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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

where we get a research paper on Nano chemistry....?
Maira Reply
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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