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Analitiese meetkunde, ook bekend as koördinaatmeetkunde en vroëer bekend as Cartesiese meetkunde,is die studie van meetkunde op grond van die beginsels van algebra en die Cartesiese koördinaatstelsel. Dit is gemoeid metdie definisie van meetkundige figure op 'n numeriese wyse en onttrek numeriese inlligting uit die voorstelling. Sommige beskoudie ontwikkeling van analitiese meetkunde as die begin van moderne wiskunde.

Afstand tussen twee punte

As ons die koördinate van die hoekpunte van 'n figuur het, dan kan ons die figuur op die Cartesiese vlak teken. Byvoorbeeld, neem die vierhoek ABCD met koördinate A(1,1), B(1,3), C(3,3) en D(1,3) en stel dit voor op die Cartesiese vlak. Dit word getoon in [link] .

Vierhoek ABCD voorgestel op die Cartesiese vlak

Om enige figuur voor te stel op die Cartesiese vlak, plaas ons 'n punt by elke gegewe koördinaat en verbind dan hierdie punte met reguitlyne. Een belangrike saak om op te let, is in die benoeming van die figuur. In bostaande voorbeeld, het ons die vierhoek ABCD genoem. Dit dui vir ons aan dat ons beweeg van punt A, na punt B, na punt C, na punt D en dan weer terug na punt A. Dus, wanneer jy gevra word om 'n figuur op die Cartesiese vlak te teken, moet jy hierdie benamingswyse gebruik. Soms word net sekere punte gegee en dan moet ons die ander punte vind deur gebruik te maak van die metodes wat ons verder in die hierdie hoofstuk gaan bespreek.

Afstand tussen twee punte

Een van die eenvoudigste dinge wat met analitiese meetkunde bereken kan word, is die afstand tussen twee punte. Afstand is a getal wat beskryf hoe ver twee punte van mekaar is. Byvoorbeeld, punt P het ( 2 , 1 ) as koördinate en punt Q het ( - 2 , - 2 ) as koördinate. Hoe ver is die punte P en Q van mekaar? In die figuur beteken dit, hoe lank is die stippellyn?

In die figuur kan gesien word dat lyn P R 3 eenhede lank is en lyn Q R 4 eenhede. P Q R het 'n regte hoek R . Dus kan die lengte van sy P Q bereken word deur Stelling van Pythagoras te gebruik:

P Q 2 = P R 2 + Q R 2 P Q 2 = 3 2 + 4 2 P Q = 3 2 + 4 2 = 5

Die lengte van P Q is gelyk aan die afstand tussen punte P en Q .

As 'n veralgemening van die idee, neem aan dat A enige punt is met ( x 1 ; y 1 ) as koördinate en B is enige ander punt met ( x 2 ; y 2 ) as koördinate.

Die formule vir die berekening van die afstand tussen twee punte word as volg afgelei. Die afstand tussen twee punte A en B is die lengte van die lyn A B . Volgens die Stelling van Pythagoras, word die lengte van A B gegee deur:

A B = A C 2 + B C 2

Ons sien

B C = y 2 - y 1 A C = x 2 - x 1

Dan is

A B = A C 2 + B C 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2

Gevolglik, vir enige twee punte, ( x 1 ; y 1 ) en ( x 2 ; y 2 ) , is die formule:

Afstand= ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2

Deur die formule te gebruik, word die afstand tussen twee punte P en Q met koördinate (2;1) en (-2;-2) as volg bereken. Gestel die koördinate van punt P is ( x 1 ; y 1 ) en die koördinate van punt Q is ( x 2 ; y 2 ) . Dan is die afstand:

Afstand = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 2 - ( - 2 ) ) 2 + ( 1 - ( - 2 ) ) 2 = ( 2 + 2 ) 2 + ( 1 + 2 ) 2 = 16 + 9 = 25 = 5

Khan akademie video oor die afstandformule

Khan akademie video oor die afstandformule

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
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Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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