<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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.”
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

( z 2 ) ( z 5 ) = 0

Got questions? Get instant answers now!

( 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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

z = 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

z = −2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Center C ( −4 , 7 , 2 ) and radius 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

P ( 0 , 10 , 5 ) and Q ( 1 , 1 , −3 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Q ( 5 , 2 , 8 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

a = 3 , −2 , 4 , b = −5 , 6 , −9

Got questions? Get instant answers now!

a = k , b = i

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

a = 4 , −3 , 6

Got questions? Get instant answers now!

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

2 62 , 7 62 , 3 62

Got questions? Get instant answers now!

a = O P , where P ( −1 , −1 , 1 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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