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Unit vectors

The unit vectors in the direction of a set of mutually orthogonal coordinate axis are defined as follows.

e ( 1 ) = [ 1 0 0 ] , e ( 2 ) = [ 0 1 0 ] , e ( 3 ) = [ 0 0 1 ]

The suffixes to e are enclosed in parentheses to show that they do not denote components. A vector, a , can be expressed in terms of its components, ( a 1 , a 2 , a 3 ) and the unit vectors.

a = a 1 e ( 1 ) + a 2 e ( 2 ) + a 3 e ( 3 )

This equation can be multiplied and divided by the magnitude of a to express the vector in terms of its magnitude and direction.

a = | a | ( a 1 | a | e ( 1 ) + a 2 | a | e ( 2 ) + a 3 | a | e ( 3 ) ) = | a | ( λ 1 e ( 1 ) + λ 2 e ( 2 ) + λ 3 e ( 3 ) )

where λ i are the directional cosines of a .

A special unit vector we will use often is the normal vector to a surface, n . The components of the normal vector are the directional cosines of the normal direction to the surface.

Scalar product – orthogonality

The scalar product (or dot product ) of two vectors, a and b is defined as

a b = | a | | b | cos θ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHIaYTcaWHIbGaeyypa0ZaaqWaaeaacaWHHbaacaGLhWUaayjcSdGaaGPaVpaaemaabaGaaCOyaaGaay5bSlaawIa7aiaaykW7ciGGJbGaai4BaiaacohacqaH4oqCaaa@49FF@

where θ is the angle between the two vectors. If the two vectors are perpendicular to each other, i.e., they are orthogonal , then the scalar product is zero. The unit vectors along the Cartesian coordinate axis are orthogonal and their scalar product is equal to the Kronecker delta.

e ( i ) e ( j ) = δ i j = { 1 , i = j 0 , i j

The scalar product is commutative and distributive. The cosine of the angle measured from a to b is the same as measured from b to a . Thus the scalar product can be expressed in terms of the components of the vectors.

a b = ( a 1 e ( 1 ) + a 2 e ( 2 ) + a 3 e ( 3 ) ) ( b 1 e ( 1 ) + b 2 e ( 2 ) + b 3 e ( 3 ) ) = a i b j δ i j = a i b i MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7B19@

The scalar product of a vector with itself is the square of the magnitude of the vector.

a a = | a | | a | cos 0 = | a | 2 a a = a i a i = | a | 2 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaqqabaGaaCyyaiabgkci3kaahggacqGH9aqpdaabdaqaaiaahggaaiaawEa7caGLiWoacaaMc8+aaqWaaeaacaWHHbaacaGLhWUaayjcSdGaaGPaVlGacogacaGGVbGaai4CaiaaykW7caaIWaaabaGaeyypa0ZaaqWaaeaacaWHHbaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaGcbaGaaCyyaiabgkci3kaahggacqGH9aqpcaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlaadggadaWgaaWcbaGaamyAaaqabaaakeaacqGH9aqpdaabdaqaaiaahggaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaaaaa@6096@

The most common application of the scalar product is the projection or component of a vector in the direction of another vector. For example, suppose n is a unit vector (e.g., the normal to a surface) the component of a in the direction of n is as follows.

a n = | a | cos θ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHIaYTcaWHUbGaeyypa0ZaaqWaaeaacaWHHbaacaGLhWUaayjcSdGaaGPaVlGacogacaGGVbGaai4CaiaaykW7cqaH4oqCaaa@45FE@

Directional cosines for coordinate transformation

The properties of the directional cosines for the rotation of the Cartesian coordinate reference frame can now be easily illustrated. Suppose the unit vectors in the original system is e ( i ) and in the rotated system is e _ ( j ) . The components of the unit vector, e _ ( j ) , in the original reference frame is l i j . This can be expressed as the scalar product.

e ¯ ( j ) = l 1 j e ( 1 ) + l 2 j e ( 2 ) + l 3 j e ( 3 ) , j = 1 , 2 , 3 e ( i ) e ¯ ( j ) = l i j , i , j = 1 , 2 , 3 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7331@

Since e _ ( j ) is a unit vector, it has a magnitude of unity.

e ¯ ( j ) e ¯ ( j ) = 1 = l i ( j ) l i ( j ) = l 1 ( j ) l 1 ( j ) + l 2 ( j ) l 2 ( j ) + l 3 ( j ) l 3 ( j ) , j = 1 , 2 , 3 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7487@

Also, the axis of a Cartesian system are orthorgonal.

e ( i ) e ( j ) = { 0 , if i j 1 , if i = j thus e ( i ) e ( j ) = d i j MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCyzamaaBaaaleaacaWHOaGaaCyAaiaahMcaaeqaaOGaeyOiGCRaaCyzamaaBaaaleaacaWHOaGaaCOAaiaahMcaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGaaGimaiaaysW7caGGSaGaaGzbVlaabMgacaqGMbGaaGjbVlaadMgacqGHGjsUcaWGQbaabaGaaGymaiaacYcacaaMf8UaaeyAaiaabAgacaaMe8UaamyAaiabg2da9iaadQgaaaaacaGL7baaaeaacaqG0bGaaeiAaiaabwhacaqGZbaabaGaaCyzamaaBaaaleaacaWHOaGaaCyAaiaahMcaaeqaaOGaeyOiGCRaaCyzamaaBaaaleaacaWHOaGaaCOAaiaahMcaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaaaa@6583@
e ¯ ( i ) e ¯ ( j ) = l k i l k j = l 1 i l 1 j + l 2 i l 2 j + l 3 i l 3 j , i , j = 1 , 2 , 3 = δ i j MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6E59@

Vector product

The vector product (or cross product ) of two vectors, a and b , denoted as a × b , is a vector that is perpendicular to the plane of a and b such that a , b , and a × b form a right-handed system. (i.e., a , b , and a × b have the orientation of the thumb, first finger, and third finger of the right hand.) It has the following magnitude where θ is the angle between a and b .

| a × b | = | a | | b | sin θ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaCyyaiabgEna0kaahkgaaiaawEa7caGLiWoacqGH9aqpdaabdaqaaiaahggaaiaawEa7caGLiWoacaaMc8+aaqWaaeaacaWHIbaacaGLhWUaayjcSdGaaGPaVlGacohacaGGPbGaaiOBaiaaykW7cqaH4oqCcqGHEisPaaa@50B4@

The magnitude of the vector product is equal to the area of a parallelogram two of whose sides are the vectors a and b .

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, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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