# 0.1 Cartesian vectors and tensors: their algebra  (Page 2/5)

 Page 2 / 5

## Unit vectors

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

${\mathbf{e}}_{\left(1\right)}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\text{}\text{}{\mathbf{e}}_{\left(2\right)}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\text{}\text{}{\mathbf{e}}_{\left(3\right)}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

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.

$\mathbf{a}={a}_{1}{\mathbf{e}}_{\left(1\right)}+{a}_{2}{\mathbf{e}}_{\left(2\right)}+{a}_{3}{\mathbf{e}}_{\left(3\right)}$

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

$\begin{array}{c}\mathbf{a}=|\mathbf{a}|\left(\frac{{a}_{1}}{|\mathbf{a}|}{\mathbf{e}}_{\left(1\right)}+\frac{{a}_{2}}{|\mathbf{a}|}{\mathbf{e}}_{\left(2\right)}+\frac{{a}_{3}}{|\mathbf{a}|}{\mathbf{e}}_{\left(3\right)}\right)\\ =|\mathbf{a}|\left({\lambda }_{1}{\mathbf{e}}_{\left(1\right)}+{\lambda }_{2}{\mathbf{e}}_{\left(2\right)}+{\lambda }_{3}{\mathbf{e}}_{\left(3\right)}\right)\end{array}$

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

$\mathbf{a}•\mathbf{b}=|\mathbf{a}||\mathbf{b}|\mathrm{cos}\theta$

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.

$\begin{array}{ccc}{\mathbf{e}}_{\left(i\right)}•{\mathbf{e}}_{\left(j\right)}& =& {\delta }_{ij}\\ & =& \left\{\begin{array}{c}1,i=j\\ 0,i\ne j\end{array}\end{array}$

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.

$\begin{array}{ccc}\mathbf{a}•\mathbf{b}& =& \left({a}_{1}{\mathbf{e}}_{\left(1\right)}+{a}_{2}{\mathbf{e}}_{\left(2\right)}+{a}_{3}{\mathbf{e}}_{\left(3\right)}\right)•\left({b}_{1}{\mathbf{e}}_{\left(1\right)}+{b}_{2}{\mathbf{e}}_{\left(2\right)}+{b}_{3}{\mathbf{e}}_{\left(3\right)}\right)\hfill \\ & =& {a}_{i}{b}_{j}{\delta }_{ij}\hfill \\ & =& {a}_{i}{b}_{i}\hfill \end{array}$

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

$\begin{array}{ccc}\mathbf{a}•\mathbf{a}& =& |\mathbf{a}||\mathbf{a}|\mathrm{cos}0\hfill \\ & =& {|\mathbf{a}|}^{2}\hfill \\ \mathbf{a}•\mathbf{a}& =& {a}_{i}{a}_{i}\hfill \\ & =& {|\mathbf{a}|}^{2}\hfill \end{array}$

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.

$\mathbf{a}•\mathbf{n}=|\mathbf{a}|\mathrm{cos}\theta$

## 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 ${\mathbf{e}}_{\left(i\right)}$ and in the rotated system is ${\stackrel{_}{\mathbf{e}}}_{\left(j\right)}$ . The components of the unit vector, ${\stackrel{_}{\mathbf{e}}}_{\left(j\right)}$ , in the original reference frame is ${l}_{ij}$ . This can be expressed as the scalar product.

$\begin{array}{l}{\overline{e}}_{\left(j\right)}={l}_{1j}{e}_{\left(1\right)}+{l}_{2j}{e}_{\left(2\right)}+{l}_{3j}{e}_{\left(3\right)},\text{}j=1,2,3\\ {e}_{\left(i\right)}•{\overline{e}}_{\left(j\right)}={l}_{ij},\text{}i,j=1,2,3\end{array}$

Since ${\stackrel{_}{\mathbf{e}}}_{\left(j\right)}$ is a unit vector, it has a magnitude of unity.

${\overline{e}}_{\left(j\right)}•{\overline{e}}_{\left(j\right)}=1={l}_{i\left(j\right)}{l}_{i\left(j\right)}={l}_{1\left(j\right)}{l}_{1\left(j\right)}+{l}_{2\left(j\right)}{l}_{2\left(j\right)}+{l}_{3\left(j\right)}{l}_{3\left(j\right)},\text{}j=1,2,3$

Also, the axis of a Cartesian system are orthorgonal.

$\begin{array}{l}{\mathbf{e}}_{\left(i\right)}•{\mathbf{e}}_{\left(j\right)}=\left\{\begin{array}{c}0,\text{}\text{if}i\ne j\\ 1,\text{}\text{if}i=j\end{array}\\ \text{thus}\\ {\mathbf{e}}_{\left(i\right)}•{\mathbf{e}}_{\left(j\right)}={d}_{ij}\end{array}$
$\begin{array}{ccc}{\overline{e}}_{\left(i\right)}•{\overline{e}}_{\left(j\right)}& =& {l}_{ki}{l}_{kj}={l}_{1i}{l}_{1j}+{l}_{2i}{l}_{2j}+{l}_{3i}{l}_{3j},i,j=1,2,3\hfill \\ & =& {\delta }_{ij}\hfill \end{array}$

## 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 .

$|\mathbf{a}×\mathbf{b}|=|\mathbf{a}|\text{}|\mathbf{b}|\text{}\mathrm{sin}\theta$

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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!