<< Chapter < Page Chapter >> Page >

Definition

A sphere    is the set of all points in space equidistant from a fixed point, the center of the sphere ( [link] ), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the radius .

This image is a sphere. It has center at (a, b, c) and has a radius represented with a broken line from the center point (a, b, c) to the edge of the sphere at (x, y, z). The radius is labeled “r.”
Each point ( x , y , z ) on the surface of a sphere is r units away from the center ( a , b , c ) .

The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.

Rule: equation of a sphere

The sphere with center ( a , b , c ) and radius r can be represented by the equation

( x a ) 2 + ( y b ) 2 + ( z c ) 2 = r 2 .

This equation is known as the standard equation of a sphere    .

Finding an equation of a sphere

Find the standard equation of the sphere with center ( 10 , 7 , 4 ) and point ( −1 , 3 , −2 ) , as shown in [link] .

This figure is a sphere centered on the point (10, 7, 4) of a 3-dimensional coordinate system. It has radius equal to the square root of 173 and passes through the point (-1, 3, -2).
The sphere centered at ( 10 , 7 , 4 ) containing point ( −1 , 3 , −2 ) .

Use the distance formula to find the radius r of the sphere:

r = ( −1 10 ) 2 + ( 3 7 ) 2 + ( −2 4 ) 2 = ( −11 ) 2 + ( −4 ) 2 + ( −6 ) 2 = 173 .

The standard equation of the sphere is

( x 10 ) 2 + ( y 7 ) 2 + ( z 4 ) 2 = 173 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the standard equation of the sphere with center ( −2 , 4 , −5 ) containing point ( 4 , 4 , −1 ) .

( x + 2 ) 2 + ( y 4 ) 2 + ( z + 5 ) 2 = 52

Got questions? Get instant answers now!

Finding the equation of a sphere

Let P = ( −5 , 2 , 3 ) and Q = ( 3 , 4 , −1 ) , and suppose line segment P Q forms the diameter of a sphere ( [link] ). Find the equation of the sphere.

This figure is the 3-dimensional coordinate system. There are two points labeled. The first point is P = (-5, 2, 3). The second point is Q = (3, 4, -1). There is a line segment drawn between the two points.
Line segment P Q .

Since P Q is a diameter of the sphere, we know the center of the sphere is the midpoint of P Q . Then,

C = ( −5 + 3 2 , 2 + 4 2 , 3 + ( −1 ) 2 ) = ( −1 , 3 , 1 ) .

Furthermore, we know the radius of the sphere is half the length of the diameter. This gives

r = 1 2 ( −5 3 ) 2 + ( 2 4 ) 2 + ( 3 ( −1 ) ) 2 = 1 2 64 + 4 + 16 = 21 .

Then, the equation of the sphere is ( x + 1 ) 2 + ( y 3 ) 2 + ( z 1 ) 2 = 21 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the equation of the sphere with diameter P Q , where P = ( 2 , −1 , −3 ) and Q = ( −2 , 5 , −1 ) .

x 2 + ( y 2 ) 2 + ( z + 2 ) 2 = 14

Got questions? Get instant answers now!

Graphing other equations in three dimensions

Describe the set of points that satisfies ( x 4 ) ( z 2 ) = 0 , and graph the set.

We must have either x 4 = 0 or z 2 = 0 , so the set of points forms the two planes x = 4 and z = 2 ( [link] ).

This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x y-plane. The second is the y z-plane. They are perpendicular to each other.
The set of points satisfying ( x 4 ) ( z 2 ) = 0 forms the two planes x = 4 and z = 2 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Describe the set of points that satisfies ( y + 2 ) ( z 3 ) = 0 , and graph the set.

The set of points forms the two planes y = −2 and z = 3 .
This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.

Got questions? Get instant answers now!

Graphing other equations in three dimensions

Describe the set of points in three-dimensional space that satisfies ( x 2 ) 2 + ( y 1 ) 2 = 4 , and graph the set.

The x - and y -coordinates form a circle in the xy -plane of radius 2 , centered at ( 2 , 1 ) . Since there is no restriction on the z -coordinate, the three-dimensional result is a circular cylinder of radius 2 centered on the line with x = 2 and y = 1 . The cylinder extends indefinitely in the z -direction ( [link] ).

This figure is the 3-dimensional coordinate system. It has a vertical cylinder parallel to the z-axis and centered around line parallel to the z-axis with x = 2 and y = 1.
The set of points satisfying ( x 2 ) 2 + ( y 1 ) 2 = 4 . This is a cylinder of radius 2 centered on the line with x = 2 and y = 1 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Describe the set of points in three dimensional space that satisfies x 2 + ( z 2 ) 2 = 16 , and graph the surface.

A cylinder of radius 4 centered on the line with x = 0 and z = 2 .
This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.

Got questions? Get instant answers now!

Working with vectors in ℝ 3

Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.

Questions & Answers

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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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
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