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Consider points A ( α , 0 , 0 ) , B ( 0 , β , 0 ) , and C ( 0 , 0 , γ ) , with α , β , and γ positive real numbers.

  1. Determine the volume of the parallelepiped with adjacent sides O A , O B , and O C .
  2. Find the volume of the tetrahedron with vertices O , A , B , and C . ( Hint : The volume of the tetrahedron is 1 / 6 of the volume of the parallelepiped.)
  3. Find the distance from the origin to the plane determined by A , B , and C . Sketch the parallelepiped and tetrahedron.
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Let u , v , and w be three-dimensional vectors and c be a real number. Prove the following properties of the cross product.

  1. u × u = 0
  2. u × ( v + w ) = ( u × v ) + ( u × w )
  3. c ( u × v ) = ( c u ) × v = u × ( c v )
  4. u · ( u × v ) = 0
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Show that vectors u = 1 , 0 , −8 , v = 0 , 1 , 6 , and w = −1 , 9 , 3 satisfy the following properties of the cross product.

  1. u × u = 0
  2. u × ( v + w ) = ( u × v ) + ( u × w )
  3. c ( u × v ) = ( c u ) × v = u × ( c v )
  4. u · ( u × v ) = 0
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Nonzero vectors u , v , and w are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers α and β such that w = α u + β v . Otherwise, the vectors are called linearly independent . Show that u , v , and w are coplanar if and only if they are linear dependent.

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Consider vectors u = 1 , 4 , −7 , v = 2 , −1 , 4 , w = 0 , −9 , 18 , and p = 0 , −9 , 17 .

  1. Show that u , v , and w are coplanar by using their triple scalar product
  2. Show that u , v , and w are coplanar, using the definition that there exist two nonzero real numbers α and β such that w = α u + β v .
  3. Show that u , v , and p are linearly independent—that is, none of the vectors is a linear combination of the other two.
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Consider points A ( 0 , 0 , 2 ) , B ( 1 , 0 , 2 ) , C ( 1 , 1 , 2 ) , and D ( 0 , 1 , 2 ) . Are vectors A B , A C , and A D linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Yes, A D = α A B + β A C , where α = −1 and β = 1 .

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Show that vectors i + j , i j , and i + j + k are linearly independent—that is, there exist two nonzero real numbers α and β such that i + j + k = α ( i + j ) + β ( i j ) .

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Let u = u 1 , u 2 and v = v 1 , v 2 be two-dimensional vectors. The cross product of vectors u and v is not defined. However, if the vectors are regarded as the three-dimensional vectors u ˜ = u 1 , u 2 , 0 and v ˜ = v 1 , v 2 , 0 , respectively, then, in this case, we can define the cross product of u ˜ and v ˜ . In particular, in determinant notation, the cross product of u ˜ and v ˜ is given by

u ˜ × v ˜ = | i j k u 1 u 2 0 v 1 v 2 0 | .

Use this result to compute ( i cos θ + j sin θ ) × ( i s i n θ j c o s θ ) , where θ is a real number.

k

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Consider points P ( 2 , 1 ) , Q ( 4 , 2 ) , and R ( 1 , 2 ) .

  1. Find the area of triangle P , Q , and R .
  2. Determine the distance from point R to the line passing through P and Q .
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Determine a vector of magnitude 10 perpendicular to the plane passing through the x -axis and point P ( 1 , 2 , 4 ) .

0 , ± 4 5 , 2 5

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Determine a unit vector perpendicular to the plane passing through the z -axis and point A ( 3 , 1 , −2 ) .

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Consider u and v two three-dimensional vectors. If the magnitude of the cross product vector u × v is k times larger than the magnitude of vector u , show that the magnitude of v is greater than or equal to k , where k is a natural number.

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[T] Assume that the magnitudes of two nonzero vectors u and v are known. The function f ( θ ) = u v sin θ defines the magnitude of the cross product vector u × v , where θ [ 0 , π ] is the angle between u and v .

  1. Graph the function f .
  2. Find the absolute minimum and maximum of function f . Interpret the results.
  3. If u = 5 and v = 2 , find the angle between u and v if the magnitude of their cross product vector is equal to 9 .
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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
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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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