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Channel: Маткульт-привет! :: Алексей Савватеев и Ко

математика

Савватеев

движение прямой

математическое движение

отражение

перенос

параллельный перенос

100 уроков математики

физическое движение

движение

савватеев

100уроковматематики

геометрия

00:00:03

hello, today we will study

00:00:06

movements, what is movement,

00:00:09

physicists will tell you that movement is

00:00:12

the process of moving

00:00:14

some object in space, that is, the process itself is

00:00:19

looked at by mathematicians, but the concept of movement is

00:00:21

a little different for a mathematician,

00:00:24

movement is not a process, but roughly speaking

00:00:27

the result, that is, we look where the object

00:00:30

was at the beginning, look where it was at the end and

00:00:33

here is the result of the movement we call

00:00:36

movement, well let's give a strict

00:00:39

definition of what movement is, the

00:00:41

definition of movement is a transformation of

00:00:47

an object, that is, a change in its position

00:00:50

in space such that

00:00:54

no deformations occur, that is, the

00:00:57

distance between any two points of

00:00:59

the object

00:01:00

will be preserved if we change it at the beginning

00:01:03

and at the end it will be the same number,

00:01:05

in order to write this in a strict

00:01:07

mathematical form, we need to

00:01:10

resort to some

00:01:12

notation to terminology, first of all,

00:01:15

as in school mathematics, numbers are

00:01:19

sometimes called letters, especially when solving

00:01:21

equations x and y so further we will

00:01:24

call the movement

00:01:26

letters of the Latin alphabet, that is,

00:01:29

if we are referring to some kind of movement

00:01:31

that we are currently studying, well, let’s

00:01:33

call it, for example, a letter,

00:01:34

but we will also denote the points of the object with

00:01:38

some Latin letters

00:01:40

x and y and then what is movement

00:01:43

formally, movement is such a change in

00:01:48

position that for any this is

00:01:50

quarter as mathematicians say,

00:01:52

denoting the words any for any two

00:01:55

points from our object, the distance

00:02:02

that we will denote by the letter po

00:02:05

between the result,

00:02:10

that is, where the points ended up after the

00:02:13

same movement was applied to them

00:02:17

is equal to the distance between them before the

00:02:21

movement was applied movement can still

00:02:24

be understood drop dust cash register

00:02:26

a person comes and orders each point of

00:02:29

the object to move somewhere if such an

00:02:31

order is possible without changing the shape of the

00:02:33

object this is what we say that this order

00:02:35

is a movement let's give two

00:02:37

examples one example will be an example of

00:02:40

movement and another example will be, generally

00:02:42

speaking, an example of movement, a space that

00:02:45

is not movement, an example of movement,

00:02:48

then a plane flies from city to

00:02:50

city, here it was at the airport of one city,

00:02:52

stood at speed, flew all this time,

00:02:55

it did not change its position, that is, its

00:02:58

movement from the starting point to at any

00:03:00

point photographed during the flight,

00:03:02

as well as at the moment when he sat down, there will be a

00:03:05

transformation of movement, but if we

00:03:08

consider a bird that is flying,

00:03:11

then its movement from the point where it

00:03:13

sits to the point where the analyst

00:03:15

will not be movement because it has wings, it flaps its

00:03:19

wings and the distance there between the tip of the

00:03:21

wing, for example, and the tail will

00:03:24

change during such a flight, so

00:03:26

the movement of the bird, even the movement of the bird from the

00:03:29

moment when it was sitting to the moment when

00:03:31

it flies, is a transformation of an object, a

00:03:34

bird that is not a movement. Specifically,

00:03:38

you and I will first deal with

00:03:40

the movements of a straight line, what is a straight line from a

00:03:45

point From the perspective of mathematics, a straight line

00:03:47

is infinite in both directions, very

00:03:51

very thin, as mathematicians say, an

00:03:53

infinitely thin, even line

00:03:57

means that in order to imagine the movement of a

00:03:59

straight line, you need to imagine it from some kind of

00:04:03

material that does not allow

00:04:05

stretching, they do not bend, and if there is

00:04:09

such a wire and this wire

00:04:12

we want to somehow move the opportunity

00:04:14

to do this, well, in principle, you can

00:04:17

guess that we can take it as a

00:04:20

certain point and move the whole thing as a whole

00:04:23

to the right here is another point and

00:04:26

from point x. y y so this will move this

00:04:29

all the other points will also somehow move

00:04:32

this movement is called parallel

00:04:35

transfer and we will also give it

00:04:40

some of them we will give it a name we will designate

00:04:43

it for and to specify that

00:04:47

is, as if there is a name there is a surname

00:04:50

this surname is this indicating two points from

00:04:53

which to which should be transferred and I immediately

00:04:57

want to note that if we took some

00:05:01

other two points but at exactly the

00:05:03

same distance we moved a

00:05:08

straight line from point x3 point y stroke then this is

00:05:13

exactly the same movement as it

00:05:16

means exactly the same movement,

00:05:17

it means the order that is given to each

00:05:22

of the points on the left and right is the same

00:05:24

in general when they say that the movement is the

00:05:27

same, that two movements are equal to each

00:05:30

other, which means two movements are equal to that

00:05:31

group,

00:05:32

it means that any point is given the

00:05:34

same an order to move somewhere and

00:05:37

we can write down that these two

00:05:39

transfers are actually the same

00:05:40

transfer, okay, is it possible we are looking to

00:05:44

do something with this straight line, but if we strain

00:05:47

our imagination a little, we can

00:05:50

imagine what we took the straight

00:05:53

sharpening was reflected behind this point o and

00:05:58

like this, I was reflected, that

00:06:00

is, as if turned 180 degrees,

00:06:02

as a result the straight line as a

00:06:04

single whole coincided again with the

00:06:07

position it occupied before, but

00:06:10

now

00:06:11

all the points except point o changed their

00:06:13

position, namely they through .

00:06:15

jumped over the same distance,

00:06:18

we will call such a transformation a

00:06:21

movement relative to a point,

00:06:24

well, you can. take some other place, this

00:06:28

will be a different transformation of the reflection

00:06:30

relative to another point, but something else needs to be

00:06:34

done with a straight line or there are no more

00:06:36

movements, that is, if we

00:06:38

changed the position of the straight line so that as a

00:06:41

whole it remained there, but the

00:06:44

points could change their position then

00:06:49

The question is, is it possible that the transformation

00:06:52

we made was some kind of

00:06:53

transfer and not reflection, the answer is no and

00:06:57

there is such a theorem, classification of

00:07:00

motion, any movement of a straight line is either

00:07:02

parallel translation or reflection, we

00:07:05

will prove this theorem

00:07:08

according to the previous one, the question may arise,

00:07:12

why not the straight line as a whole

00:07:15

is rotated by approximately 30 degrees or

00:07:17

90, this will also be a movement;

00:07:20

in fact, the distance between any

00:07:22

two points will be preserved with such a

00:07:24

transformation. The answer is this:

00:07:27

we are currently studying the movement of a

00:07:31

straight line that does not lie on a plane and in

00:07:33

space,

00:07:34

but the movement of a straight line in which it as well as

00:07:38

itself as a single whole remains in place,

00:07:40

that is, we are considering such

00:07:42

transformations of the line that its points can

00:07:44

change position, but at the same time, if we

00:07:47

look at it as a drawn

00:07:49

object, we will not see any changes, and

00:07:52

in this case there is really nothing

00:07:54

except translations and reflections It’s impossible to come up with

00:07:57

and you can even prove the point

00:07:59

that there is nothing and no movement is

00:08:01

interesting because you need to take their

00:08:05

composition, that is, they allow an

00:08:08

operation such as adding or multiplying

00:08:10

numbers, but only much more complex,

00:08:13

what is the composition of movements is simply

00:08:16

performing them one after another, for example

00:08:20

first they performed some kind of reflection of a

00:08:22

straight line relative to some point and

00:08:25

then a parallel translation

00:08:27

and vice versa began a parallel translation

00:08:30

and then the reflection that the result

00:08:32

was

00:08:33

our experience of communicating with numbers tells

00:08:37

us that there should be no difference,

00:08:39

that we say a + b so that plus a this is the

00:08:42

same number and also with multiplication,

00:08:44

but here a surprise awaits us, let's

00:08:48

study this surprise in more detail and so here

00:08:52

we have it drawn.

00:08:54

and we perform a reflection relative to

00:08:57

this point and then we perform a

00:09:01

parallel transfer of attention I write these

00:09:07

two operations are performed from right to left as in the

00:09:09

Hebrew my Arabic alphabet why

00:09:13

this is also a separate interesting point in

00:09:15

mathematics it will be more convenient to act this way

00:09:19

because

00:09:21

what it is these are two sequential

00:09:24

orders

00:09:26

that are given to the points we can take a straight line

00:09:30

there and substitute it into this order,

00:09:31

let’s say point x and see what happened to it

00:09:33

. Erica

00:09:35

let's generalize some other point to let's call some point

00:09:38

lying on this line beyond z and then if

00:09:43

I want to first make a reflection and then

00:09:45

a transfer, then I write that this is a transfer

00:09:51

performed on the result of the reflection of

00:09:55

point z, that is, on the new point that

00:09:57

was obtained as a result of the reflection of the point behind

00:10:00

this relative to the point o,

00:10:01

and it turns out that this composition of

00:10:04

movements is written as through brackets in the

00:10:08

same order, then here are the letters

00:10:09

indicating the transformation, here in the same

00:10:11

order it is clear that the composition of movements

00:10:14

will also be a movement, but if

00:10:17

the distances are preserved here too they

00:10:20

were preserved and as a result of these two

00:10:22

operations the distances cannot change,

00:10:24

therefore this is some kind of movement of a straight line and

00:10:28

as you and I already know, although we have not yet

00:10:32

proven any movements of a straight line except

00:10:34

translations and reflections do not exist,

00:10:37

therefore what we wrote down is this

00:10:39

composition, it is also must be either

00:10:41

figurative or reflection, let's

00:10:44

see how the points behave to

00:10:46

guess

00:10:47

what this composition is equal to, well, for example,

00:10:50

we can see where

00:10:55

it goes. and this is simpler because at the

00:11:00

beginning it does not go anywhere;

00:11:02

when reflected, it remains in place, but when

00:11:04

transferred,

00:11:05

it shifts by this vector, a

00:11:10

vector is called such segments are directed

00:11:13

from one point to another

00:11:14

by the same longer vector to the right, so

00:11:16

it has moved, here it has moved In

00:11:19

some . and their pin probably

00:11:24

gives us a hint, probably gives us a hint

00:11:27

that this transfer will probably be

00:11:29

this transformation, but before we

00:11:31

make such hasty conclusions, let’s

00:11:34

study the behavior of some other point,

00:11:38

let’s study the behavior of, for example, a dot

00:11:42

and a stroke at this point, what

00:11:45

will happen to it these here with this

00:11:48

composition in this order if there were 90

00:11:51

she went even further tied up this

00:11:53

frame but let's see what actually wo

00:11:56

n't happen with what will

00:11:58

happen when the sentence is reflected on will

00:12:00

jump here to the left exactly the

00:12:03

same distance and then with transfer,

00:12:05

it will go exactly to .

00:12:08

and so in fact, this is clearly not a transfer

00:12:12

because your 3 passed over, then they

00:12:15

passed into,

00:12:16

but if this is not a transfer, then apparently if that

00:12:19

theory is correct which we will prove,

00:12:20

this should be a reflection, the

00:12:22

question is regarding what point this

00:12:23

will be a reflection if we have it turned out

00:12:26

that the

00:12:28

o and a stroke swapped places, well, the answer

00:12:31

should be clear - this is exactly the

00:12:34

midpoint between these two points. and that

00:12:39

is, this transformation, as it

00:12:42

seems to us, will be a reflection from point a and

00:12:47

how can we calculate this point a if we know,

00:12:50

roughly speaking, the full names and surnames of

00:12:53

these two transformations, the answer is this: we

00:13:03

add a vector to the starting point relative to which the reflection passed with o on which there

00:13:07

was a transfer exactly half as much in the

00:13:10

same direction where the transfer was and

00:13:13

thus it turns out. regarding

00:13:14

which there will be a reflection, this is true, but this

00:13:16

must be strictly proven, of course, but

00:13:19

nevertheless, our guess is correct, so

00:13:22

as an exercise, I suggest that

00:13:25

listeners

00:13:26

swap these two transformations

00:13:28

and make sure that the result will be

00:13:31

different, namely, the result will also be

00:13:34

a reflection but relative to point a the stroke

00:13:38

that is halfway here, that

00:13:42

is, which is obtained by indenting to the left by

00:13:45

half the length of

00:13:46

the vector and, as mathematicians say,

00:13:48

by adding .

00:13:50

minus this vector in half,

00:13:54

so the movement may not be

00:13:56

permutations even in such a simple case

00:13:58

as when we move a straight line,

00:14:01

but nevertheless the arithmetic of the movement can

00:14:04

be constructed, we can even create

00:14:06

such a composition, a table of the composition of

00:14:09

movements where there will be reflection transfers, we will

00:14:11

write down in each case what will happen and

00:14:13

we will do this with you after we prove the

00:14:15

classification premises

Description:

Движение в физике - это процесс, в движение в математике - это результат. Рассматриваем варианты движения прямой в уроке. Курс 100 уроков математики (плейлист обновляется): https://www.youtube.com/playlist?list=PLqBfxn8OBMGrsA_YynaQWqHKhL7kEvL4X =========================== CHILDRENScience - канал некоммерческого фонда "Дети и наука". Наша цель - улучшить качество школьного образования. Для этого мы привлекаем выдающихся учителей, создаем системные курсы из видеоуроков и заданий, готовим методические материалы для преподавателей. Подпишитесь на наш канал: https://www.youtube.com/channel/UCf053FwQD-hnTY7aaaWTaVQ?sub_confirmation=1 Наш сайт - https://classes.childrenscience.ru/seven_grade =========================== Мы в социальных сетях: Facebook - https://www.facebook.com/unsupportedbrowser Instagram - https://www.facebook.com/unsupportedbrowser VKontakte - https://vk.com/childrenscience

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