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    Topics covered in this chapter

  • Definition of a vector
  • Examples of vectors
  • Scalar multiplication
  • Addition of vectors – coplanar vectors
  • Unit vectors
  • A basis of non-coplanar vectors
  • Scalar product – orthogonality
  • Directional cosines for coordinate transformation
  • Vector product
  • Velocity due to rigid body rotations
  • Triple scalar product
  • Triple vector product
  • Second order tensors
  • Examples of second order tensors
  • Scalar multiplication and addition
  • Contraction and multiplication
  • The vector of an antisymmetric tensor
  • Canonical form of a symmetric tensor
Reading Assignment: Chapter 2 of Aris, Appendix A of BSL

The algebra of vectors and tensors will be described here with Cartesian coordinates so the student can see the operations in terms of its components without the complexity of curvilinear coordinate systems.

Definition of a vector

Suppose x i , i.e., ( x 1 , x 2 , x 3 ), are the Cartesian coordinates of a point P in a frame of reference, 0123 . Let 0 1 ¯ 2 ¯ 3 ¯ be another Cartesian frame of reference with the same origin but defined by a rigid rotation. The coordinates of the point P in the new frame of reference is x ¯ j where the coordinates are related to those in the old frame as follows.

x ¯ j = l i j x i = l 1 j x 1 + l 2 j x 2 + l 3 j x 3 x i = l i j x ¯ j = l i 1 x ¯ 1 + l i 2 x ¯ 2 + l i 3 x ¯ 3

where l ij are the cosine of the angle between the old and new coordinate systems. Summation over repeated indices is understood when a term or a product appears with a common index.

Cartesian Vector
A Cartesian vector, a, in three dimensions is a quantity with three components a 1 , a 2 , a 3 in the frame of reference 0123, which, under rotation of the coordinate frame to 0 1 ¯ 2 ¯ 3 ¯ , become components a ¯ 1 , a ¯ 2 , a ¯ 3 , where
a ¯ j = l i j a i MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadggagaqeamaaBaaaleaacaWGQbaabeaakiabg2da9iaadYgadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamyyamaaBaaaleaacaWGPbaabeaaaaa@3E14@

Examples of vectors

In Cartesian coordinates, the length of the position vector of a point from the origin is equal to the square root of the sum of the square of the coordinates. The magnitude of a vector, a , is defined as follows.

| a | = ( a i a i ) 1 / 2 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaCyyaaGaay5bSlaawIa7aiabg2da9maabmaabaGaamyyamaaBaaaleaacaWGPbaabeaakiaadggadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaa@42ED@

A vector with a magnitude of unity is called a unit vector . The vector, a /| a |, is a unit vector with the direction of a . Its components are equal to the cosine of the angle between a and the coordinate axis. Some special unit vectors are the unit vectors in the direction of the coordinate axis and the normal vector of a surface.

Scalar multiplication

If α is a scalar and a is a vector, the product α a is a vector with components, α a i , magnitude α| a |, and the same direction as a .

Addition of vectors – coplanar vectors

If a and b are vectors with components a i and b i , then the sum of a and b is a vector with components, a i + b i .

The order and association of the addition of vectors are immaterial.

a + b = b + a ( a + b ) + c = a + ( b + c )

The subtraction of one vector from another is the same as multiplying one by the scalar (-1) and adding the resulting vectors.

If a and b are two vectors from the same origin, they are colinear or parallel if one is a linear combination of the other, i.e., they both have the same direction. If a and b are two vectors from the same origin, then all linear combination of a and b are in the same plane as a and b , i.,e., they are coplanar . We will prove this statement when we get to the triple scalar product.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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