Right in front of the open space it is possible to make a difference as a line overflow of two non-parallel areas. Rivnyannya of one area, rivnyannya of another area, so rivnyannya is straightforward to ask

here non-collinear
... Tsi rivnyannya are called out-of-town people straight by the open space.

Canonical straightforward

Be it a non-zero vector, whether to lie on a straight line or parallel to it, to be called a straight line vector.

Yaksho vіdoma point
Straight Ta її Straight Vector
, then the canonical rivnyannya straight forward may look like:

. (9)

Parametric straightforward

Let me set the canonical straightforward

.

We will recognize the parametric straightforwardness:

(10)

Tsі rіvnyannya zruchnі at znakhozhennі points overflow straight that area.

Rivnyannya straight, go through two points
і
maє viglyad:

.

Kut mіzh straight

Kut mіzh straight

і

road kutu with direct vectors. Otzhe, this can be calculated by the formula (4):

Mind the parallelism straight:

.

Mind and perpendicularity of areas:

Go straight ahead

NS the mouth is given a point
that is straight

.

From the canonical rivnyans straight to the point
, which should be straight, that is a straight vector
... Todi vidstan points
from the direct path of the parallelogram induced on the vectors і
... Otzhe,

.

Umova peretinu straight

Two non-parallel straight

,

change todi and lose todi, if

.

Instead, there is a straight square.

Let the set be straight
that area. Coot among them you can know for the formula

.

Zavdannya 73. Write canonical rivnyannya straight

(11)

Decision... In order to write down the canonical straight line (9), it is necessary for the nobility to be a point to lay straight, and a straight vector straight.

We know vector , parallel straight line. Oskіlki vіn maє but are perpendicular to normal vectors in given areas, tobto.

,
, then

.

From the out-of-the-way rіvnyany straight maєmo, scho
,
... Todi

.

So yak point
be it a straight point, then the coordinates are guilty to the satisfaction of the straight line and one of them can be given, for example,
, two coordinates are known from system (11):

Zvidsi,
.

With such a rank, the canonical rіvnyannya shukanoi straight forward may viglyad:

abo
.

Zavdannya 74.

і
.

Decision. From the canonical rivnyans of the first direct view of the coordinates of the point
, which should be straight, the coordinates of the straight vector
... The canonical values ​​of other straight lines also show the coordinates of the point
the coordinates of the straight vector
.

Come out of parallel straight roads to points
through another straight line. Qia come to be counted for the formula

.

We know the coordinates of the vector
.

Numerous vector tvir
:

.

Zavdannya 75. Know the point symmetrical point
straight forward

.

Decision... Recordable area perpendicular to the center straight and pass through the point ... Yak її vector normal You can take a straight vector. Todi
... Otzhe,

We know the point
the point of cross-flow of a given straight line P.

Otzhe,
.

Come on
point is symmetric to point
schodo tsієї straight. Todi point
mid-way
... To know the coordinates of a point victorious formulas for the coordinates of the middle of the display:

,
,
.

Otzhe,
.

Zavdannya 76. Write a flat square that goes through a straight line
і

a) through the point
;

b) perpendicular to the area.

Decision. Writable zagalny rivnyannya given directly. For a wide range of doors:

Tse oznachaє scho ploschina scho shukaєtsya, nalezhit beam ploschin of utvoryuyuchimi i її rіvnyannya Mauger Buti recorded at viglyadі (8):

a) We know
і Wash, where the plane is to pass through the point
Otzhe, її coordinates guilty to the satisfaction of the rural area. Acceptable point coordinates
in the flat bunch of areas:

Know the meaning
presented for the іvnyannya (12). otrimaєmo rіvnyannya shukanoy area:

b) We know
і Wash, where the area, how to rustle, is perpendicular to the area. Normal vector of a given area
, The normal vector of the shukano area (div. Equal to the bundle of areas (12).

Two vectors are perpendicular to each other, if the scalar addition is zero. Otzhe,

Apparently know the value
flat bunch of areas (12). Otrimaєmo rіvnyannya shukanoy area:

Zavdannya for independent revision

Zavdannya 77. Bring to the canonical form of the ryvnyannya straight:

1)
2)

Zavdannya 78. Write parametric straightforward
, when:

1)
,
; 2)
,
.

Zavdannya 79... Write a flat area to pass through the point
perpendicular to straight

Zavdannya 80. Write a straight line to pass the point
perpendicular to the area.

Zavdannya 81. Know kut mіzh straight:

1)
і
;

2)
і

Zavdannya 82. Bring parallelism straight:

і
.

Zavdannya 83. Bring perpendicularity straight:

і

Zavdannya 84. Count the points
straight forward:

1)
; 2)
.

Zavdannya 85. Calculate the number of parallel straight lines:

і
.

Zavdannya 86... At rivnyannyakh straight
by parameter so, it’s right, it changed from the straight line and to know the point of the crossover.

Zavdannya 87... Show you straight
parallel to the area
and straight
lie near the square.

Zavdannya 88... Know the point symmetrical point Shodo area
, when:

1)
, ;

2)
, ;.

Zavdannya 89. Write the line of the perpendicular dropped from the point
on the straight
.

Zavdannya 90... Know the point symmetrical point
straight forward
.

NASA has launched an expedition to Mars in 2020. To deliver the spacecraft to Mars an electronic device for the names of all registered participants of the expedition.


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One of the number of options in the code needs to be copied and pasted into the code of your web site, bazhano with tags і if not directly from the tag ... For the first version of MathJax, you will get a better and less gallant side. Other options are automatically provided and updated with the new version of MathJax. Just insert the first code, it will need to be updated from time to time. If you insert a different code, then the pages will be blocked more, but you will not need to constantly follow the MathJax updates.

Add MathJax to Blogger, or WordPress: in the site's keruvannya panel, add a widget, signs for inserting third-party JavaScript code, copy it into the first one, or another version of the wrapped code presented to it, 'language-wise, the MathJax script gets bridged asynchronously). From and everything. Now grab the syntax of the layout MathML, LaTeX and ASCIIMathML, and you are ready to embed mathematical formulas on the web pages of your site.

Chergovy before the day of the New Rock ... the weather is frosty and the little girls on the shibts ... All spontaneously I didn’t know to write about ... fractals, and about those who know about Wolfram Alpha. To the third drive є tsіkava stattyа, in yakіy є butt double-sided fractal structures. Immediately we see more folding butt trivial fractals.

A fractal can be visualized (describe), like a geometric figure, like a geometric figure, like it’s like a figure. So it’s a self-similar structure, it’s easy to see the details of what’s in the case of an increase, but also the very shape, but without an increase. Todi yak u vipadku zvychaynoi geometric figuri(not a fractal), with larger details, which may be larger simple form not the figure itself. For example, when you reach the great great part of the ellipse, it looks like a straight line. It’s not possible to see this with fractals: for whatever it is, I’ll know the same fold, as with skin changes, I know and know again.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractal that mystery in the world of science: "Fractals are not geometric forms, which are, however, foldable in their details, as in their other parts. will be improved to the size of the whole, look out, like a whole, or just, or, perhaps, with a slight deformation. "

Oh-oh-oh-oh-oh ... well, hard, why did you read it to you yourself =) Well, because of relaxation, you can also buy more accessories. This can be done until the first break, I am encouraged, until the end of the statute, I will save the badges of the mood.

Roztashuvannya two straight again

That vipadok, if the hall goes in chorus. The two are straight:

1) scammed;

2) but are parallel:;

3) chi peretinatisya at the single point:.

Dovіdka for teapots : be a weasel, forget the mathematical sign you will see it more often. The record means that it is a straight line to be interlaced with a straight line at a point.

Yak visnichiti in exchange for roztashuvannya two straight?

For some reason from the first vypad:

There are two straight lines, only if they do not, if there are certain factors in proportion Tobto іsnu such a bit of "lambda",

It is easy to see straight and foldably three prices from the following factors:. From the skin viplivania, from the same date it is straightforward to start.

Dyysno, as well as all the features multiply by -1 (change signs), and all features speed by 2, then see the same rivnyannya:.

Another vipadok, if straight parallel:

Two straight parallel ones, or if they are not, if their performance is different with changing proportions: , ale.

Yak butt is visible two straight. The overwhelming proportion of the current performance during the winter:

However, it is quite obvious that.

The І third type of admission, if it changes directly:

The two are straight, sometimes it is not enough, if the performance is not proportional to the change. so DON'T have such a meaning "lambda"

So, for a straight line, the system can be stored:

From the first rivnyannya viplyaє, scho, and from another іvnyannya: the system is insane(Rishen is dumb). In such a rank, the performance for the change is not proportional.

Visnovok: straight overflow

Practical workers can have a quick look at the solution scheme. Vaughn, before the speech, the algorithm for converting vectors to collinearity, which they looked at at the level, was completely wrong. The concept of linear (non) depletion vectors. The basis of vectors... Ale іsnu bolsh civilized packaging:

Butt 1

Z'yasuvati instead of roztashuvannya straight:

Decision Snapshot on recent direct vectors in straightforward:

a) We know it directly from the vectors straight: .


Otzhe, the vectors are not collinear and straightforward.

I will put all kinds of vipadocs on the list for showers:

Інші rebuild the stone and go away, right to Kashі Bezsmertny =)

b) We know directly vectors straight:

Straight may be the most direct vector, the stench is either parallel, or zbigayutsya. There is no need for a rahuvati formatter.

Evidently, it’s the efficiency in the case of unavailable proportions, in its own.

Z'yasuєmo, what is fairness:

In such a rank,

c) We know directly vectors straight:

Numericable designator, addition of coordinates of given vectors:
Also, direct vectors are collinear. Straight or parallel or start.

The ratio of the proportionality of the "lambda" does not really matter directly from the direction of linear direct vectors. However, you can know it through the performance of the peasants themselves: .

Now z'yasuєmo, where the parity is fair. Offense to a member of zero, to that:

Otrymane meaning is satisfied with the given rivnyannyu (you are happy with the number).

Otzhe, straight away.

View:

You will soon see (or else you have already) see you asleep in just a few seconds. The connection with the cym does not bach the senseu, it is proponated for an independent solution, more importantly, we put one important point in the geometric foundation:

How can I stay direct, parallel to the tribute?

For ignorance of the most simple staff of the suvoro karak Nightingale-Rozbiynik.

Butt 2

The straight line is given to the equal. The flanks are parallel straight, yak pass through the point.

Decision: Meaningful to an unidentified direct letter. What about her said for the mind? Pass straight through the speck. And if it is straight parallel, it is obvious that the direct vector is straight forward and to induce directness.

Vityaguєmo tension vector із рівняння:

View:

Geometry to the butt of the eye

Analytical reversal of the field at the onset of the crooks:

1) Reversion, but the straight vector has a very direct vector (if the straight line is not forgiven by the proper rank, then the vector will be collinear).

2) Reversion, where the point is satisfied with the rejected rivnyannyu.

An analytic rewrite of the child is easy to fall asleep. To marvel at two equal numbers, and it’s quite a lot for you to find parallelism straight without any kind of armchair.

Apply for a self-directed version of the year to be creative. So you will have to compete with Baba-Yagoi, and you know, she's a lover of all kinds of riddles.

Butt 3

The flatness is straight, go through a point parallel to the straight, where

A simple rational and even more rational method of solution. Nykorotshy shlyakh - for a lesson.

Trochus stroked in parallel straight lines and turned to them. Vipadok straight, scho to go, little tsy, that is understandable, as well as you know well from the school programs:

Yak know the point to cross two straight lines?

I'm straight change in points, coordinates and solutions systems of linework

Yak know the point to cross straight? Virishiti system.

Axis to you i geometric zm_st system and two lіnіynykh rіvnyany two houses are unavailable- the price of doors to overturn (most often) straight on the area.

Butt 4

Know the point of overflow straight

Decision: There are two ways of solving - graphical and analytical

The graphical way of the field is in the fact that it is easy to cross the line and learn about the point of crossing without the front from the chair:

The axis is our point:. To reconcile the slid of presenting the coordinates at the skin straight, it stinks to go there, and there. In addition, it seems, the coordinates of the point є to the solution of the system. By the day we looked at a graphical method of solution systems of linework from two families, two families unavailable.

Graphic way, vividly, non-falsification, ale є pomitnі minusi. Ні, on the right is not in the way that the seventh-graders go so far, on the right in that, in the right and the exact armchair before the hour. Besides, it’s not so easy to do straight things, that very point of overflow can be found here in the thirtieth kingdom behind the borders of the zoshite leaf.

To that point, the point is repetitively shukati using the analytical method. Virishimo system:

For the revision of the system, the method of term-by-term folding of rivnyans is used. Schob napratsyuvati show you a lesson Yak to connect the system of rivnyans?

View:

The override is trivial - the coordinates of the point overflow can satisfy the skin alignment of the system.

Butt 5

Know the point of overflowing straight at times, when the stench overflows.

Tse butt of an independent solution. Zavdannya hand-crafted on a spike of steps. Analysis must be given if necessary:
1) Right side of the floor.
2) Right side of the flat.
3) Z'yasuvati instead of roztashuvannya straight.
4) If it is straight to overflow, then to know the overflow point.

Distribution of the algorithm diy typical for bagatyokh geometric buildings, and I will respect you more than once.

Outside of the decision, this is similar to the lesson:

Not yet worn out and a couple of chereviks, as they went to another section of the lesson:

Perpendicular to straight. Go from the point to the straight line.
Kut mіzh straight

For some reason, it’s the same important care... At the first part, we knew how to stay direct, parallel to the tribute, and at the same time the hatinka on the chicken nizhni flared up by 90 degrees:

Yak to stay straight, perpendicular to the tribute?

Butt 6

The straight line is given to the equal. The flanks are perpendicular to the straight line, passing through the point.

Decision: For a wash I see, scho. It is impossible to know the direct vector of straight lines. Oscillations are straight perpendicular, focus of prostitution:

Rivnyannya "know" is the normal vector:, which will be straightened by the vector of straight lines.

Equivalent straightforward by point and straight vector:

View:

Rosgornemo Geometric Study:

Hmmm ... Orange sky, orange sea, orange camel.

Analytical reconversion of solution:

1) Z rivnyan vityaguєmo direct vector the one for help scalar vectors we come to visnovka, scho straight to right perpendicular:.

Before the speech, you can vikoristovuvati vectors normal, tse simpler.

2) Perevryaєmo, chi is the point of the renounced rivnyannya .

Perevirku, I know, it’s easy to go to sleep.

Butt 7

Know the point of cross-flow of perpendicular straight lines, when vidomo rivnyannya and speck.

Tse butt of an independent solution. At the factory, the decision is made by hand for the points.

Our hustle and bustle will go up:

Go from point to straight

Before us is the swarth of the little river and our zavdannya polyagaє in that, to go to her with the shortest way. There is no crossing, and the most optimal route will be perpendicular. To go from a point to a straight line - the chain of a perpendicular direction.

Look at the geometriya traditionally means the walnut letter "ro", for example: - Look from the point "em" to the straight "de".

Go from point to straight swing the formula

Butt 8

Know from a point to a straight line

Decision: everything that is required, carefully put the numbers in the formula and carry out the calculation:

View:

Viconaєmo chair:

It is known from the point to a straight line - the price is exactly the same as the end of a heart. Yaksho arrange an armchair on cartoony papir at scale 1 od. = 1 cm (2 cells), then you can see it in a snake line.

Easy to read from the same chair:

Zavdannya polyagaє in order to know the coordinates of the point, which is symmetrical to the point, the point is straight ... I will propose the viconati on my own, I will propose the algorithm for the solution based on the preliminary results:

1) It is known to be straight, the yak is perpendicular to straight.

2) We know the point of overflow straight: .

Obidvі diї reports rosіbranі in the framework of the whole lesson.

3) Dotted є in the middle of the edge. We see the coordinates of the middle of one of the kints. per formulas for the coordinates of the middle known.

We will not change it, but we will also be available for 2.2 units.

Problems here can be found in the numbers, but in the vezha there is a miraculous micro-calculator, which allows you to use simple fractions. Having gladdened me several times, I will gladden and know.

How do you know if there are two parallel straight lines?

Butt 9

Know how to stand between two parallel straight lines

There is one buttstock for the independent solution. Crumbs for help: there are no ways to show up. Pick a favor for a lesson, or try to help yourself more beautifully, I guess, your grace is far beyond the reach of confusion.

Kut mіzh two straight

Шо ні кут, then the joint:


At the geometriya, for the cut, there are two straight lines, the LESSER cut, which is automatically dispensed, but we cannot be stupid. On a little kut, meanings like a red arc, do not get into kut mіzh straight, scho overturn. And to get involved in such a yogo "green" susid chi protesting arrangements"raspberry" kut.

As long as it is straight perpendicular, then for the kut between them it is possible to take whatever from 4 kutiv.

Why do you get kuti? orієntаtsієyu. In a way, we are important in principle є straight "scrolling" kuta. In a different way, negatively orating kut is recorded with a minus sign, for example, yaksho.

What do I want you to do? It is possible to get along well for the wicked understanding of the kuta. On the right, in the fact that the formulas, behind what they know, are kuti, you can easily get a negative result, and it’s not guilty to catch you knowing. Coot with the sign "minus" is not a gigantic and completely concrete geometric zmist. On a chair for a negative kuta, it is obov'yazkovo vkazuvati with a strut of yogo orinntation (for a year of strilkoyu).

Yak know kut mіzh two straight? Check two working formulas:

Butt 10

Know kut mіzh straight

Decisionі Sposib first

Two straight lines, set by the girls at the outward-looking view:

I'm straight not perpendicular, then organization kut between them can be calculated according to the additional formula:

I will pay respect to the beast for the banner - tse for sure scalar tvir straight vectors:

Yaksho, then the denominator of the formula will be reset to zero, and the vector will be orthogonal and straight perpendicular. The very fact is a bit shy about the non-perpendicularity of the formulas.

Vyhodyachi from the foregoing, the solution is manually issued in two crocs:

1) A numerically scalar addition of direct vectors in straight lines:
, Otzhe, straight not perpendicular.

2) Kut mіzh straight lines are known for the formula:

It is easy to know the kut itself for the auxiliary zorotnoy function. In tsyom vikoristovumo unpaired arctangent (div. Graphs and power of elementary functions):

View:

In general, the value is probably more accurate, as well as the approximation (bazhano in degrees, and in radians), calculated for an additional calculator.

Well, minus, then minus, nothing terrible. The axis is geometric illustration:

It is not surprising that kut viyshov had a negative oration, and even for the first number of the first number, the kuta respected itself from her.

I even want to reject positive kut even more, it is necessary to remember it in straight mice, so that it’s worth taking from another rivnyannya , and the efficiency of taking from the first rivnyannya. Shorter, it seems, you need to look straight .