested. For example, the rule of a trikutnik (or a parallelogram) for additional information and confirmation:
The vector is stretched or shrunk or shrinks straight when multiplied or extended by a number:
However, here we will have a power supply, well, we will be looking at the coordinates.
1. When added (seen) two vectors, the warehouse (one) is elementary coordinates. Tobto:
2. When multiplying (dividing) a vector by a number, all the coordinates are multiplied (dividing) by a whole number:
for example:
· Nay-di-ti sumu ko-or-di-nat stolittya-to-ra.
Let us know the coordinates of the skin z vectors. The offense of stink may, however, be the cob - the cob point of coordinates. Kіntsі they have a difference. Todi ,. Now the coordinates of the vector Todi are numbered.
as follows:
Now the verses are coming on their own:
Know the sum of the coordinates of a vector
revision:
Let's get it clear now: we have two points on the coordinate area. How do you know to be among them? Nekhai persha dot bude, and a friend. Significantly seen between them through. Come on, for the sake of accuracy, the offensive armchair:
Am I dead? I, in Perche, z'єdnav points i, a also from the point of the wire to the line, parallel to the axis, and from the point of the wire to the line, parallel to the axis. The stench flashed to the spot, having made a figurine with a miracle? Chim won a miracle? So we know everything about the rectangular tricycle. Well, the Pythagorean theorem - for sure. Shukaniy vidrizok is a hypotenuse of a tricycle, and vidrizki is a leg. What are the coordinates of the point? So, it’s awkward to know from the pictures: So, as the parallel axes and apparently, then it’s easy to know: if you denote the genesis when you see it through, then
Now speeding by the Pythagorean theorem. Dovzhini cathetiv we see, we know hypothesis:
In such a rank, there are two points - the root of the sum of the squares is the difference of the coordinates. For f - vіdstan mіzh two dots - tse dovzhina vіdrizka, їkh z'єdnu. It is easy to mark, so that you do not lie right in between the dots. Todi:
Zvidsy robimo three patterns:
Let's get three points in the numbers:
For example, if it is, then there is a place and a door
Anyway, in a nutshell: we know the coordinates of the vector
I know about the genius of the vector:
Yak bachish, one and the same!
Now try it yourself:
Zavdannya: know the value between the dots:
revision:
The axis is still a couple of tasks on the same formula, the truth is, the stench of a trocha sounds like this:
1. Nay-di-ti kvad-rat dozhini stolittya-to-ra.
2. Nay-di-ti kvad-rat dozhini stolittya-to-ra
I think so, did you get into trouble with them? revision:
1. And the price for respect) We already knew the coordinates of the vectors and earlier :. Todi vector maє coordinates. Square yogo dozhini bude dorivnyuє:
2. We know the coordinates of the vector
Todi square yogo dozhini dorivnyu
Nicholas foldable, right? Arithmetic is great, not more than that.
The beginning of an experience cannot be categorized unambiguously, the stench is shy of the erudite erudition and the appearance of simple little pictures.
1. Nay-di-ti sinus kuta na-clo-na vid-cut, from one-nya-yu-shch-th point, from the abscis.
і
What will we do here? It is necessary to know the sinus kuta mіzh and vіssyu. And de mi vmієmo shukati sinus? Вірно, in a rectangular tricycle. So what do we need to do? Stay tuned in!
Oskilki coordinates of the point i, then go to the road, and go to. We need to know the sinus kuta. I guess tobi, what a sinus is the price of the protolezhny leg to hypotenuse, todi
Why are we overflowing with zrobiti? Know the hypothesis. You can do it in two ways: according to the Pyphagorean theorem (the legs look like!) Or the formula is seen between two dots (for sure, one and the same, but the first way!). I go along the road:
as follows:
The next step is to be simpler. Won - at the coordinates of the point.
Zavdannya 2. From the point, the per-pen-di-ku-lar is lowered to the abs-Ciss. Nay-di-ti abs-cis-su os-no-va-nya per-pen-di-ku-la-ra.
Come on brutally little ones:
The edge of the perpendicular is the center of the point, where the abscis overlaps with the lower point. Behind the little one you can see that the coordinates are:. We are surrounded by abscissa - tobto "іksіv" of the warehouse. Vona dorivnyuє.
as follows: .
Zavdannya 3. In the minds of the foreground tasks, to know the sum of the views from the point to the coordinate axes.
Zabdannya - vzagal elementary, as well as nobility, but also seen from point to axis. Do you know? I am happy, but I will guess all the same to you:
Otzhe, on my little one, roztashovany trocha vishche, I already imagined one such perpendicular? Before what kind of win? Before the axis. And why do you want to go to your birthday party? Vona dorivnyuє. Now draw the perpendicular yourself to the axis and know your age. Wonder if it’s okay? Todi їkh sum dorіvnyu.
as follows: .
Zavdannya 4. In the minds of problems 2, know the ordinate of a point, symmetric to the point of the abscis axis.
I think it is intuitively clear, how is this symmetry? It’s even more abundant about it’s volume: richly budgetary, stoli, litaki, richly geometric figurines: kulya, cylinder, square, rhombus etc. ... This symmetry is called the axis. And what about today? Tse yakraz that line, according to yak figur it is possible, cleverly it seems, to "break" into the same halves (on the date of the picture, the symmetry is straight):
Now let's turn around to our factory. We see a point that is symmetrical to an axis. Todi tsya wis - wis simetryi. This means that we need to point out this point, so that all of them have been distributed on two separate parts. Try it yourself to see the point. Now, try to match my decisions:
Did you get it the same way? Good! At the known point we have the ordinate. won the door
as follows:
Now tell me, after thinking for a second, why would there be an abscissa of a point, a symmetric point A, which is likely to be the axis of ordinates? What is your opinion? The correct answer is:.
In a zagalnyy vipadku, the rule can be written as follows:
A point, symmetric to a point along the abscis axis, ma coordinates:
A point, symmetric to a point along the ordinate axis, ma coordinates:
Well, now the call is scary task: Know the coordinates of a point, symmetric point, and a cob of coordinates. Think about it for yourself, and then hang on to my little ones!
as follows:
now zavdannya on a parallelogram:
Zavdannya 5: Points yav-la-yut-sya ver-shi-na-mi pa-ra-le-lo-gram-ma. Nay-di-ti or-di-na-tu points.
There are two ways to check the problem: by logic and by the method of coordinates. I sometimes use the method of coordinates, and then I’ll open it up, as it is possible to see it.
It is absolutely clear what the abscissa of the point is. (Vaughn lie on a perpendicular drawn from a point to the abscis axis). We need to know the ordinate. Skoristaєmosya tim, scho our figure is a parallelogram, tse means, scho. We know, for example, that the formula is displayed in two dots:
The perpendicular is lowered to the bottom point from the top. I will mark the point with a letter.
Dovzhina vidrizka dorivnyu. (Know your own mind, de mi discussed the moment), then we know it by the theorem of Pyphagoras:
Dovzhina vidrizka - in the accuracy of being bound by the ordinate.
as follows: .
Іnshe decision (I'll just point the little guy, which is yo іlustruє)
Hid solution:
1. Conduct
2. To know the coordinates of the point i dozhinu
3. Bring, scho.
One more zavdannya for a dinner vidrizka:
Points yav-la-yut-sya ver-shi-na-mi tre-vugillya-ni-ka. Nay-di-ti to the dinner of the middle line, pa-pa-lel-noi.
Ty pam'yataєsh, what is the middle line of the trikutnik? Todi is an elementary task for you. If it’s not a memory, then I’ll guess: the middle of the line of the trikutnik is the whole of the line, as it is the middle of the other sides. This is parallel to the base and the second half.
Pidstava - tse vidrizok. Yogo was brought to us by the shukati earlier, very expensive. Todi more middle line is less and more expensive.
as follows: .
Comments: the process of production can be done in the same way, until any animal is harvested.
And pooki is the axis of a series of tasks, tinker with them, the smell is simple, or you can help "fill your hand" on the victorian method of coordinates!
1. Points yav-la-yut-sya ver-shi-na-mi tra-pe-tsії. Nay-di-ti to dinner of the middle line.
2. Points і yav-la-yut-sya ver-shi-na-mi pa-pa-le-lo-gram-ma. Nay-di-ti or-di-na-tu points.
3. Nai-di-ti dovzhinu vid-cut, z-odi-nya-yu-shch-th point i
4. Nai-di-ti area for-beautiful-noi fi-gu-ri on ko-or-di-nat-noi of flatness.
5. Otochuyuchiy-nist from the center in na-cha-le ko-or-di-nat to pass through the point. Nay-di-ti її ra-di-us.
6. Nai-di-ti ra-di-us surround-no-sti, describe-san-noi is close to the right-vugilla-ni-ka, top-shi-no-one-ro-go mayut co-op -di-na-ty z-vid-vet-but
solution:
1. Vidomo, the middle of the line of trapezia road napivsumy її pіdstav. Pidstava is one, but pіdstava. Todi
as follows:
2. Make the task simpler as follows: note, scho (the rule of the parallelogram). Calculate the coordinates of the vectors and not become difficult :. When the vectors are added, the coordinates are stored. Todi maє coordinates. The center of the coordinates is a point, the splinters of the cob of the vector are the center of the coordinates. We are ordinate. Vona dorivnyuє.
as follows:
3. Dієmo immediately following the formula of visibility between two dots:
as follows:
4. Click on the picture and tell me, is the shaded area "overwritten" with two figures? Vona is covered with two squares. Todi area shukanoi figuri dorіvnyu area of the great square minus area of the small one. The side of the small square is the end
Todi area of a small square door
Likewise, it is itself repaired with a great square: the one side is not the same
Todi area of the great square of the road
The area of shukanoi figuri is known for the formula:
as follows:
5. If the circumference is in the center of the cob of coordinates and pass through the point, then the radius will be in the exactness of the road to the edge (the little ones and the eyes, for which it is obvious). We know the following:
as follows:
6. Seemingly, the radius of the cola, described near the rectum, is half of the first diagonal. You know, be-like two diagonals for dinner
as follows:
Well, uh, are you right for us? Bulo doesn’t go well enough, can you? The rule here is the same - to smash the picture on the spot і just "rahuvati" from all of its tributes.
We are overwhelmed by the call of nonagato. There are literally two more points that I would like to discuss.
Let's try to virtual the axis of such a simple task. Let two points i be given. Know the coordinates of the middle of the display. The list of tasks is coming: hey point - shukana middle, todi maє coordinates:
Tobto: coordinates of the middle of the display = the arithmetic mean of the coordinates of the end of the display.
The whole rule is even simpler, as the rule is not wicked. Let's wonder, in what kind of political and political work and how to get used to:
1. Nai-di-ti or-di-na-tu se-re-di-no vid-cut, z-odi-nya-yu-shch-go point i
2. Points yav-la-yut-sya ver-shi-na-mi-ti-ryokh-vugillya-ni-ka. Nay-di-ti or-di-na-tu points pe-re-se-che-nya yogo dia-go-na-lei.
3. Nay-di-ti abs-cis-su center-tra of the surroundings, describe-san-noi close to the right-wugilya-ni-ka, top-shi-no-to-ro-go mayut co-op-di-na-ty z-vid-vet-no.
solution:
1. Perche zavdannya is just a classic. Dimo immediately by the designation of the middle of the display. Vona maє coordinates. Ordinate
as follows:
2. It is easy to bachiti, giving a chotirikutnik є a parallelogram (turn with a rhombus!). You can bring it yourself by counting the sides and making them different. What do I know about the parallelogram? Yogo diagonal point to cross-flow to navpil! Aha! So the point of overturning diagonals is tsesho? Be the middle of the diagonals! Viber, zokrema diagonal. Todi point MAє coordinates The ordinate of the point dorіvnyuє.
as follows:
3. What is the center of the described stake near the rectangle? Win zbigaєtsya at the point of overflow of the two diagonals. Do you know about rectangular diagonals? The stench of івні і point to overflow to the distance to navpіl. Zvdannya rang to the front. Vizma, for example, diagonal. Todi yaksho is the center of the described stake, then it is the middle. Shukayu coordinates: Abscisa dorіvnyuє.
as follows:
Now try three things on your own, I will deprive you of the view to the skin care, so that you can change yourself.
1. Nai-di-ti ra-di-us surround-no-sti, describe-san-noi is close to tre-vugillya-ni-ka, ver-shi-no-one-ro-go may be co-op-di -on you
2. Nai-di-ti or-di-na-tu center-tra of the surroundings, opi-san-noi close to tre-vugillya-ni-ka, top-shi-no-one-ro-go mayut co-op-di-na-ti
3. How-to-ra-di-u-sa must-be surrounded by the center at the point where you won ka-sa-las axis abs-Ciss?
4. Nay-di-ti or-di-na-tu points pe-re-se-che-nya axis і vіd-cut, z-odi-nya-yu-th-th point і
indications:
Is everything gone? I still hope for the price! Now it's a stop ryvok. Be especially respectful of the infection. This material, which I will explain at a glance, is not just a simple approach to the method of coordinates from B part, but it is also used everywhere in the C2 plant.
How many of my obitsyanok I haven’t seen yet? Guess what kind of operations on vectors am I suggesting to enter i into the final rakhunka vviv? I'm sure I haven't forgotten anything? Forgetting! Forgetting to explain, scho means a lot of vectors.
Є two ways to multiply vector by vector. In the opposite way, we will have a view of the beautiful nature:
Vectorniy dobutok visonuitsya to finish slyly. Yak yogo robiti and for whatever is needed, we will discuss it with you in the offensive statty. And in tsіy mi we go to the scalar creation.
Є There are already two ways to allow us to count it:
Yak ti zdogadavsya, the result is guilty but one and the same! Otzhe, let’s pick a quick and easy way:
Scalar tvir through coordinates
Know: - take the value of scalar creation
The formula for calculating the calculation is:
Tobto scalar tvir = sum of creation of coordinates of vectors!
butt:
Nay-di-ti
Decision:
We know the coordinates of the skin vectors:
Scalar tvir is calculated by the formula:
as follows:
Bachish, absolutely nothing folding!
Well, now try it yourself:
Nay-di-ti ska-lar-ve pro-z-ve-de-nya vek-to-riv i
Are you right? Maybe it's a small step? Let's change it:
Coordinate vectors, yak in the past! Suggestion :.
Krim coordinate, the first way to calculate scalar tvir, and itself, through some vectors and cosine kuta between them:
We denote kut between vectors i.
Tobto scalar add-on add-on add-on vectors to the cosine cut between them.
Now we have a different formula, as we have є Persha, as it’s more simple, in no time we accept a few cosines. And you need it in order to be able to use the first and other formulas for you, as you know there are vectors!
Come on Todi guess the formula for vector genie!
Todi, if I put the dan in the formula for scalar creation, then I reject:
Ale from the side:
With such a rank, why did they take me away from you? We now have a formula that allows you to calculate the number of vectors in two vectors! Inodi її for stiffness write it down like this:
Tobto the algorithm for calculating the kuta with the offensive vectors:
- Scalar tvir in terms of coordinates
- It is known that even the vectors і multiply їх
- Dimo the result of paragraph 1 on the result of paragraph 2
Let's try it on the butts:
1. Nai-di-ti kut mіzh vek-to-ra-mi i. Give me a message at the gra-du-sakh.
2. In the minds of the foremost task, know the cosine between the vectors
It’s so hard: I’ll help you first, and try it yourself for a friend! Is it fit? Todi will fix it!
1. The vectors are our old knowledge. Їх scalar tvіr mi already respected and wonо dorіvnuvalo. Their coordinates are:,. Todi know, dozhini:
Todi shukaєmo cosine mіzh vectors:
The cosine of a kuta dorіvnyuє? Tse kut.
as follows:
Well, now he himself is a friend of a friend, and then again! I will give you a shorter solution:
2. maє coordinates, maє coordinates.
Nekhai - kut mіzh vectors i, todі
as follows:
It is necessary to inquire that it is necessary to establish without the middle on the vectors and the method of coordinates in the part B of the replacement robot and to complete the calculation. However, the greater number of C2 buildings can be easily viewed by going down to the internal coordinate systems. So you can respect the article with the foundation, on the basis of which we will work to finish the cunning, as we need for the revision of folding buildings.
COORDINATE I VECTOR. MIDDLE U Rowen
I will prodovumo vivchati with you the method of coordinates. In the last part, a number of important formulas were introduced, which allow:
- Know the coordinates of the vector
- Know about the amount of the vector (alternatively: look like two dots)
- Dodavati, vidnimati vectors. Multiply їх by the number
- Know the middle of the
- Calculate scalar add-on vectors
- Know kut mіzh vectors
Obviously, the whole coordinate method is not included in the 6 points. Vіn lie in the basis of such a science, as analytical geometry, in order to be learned in the university. I no longer want to create a foundation that will allow you to live in a single state. іspitі. With the zavdannyi of the part B we have moved to Now it’s time to move on to a well-known new rivn! The statute will be assigned to the method of revision quietly at C2, in which case it would be reasonable to go to the method of coordinates. The reason for this is that it is necessary to know in the tasks that the figure is given. Otzhe, if I would become a zastosovuvati method of coordinates, I would ruin the food:
- Know kut mіzh two squares
- Know kut mіzh straight and square
- Know kut mіzh two straight
- Know the point from the point to the area
- Know from a point to a straight line
- Know how to go straight to the square
- Know how straight between two houses
It is given in the mind of the figure є by the wrapping (kulya, cylinder, cone ...)
With the help of figures for the method of coordinates є:
- rectangular parallelepiped
- Piramida (trikutna, chotirikutna, sixkutna)
So it’s for me underestimated vicoristovuvati method of coordinates for:
- Znakhodzhennya square peretiniv
- Calculated ob'єmіv til
However, it immediately means that there are three "unimaginable" for the method of coordinates of the situation in practice to complete the situation. In most cases, the staff can become your rivals, especially if they are not even stronger in trivial impulses (as they spend an hour to beat the wise ones).
Yakims є all the figuri pererakhovanі by me? The smell is no longer flat, yak, for example, square, tricycle, colo, but ob'єmnі! Apparently, we need to look at not two-world, but a trivial coordinate system. It will be easy to get there: just the edge of the abscis and ordinates, we introduce one more, the applique. The little one schematically depicts їх roztashuvannya in my place:
All the stench is mutually perpendicular, rewind in one point, as we will call it a cob of coordinates. We will see the abscis, as before, we will begin, we will see the ordinates, and I will introduce the whole aplikat -.
Even earlier, a dermal point on the area was characterized by two numbers - an abscissa and an ordinate, then a dermal point in the open space was also described by three numbers - an abscissa, an ordinate, an aplikata. for example:
The abscissa of the point is displayed, the ordinate is, and the applicate is.
One abscissa of a point is also called the projection of a point onto the vertical axis, the ordinate is the projection of a point onto the axis of ordinates, and the aplikat is the projection of a point onto the vertical axis. Apparently, when a point is set, a point with coordinates:
call the projection of the point onto the area
call the projection of the point onto the area
Fasting natural power: how are all the formulas true, vivedeni for the two-worldly vypadnu, in space? It seems to be true, the smell is fair and may look the same. For a small detail. I think you've already wondered for yourself. All guilt formulas will have one term, which will be considered for the whole application. And itself.
1. If two points are given: then:
- Vector coordinates:
- View between two dots (or even a vector)
- The middle of the vidrizka maє coordinates
2. Given two vectors: i, then:
- Їх scalar add-on file:
- The cosine of the cut between the vectors of the paths:
However, the spaciousness is not so simple. Yak ty rosumієsh, adding more one coordinate to bring in a hundred versatility in the spectrum of figures, "live" in a whole space. And for a fake notification, it is necessary for me to introduce a deyakiy, roughly seemingly, “uzagalnennya” straight. Tsim "zagalnennyam" will be the area. Do you know about the area? Try to adapt to food, but what about the area? Says it neatly. However, everything is intuitively represented, like a viglyad:
Roughly, seemingly, a whole, uninterrupted "leaf", shoved into space. "Unscendence" is a trace of innovation, but the area is widened on all sides, so that the area is not limited. However, the price of the explanation "on the fingers" does not give the least notice about the structure of the area. And we will be tsіkaviti itself won.
Let's guess one of the main axioms of geometry:
- through two points on the area to pass straight, before that there is only one:
Abo її analogue in space:
Surely, ty pam'yatash, yak for two set points to bring the straight forward, it’s not important: if the first point is the coordinate: but a friend, then we’ll step straight forward:
The chain took place in the 7th grade. In the vastness of the straight line, the axis is as follows: let us have two points with coordinates: then straight line, pass through them, maє viglyad:
On the butt, through the points, go straight:
Yak tse slіd rosumіti? Tse slіd size yak axis: the point lies on a straight line, where the coordinates are satisfied with the offensive systems:
We are no worse than tsіkavitime rіvnyannya straight, but we need to brutalize respect on an even more important understanding of the directing vector of straightforwardness. - be it a non-zero vector, how to lie on a straight line or parallel.
For example, offending vectors, є straighten straight vectors. Lean is a point to lie on a straight line, and - is a directional vector. Todi straight forward can be written in the offensive view:
Once again I will repeat myself, I will not be as straightforward as it should be, if I have forgotten it, I will also direct the vector! Again: be a non-zero vector, lie on a straight line, or parallel їй.
vive Rivnyannya area by three assigned points it's not so trivial either, і please don't look at the course middle school... And dareny! Tsei priyom life is necessary, if it comes to the method of coordinates for the revision of folding buildings. However, I will let you know what new things are coming up? Moreover, you can turn your viclacage at the first-rate mortgage, if you try to use the methodology as well, as you start learning in the course of analytical geometry. Otzhe, start.
Rivnyannya area does not need to be seen as rіvnyannya right on the area, but the most vono viglyad:
deyaki numbers (not all equal to zero), but changes, for example: etc. Yak bachish, the rivnyannya of the area is no less likely to be seen as rivnyannya straight (linear function). However, guess what was mine with you? We said that if we have three points, but not to lie on one straight line, then the equal area will be unambiguously updated according to them. Ale yak? I'll try to explain.
Oskilki rіvnyannya ma viglyad area:
And the points are located in the area, then when the coordinates of the skin point are set in the equal area, then the correctness is correct:
With such a rank, fasting is necessary for three rіvnyannya already from the unavailable! Dilemma! However, it is possible to allow for allowance, but (for the whole need to be distributed). In such a rank, we recognize three rivnyannyas from three unavoidable ones:
However, we will not virishuvati such a system, but vypishemo viraz, like a znogo slid:
Rivnyannya area, go 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! Tse scho take? Yakis dusha is an unaware module! However, the object, like ti bachish in front of him, is not a good spiel with a module. Tsey ob'єkt to be called a third-order visitor. Now, if you can use the method of coordinates on the ground, if you can use the method of coordinates on the area, then the designers will often be used. Well, is this a third-order visitor? Yak is not marvelous, tse all-for-all number. Lingering intelligence, as a specific number we will be placed with the visitor.
Let us write down a copy of the third-order form in the big zebra view:
De - deyakі numbers. Moreover, the first INDECO is the number of the row, and the first INDECO is the number of the stacker. For example, it means that the number is given to be on the cross of another row and the third hundred. Let’s put on the fast track: will we count such a business card by the same rank? Tobto, how exactly the number we are going to put? For the visitor of the third order itself є heuristic (in fact) the rule of the tricot worker is in the first place by the offensive rank:
- Tvir elements of the head diagonal (from the upper left cut to the lower right) tvir elements, which make the first tricycle "perpendicular" to the head diagonal tweeter elements, which "set up" other tricycle
- Tvir elements of the side-by-side diagonal (from the upper right cut to the bottom left), the set of elements, to create the first tricycle "perpendicular" of the side-to-side diagonals to the set of the other triangular
- Todi viznachnik dorivnuyu rіvnyu value, trimmed in crotch and
If you write down everything in numbers, then we can recognize such a viraz:
In fact, remembering the way of calculating in such a view is not necessary, there is enough in the head just trimmings and the very idea of what to build and what to see in the future).
Let's illustrate the method of tricycles on the butt:
1. Calculate the visnatnik:
Let's get it out of the way, it’s well stocked, but it’s obvious:
Warehouses, which go with the "plus":
Price golovna diagonal: twir elements one
First tricycle, "perpendicular head diagonal: twir of elements one
Another tricycle, "perpendicular to the head diagonal: twir of elements one
There are three numbers in the warehouse:
Warehouses that go with the "minus"
Tse pobichna diagonal: tvir elements one
The first tricycle, "perpendicular to the side-by-side diagonal: the set of elements is one
Another tricycle, "perpendicular to the side-by-side diagonal: twir of elements one
There are three numbers in the warehouse:
Everything that has become overwhelmed with zrobitis is to take from the sum of dodankiv "with plus" the sum of dodankiv "z minus":
In such a rank,
Yak bachish, nothing folding and supernatural in the numbered third-order forms is not. It is just important to remember about trikutniki and not to allow arithmetic pardons. Now try calculating yourself:
revision:
- First tricycle, perpendicular to the head diagonal:
- Another tricycle, perpendicular to the head diagonal:
- Suma dodankiv with plus:
- The first tricycle, perpendicular to the side-by-side diagonal:
- Another tricycle, perpendicular to the bitwise diagonal:
- Suma dodankiv with minus:
- Sum dodankiv with plus minus sum dodankiv with minus:
The axis is also a pair of forms, numbered independently and in accordance with the indications:
indications:
Well, is it all gone? Apparently, it’s possible to collapse far! If there are difficulties, then I am glad of such: in the Internet there is a set of programs for calculating a business card online. Everything you need is to come up with your own business card, calculate it independently, and then take it into account, how to use the program. And so until quietly feast, as long as the results do not come back to waste. Revelations, do not sniff yourself for a moment!
Now let’s turn to that determinant, as if I’ve written, if I’ve talked about the flat area, I’ll go through three given points:
Everything that is necessary is to calculate the value without the average (by the method of triplets) and set the result to zero. Naturally, the splinters are wintry, then the deyakiy viraz is removed from them. The very same hangs and will be equal to the area, how to pass through three given points, but not to lie on one straight line!
Let's illustrate it in a simple butt:
1. Pobuduvati rіvnyannnya area, scho pass through the points
Warehouseє for qix three points viznachnik:
I'll just say:
Now, it is numbered without preceding the rule of tricycles:
\ [(\ Left | (\ begin (array) (* (20) (c)) (x + 3) & 2 & 6 \\ (y - 2) & 0 & 1 \\ (z + 1) & 5 & 0 \ end (array)) \ 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 such a rank, equal to the area, to pass through the points, maє viglyad:
Now try one problem on your own, and then we will discuss:
2. Know the area of \ u200b \ u200bthe area to pass through the points
Well, now, let's discuss the solution now:
Warehouse business card:
I numbered value:
Todi Rivnyannya of the ma viglyad area:
For well, having passed on, we will otrimaєmo:
Now there are two points for self-control:
- Pobuduvati rіvnyannya area, scho pass through three points:
indications:
Is everything gone? I know, since it’s difficult, my joy is this: I’ll take three points from my head (the stink will not lie on one straight line with the great step of immobility), I’ll be on them a flat. And then you can change yourself online. For example, on the site:
However, for the help of a visnichnik, we will not only be in a rural area. Guess, as I say, it's not just scalar TV that is meant for vectors. Є More vector, as well as change TV. If a scalar product is two vectors and if there are a number, then there are two vectors in a vector, where the vector will be perpendicular to the given ones:
Moreover, the module will be built on the vectors i. We know the Danish vector for calculating the distance from the point to the straight line. How can we value the vector add-on vectors і, where їх coordinates are given? A third-order visitor will come to help us. However, first I will go down to the algorithm for calculating the vector creation, I will move a small lyric entry.
You are not allowed to enter basic vectors.
Schematically, the stench of images on a baby:
Yak ti thinksh, and why should the stench be called basic? On the right in that scho:
For the images:
The validity of this formula is obvious, even:
Vectorniy vitvir
Now I can get down to creating vector art:
The vector production of two vectors is called a vector, which is calculated according to the offensive rule:
Now, let’s do it with the help of the vector calculator:
Application 1: Know the vector add-on vectors:
Solution: I am putting the card holder:
I count yogi:
Now I will write through the base vector, I will turn to the base vector:
In this rank:
Now try it.
Ready? revision:
І traditionally dvі zavdannya for control:
- Know the vector of the offensive vectors:
- Know the vector of the offensive vectors:
indications:
Zmіshane tvіr three vectors
Remaining the construction, as I know it - all the changes are made of three vectors. Wono, yak і scalar, є number. Є two ways of calculating. - through the viznachnik, - through the zmіshane tvіr.
But hey, let us have three vectors:
Todi minus add three vectors, which can be denoted through you can calculate yak:
1. - tobto change tvir - create scalar vectors for vector add two vectors
For example, you want to add three vectors to the door:
Try on your own to calculate it through a vectorial add-on, and go over it, so the results and spivpadut!
I know - two butts for independent solution:
indications:
Vibir coordinate systems
Well, the axis, now we have all the necessary foundation of knowledge, how to virishuvati folding stereometric design of geometries. However, the first step is not going ahead before applying to the algorithms of the first time, I am going to do it, if there will be a corny axis on a given power supply: yak the same select a coordinate system for this chi іnshoї figuri. Ade the same vibration of the interchangeable system of coordinates and figuri in the open space in the kintsev rakhunka mean, how bulky will be the calculation.
I will guess, that in the whole distribution of me you will see these figures:
- rectangular parallelepiped
- Straight prism (tricutna, sixkutna ...)
- Piramida (trikutna, chotirikutna)
- Tetrahedron (one і the same, scho і trikutna pіramida)
For a straight-sided parallelepiped or a cube, I recommend that you step on it:
Tobto the figuru I will help "in the kut". Cube and parallelepiped - even better than figurines. For them, you can easily know the coordinates of your vertices. Naprilad, yaksho (yak shown on the baby)
then the coordinates of the vertices are:
Zapam'yatovuvati tse, zychayno, not required, however, pam'yatati, yak more beautifully roztashovuvati cube or straight-sided parallelepiped - bazhano.
straight prism
The prism is a big shkidliva figure. Roztashovuvati її in the open space is possible in a smart way. However, the most acceptable for me is an offensive option:
Tricutna prism:
So that one of the sides of the tricycle is put on top of each other, and one of the vertices is set on the cob of coordinates.
Six-foot prism:
So that one of the vertices is formed by a cob of coordinates, and one of the sides is to lie on the axis.
Chotirikutna and six-kilometer piramida:
The situation is analogous to a cube: two sides are placed one by one with the axes of coordinates, one of the vertices is one by one with a cob of coordinates. In one small folding, the coordinates of the point will be raised.
For a six-sided piracy - similarly, a yak for a six-sided prize. Mostly zavdannya know the coordinates of the vertex in a joke.
Tetrahedron (trikutna piramida)
The situation is even more similar to the one that I grafted for tricot prisms: one vertex is set on the cob of coordinates, one side lies on the coordinate axis.
Well, now, we are with you, nareshty, close before you start before the final date. From what I have said on the very ear of the statty, the axis of the building is the most important: there are more C2 buildings to be divided into 2 categories: the building for the cut and the building for the building. With the help of you, I can see the zest for the knowledge of the kuta. Stink into your own room to go on the offensive categories (in the world of increasing folding):
Zavdannya on poshuk kutiv
- Znahozhennya kuta mіzh two straight
- Znahodzhennya kuta with two houses
Let's take a look at the last one: it’s almost always straight. Come on, guess, and do not tell me about you, but do it earlier? Probably, even we already had some more ... We used to joke around with two vectors. I will guess if there are two vectors: i, then there are two vectors between them:
Now we have a meta - znakhozhennya kuta mіzh two straight lines. Let's get the beast to the "flat picture":
How much did we get kutіv when we crossed two straight lines? As many things. True, not only two of them are not equal, but they are vertical to them (and besides, they are lost). That same kut we vvazhati kut mіzh two straight lines: abo? Here the rule is as follows: kut mіzh two houses straight lines at least not more than nіzh degrees... Tobto with two kutіv we will always choose kut with the lowest degree world. Tobto on the day of karting kut between two houses with direct roads. If you do not want to fool around with the smack of the smallest two kutiv, the cunning mathematicians have propounded the vicoristovuvati module. This rank kut mіzh two straight lines begin with the formula:
Do you, like a respectable reader, are guilty of a food fault: and the stars, no matter how many numbers, how do we need to calculate the cosine of the cut? Suggestion: we will be brothers from the guiding vectors straight! In such a rank, the algorithm for the knowledge of the kuta between two straight lines is the next rank:
- Zastosovuєmo formula 1.
Abo big report:
- Shukaєmo coordinates of the directing vector of the straight line
- Shukaєmo coordinates of the direction vector other straight
- Computably module їх scalar creation
- Shukaєmo for the first time vector
- Shukaєmo for another vector
- Multiply the result in paragraph 4 by the result in paragraph 5
- Dimo the result from point 3 to the result from point 6. Otrimuєmo cosine kuta with straight lines
- Yaksho daniy the result allows you to accurately count the kut, shukahmo yogo
- Inakse is written through the inverse cosine
Well, now it’s an hour to go to the building: I will demonstrate the solution of the first two in a lecture, I will present the solution to the first two in a simple view, and until the rest of the two employees I will deprive them of all, I’m guilty to carry out all the settings before them.
zavdannya:
1. At the great-Vіl-nom tet-ra-ed-re nai-di-ti kut mіzh vi-so-thіy tet-ra-ed-ra і me-di-a-noi bo-ko-viy border.
2. At the great-Vіl-noi six-sti-vugillya-niy pi-ra-mi-de-ro-ni os-no-va-nya ko-to-rіy rіvnі, and bo-ko-wі ribs rіvnі, nai-di-ti kut mіzh straight-mi i.
3. Dovzhini of all ribs is right-Vіl-niy four-ti-ryokh-vugillya-niy pi-ra-mi-di rivnі mіzh. Nai-di-ti kut mіzh straight-mi і yakshо vіd-re-zok - vi-so-ta dan-noi pi-ra-mi-di, point - se-re-di-na її bo-ko po th rib
4. On the edges of the cube, there is a point so that Nay-di-ti kut mіzh straight-mi
5. Point - se-re-di-on the edges of the cube Nay-di-ti kut mіzh straight-mi i.
I am not vypadkovo rostashuvav zavdannya in this order. Leave still didn’t get to start looking into the method of coordinates, I myself will select the most “problematic” figuri, and for that I will pick the simplest cube! Proceedingly, you will be able to see a lot of figures, the foldability of the factory, I will be from those to those.
Start up until the date of arrival:
1. A small tetrahedron, which is placed in the coordinate system in such a way that I had passed it earlier. Oskіlki tetraed is correct - then all of the two lines (the figure includes the text) are correct tricytes. Oskilka we have not been given a dinner party, then I can accept it її їівnoyu. I think, ti mind, why don't you get lost because of the fact that our tetrahedron is going to be "stretched out"? I will also hold a median in the tetrahedron. Along the way, I will paint yogo before (we may be in the middle of now).
I need to know kut mij i. What do we see? We see only the coordinate of the point. This means that you need to know the coordinates of the points. Now I think: point - tse point overretin the height (abo bisectriss abo median) of the trikutnik. And the point is the chain of the point. Point w is the middle of the vidrizka. Todi it is enough for us to know: coordinates of points :.
Possibly from the simplest: the coordinates of a point. Marvel at the little ones: It is clear that the point is to be applied to zero (the point is to lie on the square). Її ordinate dorіvnyuє (so yak - mediana). It is better to know the abscissa. However, it’s easy to be afraid of presenting Pythagoras’s theorems: It’s easy to understand. Yogo hypothesis for the door, and one of the cathetes for the door Todi:
Remaining money :.
Now we know the coordinates of the point. It is clear that I know it is zero, and the ordinate is also near the point, tobto. We know the abscissa. Don't be afraid to finish it trivially, as a memory, The length of the equilateral tricycle with a point of overturning is divided in proportion, Mending from the top. So yak: then the shukana of the abscissa of the point In this rank, the coordinates of the point рівні:
We know the coordinates of the point. It is clear that the abscissa and ordinate are bound from the abscissa and ordinate of the point. And the applicants are ready to go. - tse one of the legs of the tricycle. Hypotenuse of a tricycle - tse vidrizok - cathetus. Vіn whisper about mіrkuvan, as I saw in bold:
The point is the middle of the direction. Todi we need to guess the formula for the coordinates of the middle of the display:
All right, now we can shukati coordinates of direct vectors:
Well, everything is ready: we put all the data into the formula:
In such a rank,
as follows:
You are not guilty of such "terrible" statements: for C2 tasks, practice is very important. I would soon greet the "beautiful" view in the whole part. Also, having said that, I practically did not go into anything, except for the Pythagorean theorem and the power of the one-sided tricycle. So for the definition of stereometric measurements, I will be victorious in the very minimum of stereometry. Vigrash in the whole section "to be extinguished" to finish up with bulky enumerators. Behold the stench to finish algoritmіchno!
2. Imaginably correct six-day parade at the same time with the coordinate system, as well as given:
We need to know kut mіzh straight i. In such a rank, our task is to create the coordinates of the points: The coordinates of the remaining three are known by the small baby, and the code of the vertices is known through the coordinate of the point. Robots in bulk, ale treba until she starts!
a) Coordinate: it is clear that it should be zero. We know the abscissa. For tsyogo clear rectangular tricycle. It’s a pity, in the new house we only have hypotenuse, which is expensive. Cathetus we will namagatisya v_dshukati (it is clear that we will give us the abscissa of the point). How can we її shukati? Come on, guess what, for the figurine we have to lie in the basis of the piracy? Tse is the correct six-walker. And what does it mean? Tse means that everything is on the side and in all kuti ryvn. It is required to know one such kut. Є іdeї? Idey masa, ale є formula:
suma kutiv correct n-kutnik door .
In such a rank, the sum of kutiv of the correct six-kutnik dorivnyuє degrees. Todi kozen z kutiv dorivnyu:
I know I am amazed at the picture. It is clear that it is a bisectrum of a kuta. Todi kut to degrees. Todi:
Todi, stars.
In such a rank, maє coordinates
b) Now it is easy to know the coordinate of the point :.
c) We know the coordinates of the point. So, like її abscissa, it’s going to be a good thing. Knowing the ordinate may not be very easy: where the point is, and the point is crossed straight, it is meaningful, say for. (Zrobi himself is awkwardly pobudova). Todi In such a rank, the ordinate of point B is dorіvnyu sumі dovzhin vіdrіzkіv. I know the animal to the tricycle. Todi
Todi so yak todi point maє coordinates
d) Now we know the coordinates of the point. Look up the rectangle and bring it, so in this rank, the coordinates of the point:
e) It is too late to know the coordinates of the vertex. It is clear that the abscissa and ordinate are located from the abscissa and ordinate of the point. We know the applikat. So yak, then. Rectangular tricycle is clearly visible. For the mind, there is a rib. Tse hypotenuse of my trikutnik. Todi visota pіramidi - cathetus.
Todi point maє coordinates:
Well, everything, at me є coordinates of all points less points. I shuffle the coordinates of direct vectors in straight lines:
Shukaєmo kut mіzh qimi vectors:
as follows:
I know, after all, when I go viral, I didn’t vikoristovuvvvav any extravagant priyomy, except the formulas of sumi kutiv of the correct n-kutnik, as well as the value of the cosine and sine of the rectangular tricycle.
3. Oskilka we know not given to the ribs in the pirida, then I will respect all the same ones. In such a rank, all the ribs are splintered, and not only bichni, equal to oneself, then the basis of the frame and me is the square, and the bichny edges are the correct tricycles. Imaginably, such a piracy, as well as a show on the square, signifying all the tributes that were brought into the text of the plant:
Shukaimo kut mіzh i. I will be robust even for a short wiki, if I am busy fumbling with the coordinates of the points. You need to "decrypt" їх:
b) - midway through. Її coordinates:
c) I know Dovzhin by Pyfagor's theorem in a trikutnik. I know from Pythagoras' theorem in a trikutnik.
coordinates:
d) - midway through. Її coordinates рівні
e) Vector coordinates
f) Vector coordinates
g) Sukaєmo kut:
The cube is the simplest figure. I vpevneniy, that you will sort out with her independently. Visited before the start of the 4th and 5th stages:
Significant kuta between straight and square
Well, the hour of simple tasks is over! Now, lay it down even more foldable. For the formation of a kuta between a straight line and an area, we will act in the following order:
- Three points will be rivnyannya area
,
vikoristovuchi a third-order visitor. - On two points shukaєmo coordinates of the directing vector of a straight line:
- Zastosovuєmo formula for calculating the kuta with a straight and an area:
Yak bachish, the formula is even more similar to the one that was used for the joke of kut_v m_zh two straight lines. The structure of the right part is simply the same, and the evil is now a sine, not a cosine, as before. Well, and it was given to one disagreeable thing - the buzz of the rural area.
We will not add to your mailbox solution of applications:
1. Os-no-va-ni-mu direct-my prizes are-la-et-sya equally-bid-ren-ny tre-vugillya-nik Vi-so-ta prizes-mi dorivnyu. Nay-di-ti kut mіzh straight-my and flat-to-stu
2. At direct-mo-vugillya-nom pa-ra-le-le-pi-ne-de-vest-ni Nay-di-ti kut mіzh straight and flat-to-stu
3. The great-Vіl-noi six-sti-vugіllа-ny have all the ribs of the rіvnі. Nay-di-ti kut mіzh straight and flat-to-stu.
4. At the great-Vіl-niy tre-vugillya-niy pi-ra-mi-de z os-no-va-ni-em z-west-ni ribs Nai-di-ti kut, about ra zo-van-ny flat-to-stu os-no-va-nya і straight-mіy, pro-ho-dya-schuya through gray-re-di-no ribs і
5. Dovzhini of all the ribs of the great-Vil-noi four-angled pi-ra-mi-di from the top-shi-noi equal to oneself. Nai-di-ti kut mіzh straight and flat-to-stu, where the point is se-re-di-na bo-ko-th ribs pi-ra-mi-di.
I know the first two in detail, the third - briefly, and the rest of the two I'll leave you for an independent decision. Until then, the mother was already brought to the right with tricut and chotiric pyramids, and the axis with prisms is still dumb.
solution:
1. Imaginable prism, as well as її pіdstavu. Summaries of the system of coordinates and of all the data, which are in the minds of the tasks:
I ask again for a deyakiy nedotrimannya proportions, but for a verishennya zavdannya tse, on the day, is not so important. Ploshchina is just a "back wall" of my prizes. It’s just good luck to finish it, well, this is the area of the ma viglyad:
However, it is possible to show and without any average:
Viberemo are three points on the whole area: napril ,.
Warehouse area:
To the right you: self-count the design card. Did you get it? Todi equalized area of ma viglyad:
It's just
In such a rank,
For the revision of the butt, I need to know the coordinates of the directing vector of the straight line. Since the point has coincided with the cob of coordinates, then the coordinates of the vector are simply sp_vpadut with the coordinates of the point.
For tsyogo tricycle is visible. Conducted to the top (won f - median and bisectriss) from the top. So yak, then the ordinate of the point is road. In order to know the abscissa of the central point, we need to calculate the total number of points. According to the theorem of Pyphagoras, maєmo:
Todi point maє coordinates:
Point - tse is "transferred" to the point:
Todi vector coordinates:
as follows:
Yak bachish, nothing is foldable in principle, when such buildings are violated. For the sake of the process, I will forgive the "directness" of such a figuri, like a prism. Now let's go over to the offensive butt:
2. Small parallelepiped, carried out in a new area and in a straight line, as well as around the lower part of the area:
The list of known areas is: Coordinate three points that lie in them:
(The first two coordinates are recognized in an obvious way, but the remaining coordinates are easily known from the pictures from the point). Todi warehouse area:
Rahuєmo:
Shukaєmo coordinates of the directional vector: Clearly, the coordinates of the point are taken from the coordinates of the point, why is it wrong? Yak to know the coordinates? The same coordinates of the point, taken along the axis of the applikat on one! ... Todi Shukaymo shukaniy kut:
as follows:
3. A small, correct six-minute pire, and at the same time it is carried out in a straight area.
Here it is problematic to navigate the area on a small scale, it doesn’t seem like it’s about the solution, however, the method of coordinates is all the same! Itself in yogo versatility and in the field of yogo is the main change!
The flank pass through three points: Shukaєmo їх coordinates:
1). Vivedi itself coordinates for the remaining two points. To be familiar for the whole solution of the tasks in the six-foot parade!
2) There will be a rural area:
Shukaєmo vector coordinates :. (I know the wonder of the tricky feast!)
3) Shukaєmo kut:
as follows:
Yak bachish, nothing above-nature folding in the cich zavdannyah nemaє. It is necessary to deprive them of being even more respectable from the roots. Until the rest of the two employees, I will give you more information:
Yak ti mіg perekonatisya, the technology of the revision of the work is the same: it is the main task to know the coordinates of the peaks and to put them in the formulas. We have lost more of one class of problems for the number of buckets, and itself:
Enumeration of kut_v between two houses
The solution algorithm will be as follows:
- On three points of the shukaєmo ryvnyannya of the first area:
- Behind these three points there is another area:
- Zastosovuєmo formula:
Yak bachish, the formula is even more similar to the two in front, behind the help of which they whispered kuti both straight and straight and square. So it’s not a warehouse of special difficulties. Immediately pass to the selection of items:
1. One hundred-ro-na os-no-va-nya pra-vil-niy tre-vugillya-niy prizes-mi dorivnyu, and dia-go-nal bo-kovy mezhi dorivnyu. Nay-dі-tі kut mіzh flat-to-stu and flat-to-stu os-no-va-nya prizes.
2. At the great-Vіl-niy four-ti-ryokh-vugillya-niy pi-ra-mi-de, all the ribs to that pіy rіvnі, nai-di-tі sinus kuta mіzh flat-to-stu and flat- stu, pro-ho-dya-shchey through the point per-pen-di-ku-lar-but straight-my.
3. At the correct four-ryokh-vugillya-niy prism of the side-ro-no of the os-no-va-nya rivni, and the bo-ko-vi of the ribs. On the edge, there is a point so, so. Know kut mіzh flat-to-sti-mi i
4. At the great-Vіl-noi four-cornered, the prize is me st-ro-no os-no-va-nya rіvnі, but bo-ko-vі ribs are rіvnі. On the edge, there is a point like this, so Nay-di-ti kut mіzh flat-to-sti-mi i.
5. In cubi nai-di-ti ko-si-npu kuta mіzh flat-to-ti-mi
Solution of tasks:
1. A small correct (in the base - a one-sided triplet) triple prism and bare on a new area, which figures in the mind of the manager:
We need to know the two areas:
Now we know івняння Point is a coordinate Point - So yak is the median and the height of the tricycle, it is easy to be found in the tricut by the theorem of Pythagoras. Todi point maє coordinates: Known applikat of the point
Today the axis of such coordinates is recognizable: warehouse area.
Numerous kut between areas:
as follows:
2. Robimo babies:
Nayskladnіshe - a sense of vision, so tse taka beyond the tamnichaya area, so pass through the point perpendicularly. Well, well, smut, tse scho? The head is the price of respect! True, it is straight perpendicular. The straight line is also perpendicular. That is the area that passes through the straight lines, if it is perpendicular to the straight lines, and, before the speech, passes through the point. Qia area also passes through the top of the pіramіdi. Todi shukana area - And the area is already given to us. Shukaєmo coordinates of points.
The coordinate of the point is known through the point. It’s easy to get a small baby, but the coordinates at the point will be like this: So now it’s too much to know, why should I know the coordinates of the top of the piracy? It is also necessary to calculate the height. Try to get help from all these theorems of Pythagoras: bring a handful of things (trivially from small trikutniks, how to make a square in the base). Oskilki for the mind, then maєmo:
Now everything is ready: vertex coordinates:
Warehouse area:
There are also specials in numbered viznachnikiv. Without pratsi ti otrimaєsh:
Abo inakshe (which is multiplying of offense of a part on a root of two)
Now we know the area:
(Well, without forgetting, how can I fix the area, right? If it’s not sound, there’s a single minus, then turn around until you see the area!
Numeric designator:
(You can do it, so that the flat area is going from straight ahead, you can pass through the points i! Think, why!)
Now the number is calculated:
We need to know the sine:
as follows:
3. tricky food: And why is it a rectangular prism, what do you think? All the good news to be parallelepiped! Immediately robimo armchair! You can go to the place without imagining it, because it’s not big here:
Ploshchina, as they already remembered earlier, sign up with the viglyadi rivnyannya:
Now warehouse
Immediately from the warehouse:
Sukaєmo kut:
Now the last two buildings have been updated:
Well, now it’s the very hour to re-read it, and they made a great job with you and made a great robot!
Coordinates and vectors. Prosunutiy Rivn
In the case of the statistics, another class of tasks is discussed for you, as it is possible to check for an additional method of coordinates: assignment to the calculation of results. But myself, I can see the following vidgets with you:
- Calculating the number of alternate straight lines.
I have ordered the data of the plant in the world of increasing folding. Naybils just appear to know go from point to area, And most of all - to know look like alternate straight lines... I want, zychayno, dumb no unhappy! It is too much to add to a new box and immediately proceed to the first class of tasks:
Calculating the number of points from the point to the area
What will we need for a revision?
1. Point coordinates
Otzhe, as only we can accept all the necessary data, then there is a formula:
As I will be ryvnyannya area, you will also be guilty of being seen from the front buildings, which I have picked in the last part. Let's start right away to the building. The scheme is offensive: 1, 2, I will help you to check it out in detail, 3, 4 - only when you see it, the decision is carried out by yourself. Let’s go!
zavdannya:
1. Given a cube. Dovzhina cube ribs Nay-di-ti ras-sto-i-nya from se-re-di-no from-cut to flat
2. Dana pra-Vil-va che-ti-rhokh-vugillya-va pi-ra-mi-da Bo-ko-she edge sto-ro-na os-no-va-nya dorivnyu. Nay-di-ti ras-sto-i-nya from point to flat-to-sti de - se-re-di-on ribs.
3. The great-Vіl-niy tre-vugіllja-niy pi-ra-mi-de z os-no-va-ni-mu bo-koshe has one edge, and a hundred-ro-na os-no-va nya door. Nay-di-ti ras-sto-i-nya from ver-shi-no to flatness.
4. The great-Vіl-noi six-sti-vugіllа-ny have all the ribs of the rіvnі. Nay-di-ti distance from point to flatness.
solution:
1. A small cube with single edges, it will be shaped like a square, the middle will be like a letter
.
Let's get a tip from the easy: we know the coordinates of the point. So yak that (guess the coordinates of the middle of the vidrizka!)
Now warehouse in three locations
\ [\ Left | (\ Begin (array) (* (20) (c)) x & 0 & 1 \\ y & 1 & 0 \\ z & 1 & 1 \ end (array)) \ right | = 0 \]
Now I can start before I see it:
2. I know how to fix it from the armchair, for every single meaning of all the tributes!
For piradi, bulo b koryno okremo malyuvati її pіdstavu.
To build on the fact that I paint the yak with my paw, not to give us a problem with ease!
Now it's easy to know the coordinates of a point
So yak coordinates of a point, then
2. Since the coordinates of point a are the middle of the direction, then
We know without any problems the coordinates of two points on the area of the Warehouse, which is more easily understood:
\ [\ 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 \]
So as a point is maє coordinates:, then it will be calculated:
Vidpovid (even worse!):
Well, okay, rozibravsya? It’s for me to be built, everything here is so very technical, like in quiet butts, which they looked at you in the front part. So I’m in the background, as if I had opanuv with this material, then it’s not important if it’s gone viral. I will deprive you of the views:
Calculating the number of traffic straight ahead to the area
As a matter of fact, there is nothing new here. How can it grow straight and square one to one? They have є all the possibilities: to overwhelm, or straight parallel to the area. Yak you think, why is one thing going straight to the area, from which it is given to cross over? It’s clear for me to build, but it’s also clear to zero. Nets_kaviy vipadok.
Another vipadok is trickier: here you will see zero. However, since it is straight parallel to the area, the skin point is straight along the distance from the whole area:
In this rank:
And tse means, how my zavdannya rang to the front: shukaєmo coordinates be a point on a straight line, shukaєmo rivnyannya of the area, numbered from the point to the area. As a matter of fact, such an establishment in ADI is developed in the edge of the country. In the distance, I have to know only one task, and then the data in these boules is such that the method of coordinates is not so much as it is, until now!
Now let's move on to the most important class of tasks:
Calculating the point to the straight
What will we need?
1. Coordinates of the point, from which it is shown:
2. Coordinate whether you want to lie on a straight line
3. Coordinates of the directing vector
Does the yak have a formula?
This signifies the standard of the given fraction, and it is so guilty that it is clear: the chain of the directing vector is straight. There are two cunning numbers here! Viraz means a module (dovzhina) of a vector add-on of vectors and Yak enumerate a vector add-on that was inserted into the front part of the robot. Observe your knowledge, we should smell at once!
In such a rank, the algorithm for verifying the launch of the coming will be:
1. Shukaєmo coordinates of a point, from what shukaєmo seen:
2. Shukaєmo coordinates be a point on a straight line, before which shukaєmo are seen:
3. There will be a vector
4. I will direct the vector straight
5. Numerous vector add-ons
6. Shukaєmo to the genie of the rejected vector:
7. Numerously available:
We have a lot of robots, but just put it on foldable! So now I respect all the zooseredit!
1. Dana pra-Vil-va tre-vugillya-va pi-ra-mi-da z ver-shi-noi. One hundred-ro-na os-no-va-nya pi-ra-mi-di dorivnyu, vi-so-ta dorivnyu. Nay-di-ti ras-sto-i-nya from se-re-di-no more ribs to straight lines, de point i - se-re-di-no ribs і z-vіd- veiny-but.
2. Dovzhini ribs і straight-mo-vugіllа-th-pa-pa-le-le-pі-ne-da rіvnі so-іd-vet-nogo-but і Nay-di-tі ras-sto-i-nya from ver-shi-no to straight
3. At the great-Vil-noi six-sti-vugillya-ny, all the ribs are up to the point to the straight line
solution:
1. Robimo neat armchair, in which all the tributes are unmistakable:
The robots we have with you is strong! I would like to describe a collection of words that will be shukati in any order:
1. Coordinates of points i
2. Point coordinates
3. Coordinates of points i
4. Coordinates of vectors
5.Їх vector dobutok
6. Dovzhinu vector
7. Create vector dovzhin
8. View from before
Well, well, the robots are enough for us! Taking care of her, wrapping her sleeves!
1. In order to know the coordinates of the height of the road, we need to know the coordinates of the point of the application to zero, and the ordinate to the direction of the Abscis, to the road to the point of departure. That is enough, the coordinates were set off:
point coordinates
2. - midway through
3. - midway through
mid-way
4.Coordinati
vector coordinates
5. Numerous vector add-ons:
6. Dovzhina vector: the easiest way to substitute, so the vidrizok is the middle line of the trikutnik, and that means, the most important part of the delivery. So scho.
7. Create more vector graphics:
8. Nareshty, it is known to come:
Phew, that's it! Honestly, I will say: the solution of the task by the traditional methods (through motivation), it will be better. Then I zvіv all the way to the finished algorithm! I think so, is the algorithm clear? To that I will ask you to be virishiti lost two zavdannya independently. What is the correct message?
I will repeat myself again: the process of being simpler (shvidshe) is virishuvati through encouraging, and not being drawn to the coordinate method. I have demonstrated such a way to demonstrate a deprivation of time, to show that a universal method, which allows "nothing to be obtained."
Nareshty, the remaining class is visible:
Calculating the number of alternate straight lines
Here, the algorithm for verifying the project will be similar to the previous one. What we have є:
3. Be-like a vector from one point of the first and another straight line:
Yak mi shukaymo see me straight?
The formula is offensive:
The numeral is the module of great creation (it was introduced in the front part), and the denominator is in the front of the formula (the module of the vector addition of direct vectors in straight lines, become between the same ones for you, shukamo).
I will guess you, you
Todi the formula for the appearance can be rewritten in the viglyad:
This is the kind of business card for the business card! I want it, honestly, I don't want to get hot here! The formula is given, for the most part, even a bunch of і to make up to finish the folding numbers. On your misc, I would go to it only in the most extreme vipad!
Let's try it out virishiti kilka zavdan, vikoristovuchi vikladenii vishche method:
1. At the great-Vil-niy tre-vugillya-ny pr-me, all the ribs to that rіy rіvnі, nay-di-tі ras-sto-i-nya mіzh straight-mi i.
2. Dana pra-Vil-va tre-vugillya-va prism-ma all the edges of the os-no-va-nya ko-to-riy rivni Se-che-nya, pro-ho-dya-ny through bo-ko-she rib і se-re-di-well ribs yav-la-et-sya kvad-ra-te. Nay-di-ti ras-sto-i-nya mіzh straight-mi i
I fly, and spiral on her, friend of the virgin!
1. Small prism i mean straight i
Coordinates of point С: todі
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) \]
Vvazhaєmo vector dobutok mіzh vectors i
\ [\ 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 I love you for dinner:
as follows:
Now try to be careful with each other. I will tell you on this :.
Coordinates and vectors. Short description and basic formulas
Vector - conjugation of the form. - a cob of a vector, - a head of a vector.
Vector is abo.
absolute value vectors - dovzhina v_drizka, which image is a vector. Sign up, yak.
Vector coordinates:
,
de - kіntsі vector \ displaystyle a.
Suma vectors :.
Tvir vectors:
Scalar add-on vectors:
Scalar add-on of vectors into the road add-on of the absolute values per cosine of the cut between them:
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VECTOR
In physics and mathematics, the vector is a whole quantity, which is characterized by its numerical values and directly. Physics has a few important values, such as vectors, for example, force, position, speed, acceleration, torque, momentum, voltage of electric and magnetic fields. Їх can be opposed to other values, such as yak masa, volume, vise, temperature and strength, which can be described by a singular number, and the stench is called "scalars". The vector is written to be victorious when robotic with values, as it is unwise to put extraordinary numbers behind the aid. For example, I would like to describe the camp of the subject of the deyakoi point. We can say, a few kilometers from a point to an object, but I can’t increase the significance of this moment of growth, unless I know it directly, in which wine is located. In such a rank, the position of an object is characterized by numerical values (appearance in kilometers) and directly. Graphically, the vectors are displayed at the viewers of the straight lines, like in fig. 1. For example, in order to represent graphically the force in five kilograms, it is required to do it directly to five kilograms directly to the force. The arrow is injected, the force is from A to B; jakbi the strength of the dialer is from B to A, then we wrote it down; vector A and -A may be equal to the numerical value, but not directly. The numerical value of the vector A is called a modulus, or is called A, or | A |. Tse value, zychayno, scalar. Vector, the ear and the end of the way to start, to be called zero and to be called O.
Two vectors are called real (or vilny), both modules and straight lines. In mechanics and physics, however, the need for protection is required, since two strong forces, applied to the new points of the body in the backward look, will lead to better results. At the link from the digital vector, the vectors can be "tied" or "tied", in the following way: For example, the radius vector in the case of the position of the point is clearly fixed to the cob of coordinates. The vectors are connected in a similar fashion, since they do not only have modules that are straight, but they smell like a special point with programs. Light vectors are called equal vectors, roztashovani on one straight line.
Suma vectors. The idea of adding vectors to the winnickl is that we can know a single vector, which is the same inflow, but two other vectors at once. Just in order to take a point to the point, we need to go through a set of A kilometers in one straight line and then B kilometers in one straight line, then we could reach our end point by passing C kilometers on the third straight line. You can tell a tsyu sensei, scho
A + B = C.
The vector C is called the "result vector" A і B, we will be asked by design, we will show it to the little one; on vectors A і B yak on the sides of the parallelogram incentives, and C is a diagonal, which is a single ear A і kіnets B. З fig. 2 it can be seen that the addition of vectors is "commutative", that is, A + B = B + A. A similar order can be used to fold the vectors, last but not least, "without interruptions", as shown in Fig. 3 for three vectors D, E and F. 3 fig. 3 you can see it too
(D + E) + F = D + (E + F), so that folding vectors in an associative way. Pidsumuvati can be as many vectors, and the vectors need not lie in the same area. The appearance of vectors is represented as a folding with a negative vector. For example, A - B = A + (-B), de, as it started earlier, -B is a vector, like a road B in modulus, ale opposite to the right. The whole rule of folding can now be victorious as a real criterion for reversal, where the value of the vector is. Change to win over to the minds of this rule; then we can say about the news; they are stored in the same rank, as it is possible to bulo bachiti from the "trikutnik of forces". However, the figures of magnitude, like numerical values in such a straight way, do not comply with this rule, it cannot be seen as a vector. Butt є kіntsevі wrapping.
Multiply a vector by a scalar. Tvr mA abo Am, de m (m No. 0) is a scalar, and A is a nonzero vector, it starts as the first vector, which is m times more than A and may be right next to A, if the number m is positive, and more than m negatively, the yak is shown in Fig. 4, de m dorіvnyuє 2 і -1/2 as appropriate. In addition, 1A = A, so that when multiplied by 1 vector does not change. The value -1A is a vector, which is a road A along the line, a little more directly behind it, you want to write down yak -A. If A is a zero vector i (abo) m = 0, then mA is a zero vector. Multiplication distributively, tobto
We can add the number of vectors, and the order of completion does not flow into the result. Vіrno і vorotne: be it a vector to fold into two or more "components", to two vectors or more, as, being folded, as a result will give an output vector. For example, in Fig. 2, A і B - components C. Bagato mathematical diy with vectors say goodbye, as the vector is expanded into three components for three mutually perpendicular strains. The vibro-right system of Cartesian coordinates with the axes Ox, Oy and Oz yak is shown in Fig. 5. Go to the right coordinate system of the mamo on uvaz, so the x, y and z axes grow so that they can be, apparently, the great, middle and middle finger of the right hand. From one right coordinate system, it is possible to customize the right coordinate system in different wraps. In fig. 5 shows the decomposition of the vector A into three components і Vony in sum; fold the vector A, so as
already,
It is also possible to add a range of folds and get it until the end of the Projection of vector A on three coordinate axes, designated Ax, Ay and Az are called "scalar components" of vector A:
de a, b і g - kuti mіzh A і three coordinate axes. Now we introduce three vectors of a single relationship i, j і k (orti), but the same ones directly, as well as the given axes x, y and z. Todi, if Ax is multiplied by i, then we omit tvir - tse vector, equal i
Two vectors are equal to one and only to one, if equal to one scalar component. In such a rank, A = B only and only if Ax = Bx, Ay = By, Az = Bz. Two vectors can be folded, storage components:
In addition, for the Pyfagorian theorem:
Line functions. Viraz aA + bB, de a і b - scalars, called a linear function of vectors A and B. The vector, which is located in the same area, is A and B; If A and B are not parallel, then when a and b change, the vector aA + bB will move over the entire area (Fig. 6). If A, B and C do not all lie in the same area, then the vector aA + bB + cC (a, b and c changes) will move throughout the entire space. Suppose that A, B and C are single vectors i, j і k. The vector ai lies on the x-axis; vector ai + bj can be moved over the whole area xy; the vector ai + bj + ck can move around the whole space.
It is possible to vibrate the chotiri of mutually perpendicular vectors i, j, k і l but the value of the chotiry vector yak the value A = Axi + Ayj + Azk + Awl
with dozhinoy
And it is possible to prodovzhuvati up to five, six, for any number of vimiriv. If I want to visualize such a vector, it’s hard to imagine any mathematical difficulties here. Such a notation is often buvaє korisna; for example, the particle collapsing is described by a six-dimensional vector P (x, y, z, px, py, pz), the component of which is the position in space (x, y, z) and impulse (px, py, pz). Such space is called "phase space"; Whenever we look at two particles, then the phase space is 12-dimensional, sometimes three, then 18 and so far. The number of dimensions can not be interrupted by the number of sizes; At the same magnitude, we will be right, we will be rich in what it is, as it is, as it is visible in the resolution of the statistics, and in itself, the trivial vector.
Multiple two vectors. The rule for adding vectors to the bulo is defined by the way of determining the behavior of quantities represented by vectors. There are no obvious reasons, for which two vectors can be multiplied by the order of multiplication, the process of multiplying the matime sense in the same way, as it is possible to show its mathematical ability; besides, bazhano, schob tvir mav singing physical zmist. There are two ways of multiplying vectors, which convey the minds. The result of one of them is a scalar, such a tvir is called a "scalar work" abo internal cheese"Two vectors and write down AHB or (A, B). The result of this multiplication is a vector, titles" vector art "or" call art "and write A * B or []. Three Vimiriv, Todi Yak Vector Create a value only for the three Vimir.
Scalar create. As soon as the F point, to which it is applied, is moved to the position r, then the robot's Viconan add r and the component F directly to r. Tsya component door F cos bF, rc, de bF, rc - cut mіzh F і r, tobto Viroblena robot = Fr cos bF, rc. Tse is the butt of the physical priming of scalar creation, which is assigned for any two vectors A, B for an additional formula
A * B = AB cos bA, Bc.
So as all the values of the right part of the rivnyannya are scalars, then A * B = B * A; Also, scalar multiple commutative. Scalar multiplication is also the power of distributiveness: A * (B + C) = A * B + A * С. ni A, ni B is not equal to zero. We cannot do this by a vector. Supposedly, we have broken the offense of a part of the family A * B = A * C on A. It gave B = C, і, if it could be done, then the price became a single possible result. However, if we can rewrite the line A * B = A * C for the viewer A * (B - C) = 0 and guess if (B - C) is a vector, then it is clear that (B - C) is unnecessarily expensive to zero і, from the same, B is not guilty of being equal to C. It is very clear that the results show that it’s not a matter of vector. Scalar solid is even one way to write the numerical value (modulus) of the vector: A * A = AA * cos 0 ° = A2;
to that
Scalar tvir can be written in one way. For tsiogo guess, scho: A = Ax i + Ayj + Azk. Dear, scho
Todi,
Oskіlki left іvnyаnnya to revenge x, y and z in the position of lower indices, rіvnyаnnya, zdavalosya b, to deposit in a different specific coordinate system. However, it is not so, it can be seen from the value, as it does not lie in the reverse coordinate axes.
Vector create. A vector, or called a vector of vectors, is a vector, a module of which additional modules are added to the sinus cut, perpendicular to the output vectors, and at the same time become the right three of them. Tsey tvir is the easiest to introduce kutovy shvidk_styu... Persha is a vector; We now show that I can also interpret the vector. Kutova shvidkist tila, how to wrap up and start with the offensive rank: be it a point on the floor and draw a perpendicular from the center point to the wrap axis. Todi kutova shvidkist tila - the whole number of radians, on which the line turned in one hour. Yaksho kutova shvidk_st is a vector, the mother is guilty of the numerical value and directly. The numerical value rotates in radians per second, it is directly possible to vibrate the axis of the wrapping, it can be done by directing the vector in that direction, in which the right-handed gwent collapses when wrapped at once. The body can be easily wrapped around the fixed axis. As soon as it is installed in the middle of the circle, it is fixed on the axis, inserted in the middle of the circle, we can wrap up the middle of the first circle in the middle of the first circle in the middle of the first circle in the middle of the first circle in the middle of the circle. Malunok 7 I will explain the essence of the right; Circular arrows show straight wrap. Dane tilo is a solid sphere with a center O and radius r.
Small. 7. SPHERE IN THE CENTER O, wrap around in the middle circle w1 in the middle circle BC, yake, in your own circle, turn around in the middle circle DE in the core section w2. The sphere is wrapped in a kutovaya shvidkistu, rіvnіy sumі kutovykh vidkosti and all specks on the straight POP "are in the camp of mitten calm.
Dodamo tilu rukh, yake є a sum of two big kutovs. It’s obvious that it’s obvious that it’s no longer necessary to wrap around a fixed axis. However, you can still say how to wrap yourself up. To show, vibrate point P on the surface of the body, as in the moment we see it for an hour, we will be on the great number, where from the same point, in which two axes cross the surface of the sphere. The perpendiculars from P on the axis are permissible. Tsі perpendiculars become radіuses PJ і PK kіl PQRS і PTUW as appropriate. Draw a straight line POPў passing through the center of the sphere. Now point P, at the moment one o'clock, one hour moves on the stake, which sticks to point P. For a maliy interval hour Dt, P moves to the top
The price is zero, if
In general, the point P is located in the camp of the mitten calm, and just like all the specks on the straight POP. " wrap around the spheres, like before, like a wheel, like a wheel, like a wheel, like a wagon on the road at the skinny moment of the hour, wrap around its bottom point. we can see it, it’s possible to move in an hour Dt to come
By the stake of the radius r sin w1. For viznachennyam, kutova shvidkist
From the tsієї formulas і spіvіvіdnoshennya (1) mi otrimamo
In other words, if you write down the numerical values and vibrate right in the middle of the way, as the object is described, then the values are stored as vectors and may be displayed as such. Now you can enter a vector add-on; It’s easy to see how you can wrap yourself in a cage w. Vibremo be-like point P on til and be-like cob of coordinates Oh, like be on the axis of wrapping. Do not go r - a vector of straightening from Pro to P. Point P collapses on a count from shvidkistu V = w r sin (w, r). The fluidity vector V is similar to the point of insertion in the straight line shown in Fig. eight.
The price of the liquidity is the V point from the combination of two vectors w and r. Vikoristovuєmo tse spіvvіdnoshennya, in order to give a new type of creation, and it is written: V = w * r. So as the result of such a multiplication is a vector, the whole tvir is called vector. For any two vectors A and B, where A * B = C, then C = AB sin bA, Bc, and right of the vector C also, if it is perpendicular to the area, then pass through A and B and insert into a straight line, so that With a direct collapse of dextrorotatory gwent, which is parallel to C and wrap around from A to B. In other words, we can say that A, B and C, in this order, set up the right set of coordinate axes. Vector addition anticommutativity; the vector B * A has the same modulus, but not і A * B, ale the conjugations in the opposite bik: A * B = -B * A. can I bring you
Surprisingly, I will write a vector addition in terms of components and single vectors. Persh for everything, for any vector A, A * A = AA sin 0 = 0.
Also, in times of single vectors, i * i = j * j = k * k = 0 і i * j = k, j * k = i, k * i = j. Todi,
The price of parity can also be written in the viglyad_ of the card holder:
If A * B = 0, then either A or B road is 0, or A і B collinear. In such a rank, like і in the form of scalar creation, it is unwise to change it to a vector. The value of A * B is on the side of the parallelogram with sides A and B. It is easy to bachiti, since B sin bA, Bc - th height and A - pіdstava. There are many rich physical quantities, such as vector creations. One of the most important vector creations appears in the theory of electromagnetism and is called the Poyting vector P. Vector P can be viewed at square meter in be-yak_y point. Guided by the spike of the butt: the moment of force F (torque) to the cob of coordinates, which is to the point, the radius vector of the r, which starts at r * F; the part, which is located at point r, mass m і shvidk_styu V, mkutoviy moment mr * V, which is the cob of coordinates; the force, which is on the particle, which carries the electric charge q through the magnetic field B from the speed V, є qV * B.
Try to create. Three vectors in my can formulate such useful things: vector (A * B) * C; vector (A * B) * C; scalar (A * B) * C. The first type is the twir of the vector C і of the scalar A * B; we talked about such creatures. Another type is called a sub-vector vector; vector A * B perpendicular to the area, to lie A і B, and to that (A * B) * C - vector, to lie in the area A і B і perpendicular to C. Then, in the zahal vipad, (A * B) * C NOT one A * (B * C). Having written A, B і C through їх coordinates (components) along the x, y і z і axes, multiplying, we can show that A * (B * C) = B * (A * C) - C * (A * B). The third type of creation, such as winnick when opening the lattice in the physical structure of a solid body, numerically with a parallel line with edges A, B, C. So yak (A * B) * C = A * (B * C), the signs of scalar and vector multiplications are possible In small numbers, and TV is often recorded as yak (ABC). Tsey tvir one determinant
It is great that (A B C) = 0, if all three vectors lie in the same and the same area, if A = 0 or (i) B = 0 or (i) C = 0.
DIFFERENTIAL VECTOR
It is admitted that the vector U is a function of one scalar variable t. For example, U can be a radius vector, we draw coordinates to move the point, and t - an hour. Do not change t by a small value of Dt, but before increasing U by the value of DU. Tse is shown in fig. 9. Bridging DU / Dt is a vector of directing in the same direction, wi і DU. We can get U by t, yak
for wash, so this is the border. From the side, it is possible to show the U as the sum of the components along the three axes and write
If U is the radius vector r, then dr / dt is the frequency of the point, the function is rotated for the hour. Differentiating over the hour more times, we can accept it. It is acceptable that the point is shifting curved, shown in Fig. 10. Nekhai s - come, I pass by the point of bridle crooked. Stretching out a small interval of an hour Dt point, pass through Ds curved lines; the station of the radius vector change to Dr. Otzhe Dr / Ds is the vector of straightening yak Dr. far
Vector Dr is the change of the radius vector.
є single vector, dotted to crooked. It can be seen from the fact that when the point Q is close to the point P, PQ is close to the point and Dr is close to Ds. Formulas for differentiation create with formulas for differentiation create scalar functions; However, since the vectorial addition is anticommutative, the order of multiplication is guilty of saving. Tom,
In such a rank, mi bachimo, scho, if the vector is a function of one scalar change, then we can be taken away as well, as in the case of a scalar function.
Vector and scalar fields. Gradin. Physics is often brought to the right with vector or scalar values, as they change from a point to a point in a given area. Such areas are called "fields". For example, a scalar can be a temperature or a grip; vector can be shvidk_styu collapse the line or electrostatic field of the system of charges. Whenever we selected a coordinate system, be it a point P (x, y, z) in a given area, a certain radius vector r (= xi + yj + zk) and also the value of a vector quantity U (r) of a scalar f ( r), mated with him. It is acceptable that U і f values in the region are unambiguous; so that the skin point will be one or only one value of U or f, if the points can be different, obviously, the meaning is different. Supposedly, I would like to describe the speed, as U and f change when oversubscribed in the region. Simple private ones, such as dU / dx і df / dy, we will not be wiped out, so the stench lies in the concrete opposite coordinate axes. However, it is possible to introduce a vector differential operator, independent from the choice of the coordinate axes; the operator is called "grad". Let me do it right with the scalar field f. With a handful of yakosti butt I will clearly outline the map of the region of the country. In the first vipadk f - the height above the sea level; contour lines connect points with the same values f. When rus, bridling be-like z tsikh lines f do not change; If it collapses perpendicularly to the lines, then the speed of the snake will be maximized. We can put the skin point at the vector, which will set the magnitude and the maximum change in speed f; Such a map of the number of the number of vectors is shown in Fig. 11. If the vison is for the skin point of the field, then the vector field is visible, tied with the scalar field f. The whole field of the vector, called the "grad" f, which is written as grad f or Cf (the symbol C is also called "Nabla").
At once three vimiriv, contour lines become surfaces. Male substitution Dr (= iDx + jDy + kDz)
de points of designation of the members of other orders. Tsey visl_v can be recorded at the viewer of scalar creation
Rozdilimo right and left part of the chain of equality on Ds, and let Ds pragne to zero; Todi
de dr / ds is a single vector in the vibranome straight line. Viraz in the arches is a vector to lie at the opposite point. In such a rank, df / ds is the maximum value, if dr / ds is inserted in the same straight line, the verge, which stands in the arches, is a gradient. In such a rank,
- a vector, which is a road behind the value and to go directly behind the maximum speed of the change f from the coordinates. Gradієnt f often sign up at the viglyadі
Tse denotes that the operator C is itself by its own. In the case of baguettes, there is a vector and, in fact, a "vector differential operator" - one of the most important differential operators in physics. Unimportant to those who can replace single vectors i, j and k, which physical change does not lie in the other coordinate system. Yaky sound mіzh Сf і f? First for everything is permissible, but f is the initial potential in any point. For any small offset Dr, the value of f changes by
Where q is the value (for example, mass, charge), is shifted to Dr, then the robot, viconana when q is shifted to Dr road
So as Dr - displacement, then qСf - force; -Сf - tension (force per unit of number), tied with f. For example, let U - electrostatic potential; Todi E is the voltage of the electric field, given by the formula E = -CU. It is permissible that U is set by a point electric charge in q coulomb, put on the cob of coordinates. The value of U at the point P (x, y, z) with the radius vector r is given by the formula
De e0 - dielectric post-vilnogo space. Tom
viplivi, which is E diє in the right direction r і th value of the road q / (4pe0r3). Knowing a scalar field, you can refer to a vector field associated with it. So it is possible and ringing. From the point of view of mathematical processing, scalar fields are easier to operate, lower than vector ones, since the stench is set by one function of coordinates, at that time the vector field is represented by three functions, which are related to the components of the vector in three strands. In such a rank, vinnikє nutrition: given a vector field, who can write a scalar field associated with it?
Divergence and rotor. We have substituted the result for a scalar function. What will it become, how can it be from zastosuvati to the vector? Є two possibilities: let U (x, y, z) - vector; todi we can create a vector and scalarly create by an offensive rank:
Perche z tsikh viraziv - scalar, ranked by divergence U (denoted divU); the other is a vector of names of the rotor U (denoted by rotU). Tsi differential functions, divergence and rotor, widely used in mathematical physics. To find out that U is a good vector і wіn аnd thіѕ first lost іn thе region. Nekhai P is a point in the tsiy area, marked off by a small closed surface S, which will be surrounded by the DV obsyag. Nekhai n - a single vector, perpendicular to the whole surface in the skin point (n changes directly when Rus is near the surface, but sometimes it is single); Don't give n conjugations of names. Show me, scho
Here S indicates that the integrals are taken on all surfaces, da is the element of the surface S. For simplicity, I will vibrate for us the shape S in the view of a small parallelepiped (as shown in Fig. 12) with the sides Dx, Dy and Dz; point P is the center of the parallelepiped. Numerously integrated from the level (4) compiled on one side of the parallelepiped. For the front face n = i (single vector parallel to the x axis); Da = DyDz. Adding to the Integral from the front side of the road
On the opposite side n = -i; tsya edge yes additions to the integral
Vikoristovuchi Taylor's theorem, we can recognize the gallant contribution from two faces
Really great, DxDyDz = DV. A similar rank can be used to count the insertions of two pairs of faces. Povny Integral Dorivnyu
If it is satisfactory to DV (r) 0, then the members of the higher order should be known. For the formula (2), the viraz in the arches is the divU, which is used to bring the parity (4). Equality (5) can be brought to the same rank. Skoristaєmosya know rice. 12; todi of insertions from the front side in the integral
I, vikoristovuchi Taylor's theorem, we can deny that the sum of the additions to the integral from two faces of the ma viglyad
That is, there are two terms in the virase for rotU in rіvnyаnnі (3). Some of the members will go through the field of contributions from some of the edges. What, by the way, does it mean ts spivvidnoshennya? The parity is clear (4). It is permissible that U is shvidk_st (rіdini, for example). Todi nЧU da = Un da, de Un is the normal component of the vector U to the surface. To that, Un da - tse obsyag rіdini, scho passes through da at one hour, but tse obsyag rіdini, yaka vitikє through S at one o'clock. already,
Shvidk_st expansion of one volume around point P. Zvidsi divergence has given up its name; it will show the speed, from where the line expands from (to diverge from) P. Explain the physical value of the rotor U, the visible surface integral along the small cylindrical length h, near the point P; plane-parallel surfaces can be arranged in any direct way, like vibrating. Let k is the unit vector perpendicular to the skin surface, and do not let the skin surface area DA; todi new obsyag DV = hDA (Fig. 13). Displayed now integraral
value In order, the number of (x 1, x 2, ..., x n) n available numbers is called n-dimensional vector, And the numbers x i (i = 1, ..., n) - components, abo coordinates,
Butt. Also, for example, the car plant is responsible for the replacement of 50 passenger cars, 100 vans, 10 buses, 50 sets of spare parts for passenger cars and 150 kits for vans and buses in the , 10, 50, 150), there are five components.
Designation. The vectors are designated in bold small letters or letters from the border or arrow uphill, for example, a abo. Two vectors are called rivnim, If the stink is the same, the number of components is the same.
The components of the vector can be minimized, for example, (3, 2, 5, 0, 1) and (2, 3, 5, 0, 1) different vectors.
Operations over vectors. cheesex= (X 1, x 2, ..., x n) on the device the number λ is called the vector λ x= (Λ x 1, λ x 2, ..., λ x n).
bagx= (X 1, x 2, ..., x n) і y= (Y 1, y 2, ..., y n) is called the vector x + y= (X 1 + y 1, x 2 + y 2, ..., x n + + y n).
Space vectors. N-world vector space R n It starts as a functionless all n-world vectors, for which the value of the operation is multiplied on the date and folding.
Economical illustration. Economical illustration of the n-world vector space: space of blessings (goods). pid commodity we will be smart, good for the service, that we needed sales at the singing hour in singing song... It is admissible that the number of obvious goods is n; number of skin from them, pribani we live, are characterized by a set of goods
x= (X 1, x 2, ..., x n),
de x i denote the number of the i-th good stuffed by the resident. We will respect, that all comrades may have the power of their own identity, so that can be bought, be it not for me to have some of the skin from them. To all you can pick up goods є vectors to the space of goods C = ( x= (X 1, x 2, ..., x n) x i ≥ 0, i = 1, ..., n).
Lineage independence. system e 1 , e 2 , ... , e m n-dimensional vectors are called line-fallow, If you know the numbers λ 1, λ 2, ..., λ m, you want to see one from zero, if you want to see the parity of λ 1 e 1 + λ m e m = 0; In the given view, a system of vectors is given to be called linear square, So that no parity can be specified more than once, if all λ 1 = λ 2 = ... = λ m = 0. R 3, interpreted as directing images, explain these theorems.
Theorem 1. The system, which can be stored from one vector, is linearly deposited only once and only if the vector is zero.
Theorem 2. In order for two vectors to be lined with fallow, it is necessary and sufficient that the stinks are collinear (parallel).
Theorem 3 ... In order for three vectors to be lined with fallow trees, it is necessary and sufficient that the stench will be coplanar (lay in the same area).
Rights and lіva three vectors. Three non-coplanar vectors a, b, c be called right, I will sposter from the rykhny zagalnogo cob bypassing the vectors a, b, c in the designated order, it should be done behind the year's line. B іnshomu vipad a, b, c -liva trika... All rights (chi livi) three vectors are called equally orієntovanimi.
Basis and coordinates. Triyka e 1, e 2 , e 3 non-coplanar vectors in R 3 be called basis, And the same vectors e 1, e 2 , e 3 - baseline... be-vector a can be a single rank of expansions in basis vectors, so that the representations of the viewer
a= X 1 e 1 + x 2 e 2 + x 3 e 3, (1.1)
numbers x 1, x 2, x 3 in the spreadsheet (1.1) are called coordinatesa in basis e 1, e 2 , e 3 i start a(X 1, x 2, x 3).
Orthonormal basis. yaksho vector e 1, e 2 , e 3 pairwise perpendicular and perpendicular to the skin of them, then the basis is called orthonormal, And coordinates x 1, x 2, x 3 - rectangular. base vectors orthonormalized to a basis we will mean i, j, k.
Let’s allow it to be in space R 3 vibrano right Cartesian rectangular coordinate system (0, i, j, k}.
Vector Vitvir.vector cottage cheesea per vector b be called vector c How to start with the next three minds:
1. Dovzhin vector c numerically the area of the parallelogram induced on vectors aі b, i.e.
c= | A || b | sin ( a^b).
2. Vector c perpendicular to skin vectors aі b.
3. Vector a, bі c, Taken in the designated order, validate the right of the three.
For vector create c enter the value c =[ab] abo
c = a × b.
yaksho vector aі b collinear, then sin ( a ^ b) = 0 і [ ab] = 0, zokrem, [ aa] = 0. Create vector vectors: [ ij]=k, [jk] = i, [ki]=j.
yaksho vector aі b given in basis i, j, k coordinates a(A 1, a 2, a 3), b(B 1, b 2, b 3), then
Zmіshane tvіr. Yaksho vector dobutok two vectors aі b scalar multiplied by the third vector c, then such a set of three vectors will be called with cheese i denoted by the symbol a b c.
yaksho vector a, bі c in basis i, j, k given by your coordinates
a(A 1, a 2, a 3), b(B 1, b 2, b 3), c(C 1, c 2, c 3), then
.
Zmіshane tvіr is simpler geometrically tlumachennya - a whole scalar, in absolute value, dorіvnyuє volume of parallelepiped, prompted on three given vectors.
If the vector confirms the right of the three, then the number is more positive, equal to the stated volume; what a trika a, b, c - liva, then a b c<0 и V = - a b c, Even V = | A b c |.
The coordinates of the vectors, which are used in the tasks of the first distribution, are transferred by the given to the right orthonormal basis. Single vector codirectional to vector a, denoted by the symbol a O. symbol r=OM the radius vector of the point M is denoted by the symbols a, AB or | A |, |AB | are called moduli vectors aі AB.
butt 1.2. Know kut mіzh vectors a= 2m+4nі b= m-n, de mі n - single vectors and cut mij mі n road 120 o.
Decision... Mєmo: cos φ = ab/ Ab, ab =(2m+4n) (m-n) = 2m 2 - 4n 2 +2mn=
= 2 - 4 + 2cos120 o = - 2 + 2 (-0.5) = -3; a = ; a 2 = (2m+4n) (2m+4n) =
= 4m 2 +16mn+16n 2 = 4 + 16 (-0.5) + 16 = 12, meaning a =. b = ; b 2 =
= (M-n)(m-n) = m 2 -2mn+n 2 =
1-2 (-0.5) +1 = 3, even b =. Residual amount: cos φ == -1/2, φ = 120 o.
Application 1.3. knowchi vector AB(-3, -2.6) i BC(-2,4,4), count up to the length of the AD of the tricycle ABC.
Decision... Meaning of the area of the tricycle ABC through S, we can say:
S = 1/2 BC AD. Todi AD = 2S / BC, BC = = 
= 6,
S = 1/2 | AB ×AC |. AC = AB + BC, Mean vector AC maє coordinates
.