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On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to $2 per package. They also changed suppliers for their invitations, and are now able to purchase invitations for only 10¢ per package. All their other costs and prices remain the same. If AAA sells 1408 invitations, 147 party favors, 2112 decorations, and 1894 food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.

Sales = $15,685.50; profit = $14,073.15

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As we have seen, addition combines two vectors to create a resultant vector. But what if we are given a vector and we need to find its component parts? We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection. For example, if a child is pulling the handle of a wagon at a 55° angle, we can use projections to determine how much of the force on the handle is actually moving the wagon forward ( [link] ). We return to this example and learn how to solve it after we see how to calculate projections.

This figure is the image of a wagon with a handle. The handle is represented by the vector “F.” The angle between F and the horizontal direction of the wagon is 55 degrees.
When a child pulls a wagon, only the horizontal component of the force propels the wagon forward.


The vector projection    of v onto u is the vector labeled proj u v in [link] . It has the same initial point as u and v and the same direction as u , and represents the component of v that acts in the direction of u . If θ represents the angle between u and v , then, by properties of triangles, we know the length of proj u v is proj u v = v cos θ . When expressing cos θ in terms of the dot product, this becomes

proj u v = v cos θ = v ( u · v u v ) = u · v u .

We now multiply by a unit vector in the direction of u to get proj u v :

proj u v = u · v u ( 1 u u ) = u · v u 2 u .

The length of this vector is also known as the scalar projection    of v onto u and is denoted by

proj u v = comp u v = u · v u .
This image has a vector labeled “v.” There is also a vector with the same initial point labeled “proj sub u v.” The third vector is from the terminal point of proj sub u v in the same direction labeled “u.” A broken line segment from the initial point of u to the terminal point of v is drawn and is perpendicular to u.
The projection of v onto u shows the component of vector v in the direction of u .

Finding projections

Find the projection of v onto u.

  1. v = 3 , 5 , 1 and u = −1 , 4 , 3
  2. v = 3 i 2 j and u = i + 6 j
  1. Substitute the components of v and u into the formula for the projection:
    proj u v = u · v u 2 u = −1 , 4 , 3 · 3 , 5 , 1 −1 , 4 , 3 2 −1 , 4 , 3 = −3 + 20 + 3 ( −1 ) 2 + 4 2 + 3 2 −1 , 4 , 3 = 20 26 −1 , 4 , 3 = 10 13 , 40 13 , 30 13 .
  2. To find the two-dimensional projection, simply adapt the formula to the two-dimensional case:
    proj u v = u · v u 2 u = ( i + 6 j ) · ( 3 i 2 j ) i + 6 j 2 ( i + 6 j ) = 1 ( 3 ) + 6 ( −2 ) 1 2 + 6 2 ( i + 6 j ) = 9 37 ( i + 6 j ) = 9 37 i 54 37 j .
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Sometimes it is useful to decompose vectors—that is, to break a vector apart into a sum. This process is called the resolution of a vector into components . Projections allow us to identify two orthogonal vectors having a desired sum. For example, let v = 6 , −4 and let u = 3 , 1 . We want to decompose the vector v into orthogonal components such that one of the component vectors has the same direction as u .

We first find the component that has the same direction as u by projecting v onto u . Let p = proj u v . Then, we have

p = u · v u 2 u = 18 4 9 + 1 u = 7 5 u = 7 5 3 , 1 = 21 5 , 7 5 .

Now consider the vector q = v p . We have

q = v p = 6 , −4 21 5 , 7 5 = 9 5 , 27 5 .

Clearly, by the way we defined q , we have v = q + p , and

Questions & Answers

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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