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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

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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