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Key equations

  • Distance between two points in space:
    d = ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 + ( z 2 z 1 ) 2
  • Sphere with center ( a , b , c ) and radius r :
    ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = r 2

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point A ( 2 , 3 , 5 ) is the opposite vertex to the origin, then find

  1. the coordinates of the other six vertices of the box and
  2. the length of the diagonal of the box determined by the vertices O and A .
This figure is the first octant of the 3-dimensional coordinate system. It has a point labeled “A(2, 3, 5)” drawn.

a. ( 2 , 0 , 5 ) , ( 2 , 0 , 0 ) , ( 2 , 3 , 0 ) , ( 0 , 3 , 0 ) , ( 0 , 3 , 5 ) , ( 0 , 0 , 5 ) ; b. 38

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Find the coordinates of point P and determine its distance to the origin.

This figure is the first octant of the 3-dimensional coordinate system. It has a point drawn at (2, 1, 1). The point is labeled “P.”
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For the following exercises, describe and graph the set of points that satisfies the given equation.

( y 5 ) ( z 6 ) = 0

A union of two planes: y = 5 (a plane parallel to the xz -plane) and z = 6 (a plane parallel to the xy -plane)
This figure is the first octant of the 3-dimensional coordinate system. It has two planes drawn. The first plane is parallel to the x y-plane and is at z = 6. The second plane is parallel to the x z-plane and is at y = 5. The planes are perpendicular.

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( z 2 ) ( z 5 ) = 0

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( y 1 ) 2 + ( z 1 ) 2 = 1

A cylinder of radius 1 centered on the line y = 1 , z = 1
This figure is the first octant of the 3-dimensional coordinate system. It has a cylinder drawn. The axis of the cylinder is parallel to the x-axis.

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

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Write the equation of the plane passing through point ( 1 , 1 , 1 ) that is parallel to the xy -plane.

z = 1

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Write the equation of the plane passing through point ( 1 , −3 , 2 ) that is parallel to the xz -plane.

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Find an equation of the plane passing through points ( 1 , −3 , −2 ) , ( 0 , 3 , −2 ) , and ( 1 , 0 , −2 ) .

z = −2

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Find an equation of the plane passing through points ( 1 , 9 , 2 ) , ( 1 , 3 , 6 ) , and ( 1 , −7 , 8 ) .

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For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions.

Center C ( −1 , 7 , 4 ) and radius 4

( x + 1 ) 2 + ( y 7 ) 2 + ( z 4 ) 2 = 16

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Center C ( −4 , 7 , 2 ) and radius 6

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Diameter P Q , where P ( −1 , 5 , 7 ) and Q ( −5 , 2 , 9 )

( x + 3 ) 2 + ( y 3.5 ) 2 + ( z 8 ) 2 = 29 4

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Diameter P Q , where P ( −16 , −3 , 9 ) and Q ( −2 , 3 , 5 )

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For the following exercises, find the center and radius of the sphere with an equation in general form that is given.

P ( 1 , 2 , 3 ) x 2 + y 2 + z 2 4 z + 3 = 0

Center C ( 0 , 0 , 2 ) and radius 1

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x 2 + y 2 + z 2 6 x + 8 y 10 z + 25 = 0

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For the following exercises, express vector P Q with the initial point at P and the terminal point at Q

  1. in component form and
  2. by using standard unit vectors.

P ( 3 , 0 , 2 ) and Q ( −1 , −1 , 4 )

a. P Q = −4 , −1 , 2 ; b. P Q = −4 i j + 2 k

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P ( 0 , 10 , 5 ) and Q ( 1 , 1 , −3 )

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P ( −2 , 5 , −8 ) and M ( 1 , −7 , 4 ) , where M is the midpoint of the line segment P Q

a. P Q = 6 , −24 , 24 ; b. P Q = 6 i 24 j + 24 k

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Q ( 0 , 7 , −6 ) and M ( −1 , 3 , 2 ) , where M is the midpoint of the line segment P Q

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Find terminal point Q of vector P Q = 7 , −1 , 3 with the initial point at P ( −2 , 3 , 5 ) .

Q ( 5 , 2 , 8 )

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Find initial point P of vector P Q = −9 , 1 , 2 with the terminal point at Q ( 10 , 0 , −1 ) .

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For the following exercises, use the given vectors a and b to find and express the vectors a + b , 4 a , and −5 a + 3 b in component form.

a = −1 , −2 , 4 , b = −5 , 6 , −7

a + b = −6 , 4 , −3 , 4 a = −4 , −8 , 16 , −5 a + 3 b = −10 , 28 , −41

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a = 3 , −2 , 4 , b = −5 , 6 , −9

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a = k , b = i

a + b = −1 , 0 , −1 , 4 a = 0 , 0 , −4 , −5 a + 3 b = −3 , 0 , 5

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a = i + j + k , b = 2 i 3 j + 2 k

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For the following exercises, vectors u and v are given. Find the magnitudes of vectors u v and −2 u .

u = 2 i + 3 j + 4 k , v = i + 5 j k

u v = 38 , −2 u = 2 29

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u = 2 cos t , −2 sin t , 3 , v = 0 , 0 , 3 , where t is a real number.

u v = 2 , −2 u = 2 13

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u = 0 , 1 , sinh t , v = 1 , 1 , 0 , where t is a real number.

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For the following exercises, find the unit vector in the direction of the given vector a and express it using standard unit vectors.

a = 3 i 4 j

a = 3 5 i 4 5 j

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a = 4 , −3 , 6

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a = P Q , where P ( −2 , 3 , 1 ) and Q ( 0 , −4 , 4 )

2 62 , 7 62 , 3 62

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a = O P , where P ( −1 , −1 , 1 )

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a = u v + w , where u = i j k , v = 2 i j + k , and w = i + j + 3 k

2 6 , 1 6 , 1 6

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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
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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
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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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