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This module covers mathematical distance in preparation for later modules on conic sections.

The key mathematical formula for discussing all the shapes above is the distance between two points.

Many students are taught, at some point, the “distance formula” as a magic (and very strange-looking) rule. In fact, the distance formula comes directly from a bit of intuition...and the Pythagorean Theorem.

The intuition comes in finding the distance between two points that have one coordinate in common.

The distance between two points that have one coordinate in common

The drawing shows the points (2,3) and (6,3). Finding the distance between these points is easy: just count! Take your pen and move it along the paper, starting at (2,3) and moving to the right. Let’s see…one unit gets you over to (3,3); the next unit gets you to (4,3)...a couple more units to (6,3). The distance from (2,3) to (6,3) is 4.

Of course, it would be tedious to count our way from (2,3) to (100,3). But we don’t have to—in fact, you may have already guessed the faster way—we subtract the x coordinates.

  • The distance from (2,3) to (6,3) is 6 - 2 = 4
  • The distance from (2,3) to (100,3) is 100 - 2 = 98

And so on. We can write this generalization in words:

Whenever two points lie on a horizontal line, you can find the distance between them by subtracting their x -coordinates.

This may seem pretty obvious in the examples given above. It’s a little less obvious, but still true, if one of the x coordinates is negative.

The drawing above shows the numbers (-3,1) and (2,1). You can see that the distance between them is 5 (again, by counting). Does our generalization still work? Yes it does, because subtracting a negative number is the same as adding a positive one.

The distance from (-3,1) to (2,1) is 2 - ( -3 ) = 5

How can we express this generalization mathematically? If two points lie on a horizontal line, they have two different x-coordinates: call them x 1 and x 2 . But they have the same y-coordinate, so just call that y. So we can rewrite our generalization like this: “the distance between the points ( x 1 , y ) and ( x 2 , y ) is x 2 x 1 .” In our most recent example, x 1 = –3 , x 2 = 2 , and y = 1 . So the generalization says “the distance between the points (-3,1) and (2,1) is 2 - ( -3 ) ”, or 5.

But there’s one problem left: what if we had chosen x 2 and x 1 the other way? Then the generalization would say “the distance between the points (2,1) and (-3,1) is ( –3 ) -2 ”, or -5. That isn’t quite right: distances can never be negative. We get around this problem by taking the absolute value of the answer. This guarantees that, no matter what order the points are listed in, the distance will come out positive. So now we are ready for the correct mathematical generalization:

Distance between two points on a horizontal line

The distance between the points ( x 1 , y ) and ( x 2 , y ) is | x 2 x 1 |

You may want to check this generalization with a few specific examples—try both negative and positive values of x 1 and x 2 . Then, to really test your understanding, write and test a similar generalization for two points that lie on a vertical line together. Both of these results will be needed for the more general case below.

The distance between two points that have no coordinate in common

So, what if two points have both coordinates different? As an example, consider the distance from (–2,5) to (1,3).

The drawing shows these two points. The (diagonal) line between them has been labeled d : it is this line that we want the length of, since this line represents the distance between our two points.

The drawing also introduces a third point into the picture, the point (–2,3). The three points define the vertices of a right triangle. Based on our earlier discussion, you can see that the vertical line in this triangle is length | 5 3 | = 2 . The horizontal line is length | 1 ( –2 ) | = 3 .

But it is the diagonal line that we want. And we can find that by using the Pythagorean Theorem, which tells us that d 2 = 2 2 + 3 2 . So d = 13

If you repeat this process with the generic points ( x 1 , y 1 ) and ( x 2 , y 2 ) you arrive at the distance formula:

Distance between any two points

If d is the distance between the points ( x 1 , y 1 ) and ( x 2 , y 1 ), then d 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

x 2 x 1 is the horizontal distance, based on our earlier calculation. y 2 y 1 is the vertical distance, and the entire formula is simply the Pythagorean Theorem restated in terms of coordinates.

And what about those absolute values we had to put in before? They were used to avoid negative distances. Since the distances in the above formulae are being squared , we no longer need the absolute values to insure that all answers will come out positive.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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