<< Chapter < Page Chapter >> Page >
  • Describe three-dimensional space mathematically.
  • Locate points in space using coordinates.
  • Write the distance formula in three dimensions.
  • Write the equations for simple planes and spheres.
  • Perform vector operations in 3 .

Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.

Three-dimensional coordinate systems

As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x -axis and the vertical y -axis. We can add a third dimension, the z -axis, which is perpendicular to both the x -axis and the y -axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.

Definition

The three-dimensional rectangular coordinate system    consists of three perpendicular axes: the x -axis, the y -axis, and the z -axis. Because each axis is a number line representing all real numbers in , the three-dimensional system is often denoted by 3 .

In [link] (a), the positive z -axis is shown above the plane containing the x - and y -axes. The positive x -axis appears to the left and the positive y -axis is to the right. A natural question to ask is: How was arrangement determined? The system displayed follows the right-hand rule    . If we take our right hand and align the fingers with the positive x -axis, then curl the fingers so they point in the direction of the positive y -axis, our thumb points in the direction of the positive z -axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.

This figure has two images. The first is a 3-dimensional coordinate system. The x-axis is forward, the y-axis is horizontal to the left and right, and the z-axis is vertical. The second image is the 3-dimensional coordinate system axes with a right hand. The thumb is pointing towards positive z-axis, with the fingers wrapping around the z-axis from the positive x-axis to the positive y-axis.
(a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the z -axis, that is perpendicular to both the x -axis and the y -axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.

In two dimensions, we describe a point in the plane with the coordinates ( x , y ) . Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, z , is appended to indicate alignment with the z -axis: ( x , y , z ) . A point in space is identified by all three coordinates ( [link] ). To plot the point ( x , y , z ) , go x units along the x -axis, then y units in the direction of the y -axis, then z units in the direction of the z -axis.

This figure is the positive octant of the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn with broken lines. One corner is labeled (x, y, z). The height of the box is labeled “z units,” the width is labeled “x units” and the length is labeled “y units.”
To plot the point ( x , y , z ) go x units along the x -axis, then y units in the direction of the y -axis, then z units in the direction of the z -axis.

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask