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Sometimes we find it useful to use a different definition of distance, corresponding to an alternate norm for vectors. For example, consider the l-norm defined as

| | x | | 1 = ( | x 1 | + | x 2 | + + | x n | ) ,

where | x i | is the magnitude of component x i . There is also the sup-norm , the “supremum” or maximum of the components x 1 , ... , x n :

| | x | | sup = max ( | x 1 | , | x 2 | , ... , | x n | ) .

The following examples illustrate what the Euclidean norm, the l-norm, and the sup-norm look like for typical vectors.

Consider the vector x = - 3 l 2 . Then

  1. | | x | | = [ ( - 3 ) 2 + ( 1 ) 2 + ( 2 ) 2 ] 1 / 2 = ( 14 ) 1 / 2 ;
  2. | | x | | 1 = ( | - 3 | + | 1 | + | 2 | ) = 6 ; and
  3. | | x | | sup = max ( | - 3 | , | 1 | , | 2 | ) = 3 .

Figure 1 shows the locus of two-component vectors x = x 1 x 2 with the property that | | x | | = 1 , | | x | | 1 = 1 , or | | x | | sup = 1 .

Figure one is a diagram with a vertical axis labeled x_2 and horizontal axis labeled x_1. A large square centered at the origin with its diagonals on the axes is drawn, and around this is a circle centered at the origin and of the same diameter as the diagonals of the square. In addition, a square centered at the origin with the same width as the diagonal of the first square is the outermost shape of the diagram. The side of the smaller square that lies in the fourth quadrant is labeled locus of x such that  ||x||_1 = 1. The section of the circle that lies in the first quadrant is labeled locus of x such that ||x||_2 = 1. The section of the larger square that lies in the second quadrant is labeled locus of x such that  ||x||_sup = 1. Figure one is a diagram with a vertical axis labeled x_2 and horizontal axis labeled x_1. A large square centered at the origin with its diagonals on the axes is drawn, and around this is a circle centered at the origin and of the same diameter as the diagonals of the square. In addition, a square centered at the origin with the same width as the diagonal of the first square is the outermost shape of the diagram. The side of the smaller square that lies in the fourth quadrant is labeled locus of x such that  ||x||_1 = 1. The section of the circle that lies in the first quadrant is labeled locus of x such that ||x||_2 = 1. The section of the larger square that lies in the second quadrant is labeled locus of x such that  ||x||_sup = 1.
Locus of Two-Dimensional Vectors Whose Various Norms Are 1

The next example shows how the l-norm is an important part of city life.

The city of Metroville was laid out by mathematicians as shown in Figure 2 . A person at the intersection of Avenue 0 and Street - 2 (point A ) is clearly two blocks from the center of town (point C). This is consistent with both the Euclidean norm

| | A | | = 0 2 + ( - 2 ) 2 = 4 = 2

and the l-norm

| | A | | 1 = ( | 0 | + | - 2 | ) = 2 .

But how far from the center of town is point B at the intersection of Avenue-2 and Street 1? According to the Euclidean norm, the distance is

| | B | | = ( - 2 ) 2 + ( 1 ) 2 = 5 .
Figure two is a diagram of metroville. North points upward. There are five streets that travel north-south, labeled from left to right, street -2, street -1, street 0, street 1, and street 2. There are also five streets that run east-west labeled from top to bottom, avenue 2, avenue 1, avenue 0, avenue -1, avenue -2. At the intersection of avenue 0 and street -2 is point A. At the intersection of street 0 and avenue 0 is point C. At the intersection of avenue -2 and street 1 is point B. Figure two is a diagram of metroville. North points upward. There are five streets that travel north-south, labeled from left to right, street -2, street -1, street 0, street 1, and street 2. There are also five streets that run east-west labeled from top to bottom, avenue 2, avenue 1, avenue 0, avenue -1, avenue -2. At the intersection of avenue 0 and street -2 is point A. At the intersection of street 0 and avenue 0 is point C. At the intersection of avenue -2 and street 1 is point B.
Metroville, U.S.A.

While it is true that point B is 5 blocks from C , it is also clear that the trip would be three blocks by any of the three shortest routes on roads. Theappropriate norm is the l-norm:

| 1 B | | 1 = ( | - 2 | + | 1 | ) = 3 .

Even more generally, we can define a norm for each value of p from 1 to infinity. The so-called p-norm is

| I x | | p = ( | x 1 | p + | x 2 | p + + | x n | p ) 1 / p .

DEMO 4.1 (MATLAB). From the command level of MATLAB, type the following lines:

Check to see whether the answer agrees with the definition of vector subtraction. Now type

Check the answer to see whether it agrees with the definition of scalar multiplication. Now type

This is how MATLAB does the inner product. Check the result. Type

Now type your own MATLAB expression to find the cosine of the angle between vectors x and y . Put the result in variable t . Then find the angle θ by typing

The angle θ is in radians. You may convert it to degrees if you wish by multiplying it by 180 / π :

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
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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.
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characteristics of micro business
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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
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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