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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 .
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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

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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 .

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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

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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 .

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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.

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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 .
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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.

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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

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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RAW Reply
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Damian
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Professor
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Professor
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Brian Reply
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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
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LITNING Reply
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
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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
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research.net
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Introduction about quantum dots in nanotechnology
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Loga
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Practice Key Terms 6

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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