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The right-hand rule

For a "right-handed" coordinate system, the direction of the resultant vector for AxB can be determined as follows:

Point the forefinger of the right hand in the direction of A and point the second finger in the direction of B. The thumb will then point in the directionof the resultant vector.

The cross product is not commutative

If you think about this, you should realize that the cross product is not commutative. That is to say that AxB is not the same as BxA because thedirection of the resultant vector would not be the same.

Create a vector diagram on your graph board

Once again, in order for you to better understand the nature of a vector cross product, I recommend that you create a Cartesian coordinate system on yourgraph board, and draw the following two vectors.

A vector diagram for your graph board

Draw the first vector from the origin to a point at x = 1y = 1.73 Label this vector A.Draw a second vector from the origin to a point at x = 2.9y = 0.78 Label this vector B.

The cross product

The cross product, AxB is defined as

AxB = Amag*Bmag*sin(angle)

where

  • Amag is the magnitude of the vector A
  • Bmag is the magnitude of the vector B
  • angle is the angle between the two vectors, which must be less than or equal to180 degrees

The area of the parallelogram

Use the vectors that you have drawn on your graph board to construct a parallelogram and see if you can estimate the area of that parallelogram.

Even if you were a sighted student having the parallelogram drawn on high-quality graph paper, it would be something of a chore to manuallydetermine the area of the parallelogram.

Let's work through some numbers

Let's use the cross product to determine the area of the parallelogram.

Given the definition of the cross product, we see that there are three values that we need:

  • Amag
  • Bmag
  • angle

Same vectors as before

If we were starting out with two new vectors, we could compute the magnitude of each vector using the Pythagorean theorem. We could also determine the angleby computing the vector dot product that I explained earlier in this module.

As you may have noticed, these are the same two vectors that we used earlier, and we computed those three values earlier. Going back and recovering thosethree values, we have

  • Amag = 2.0
  • Bmag = 3.0
  • angle = 45 degrees (at least that is what I intended for it to be)

The area of the parallelogram

Using the earlier definition and the nomenclature for the Google calculator,

AxB = Amag*Bmag*sin(angle), or

AxB = 2.0*3.0*sin(45 degrees), or

AxB = 4.24 square units

The direction of the resultant vector

If you place the end of your thumb at the origin of your Cartesian coordinate system, you should be able, with reasonable comfort, to point your forefinger inthe direction of A and your second finger in the direction of B.

According to the right-hand rule , this means that the direction of the resultant vector is the direction that your thumb ispointing, or straight down into the graph board.

Perpendicular or parallel vectors

Now consider what happens as the angle varies between 90 degrees (perpendicular vectors) and 0 degrees (parallel vectors) for a given pair ofvectors.

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
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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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