# 2.3 The dot product  (Page 5/16)

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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

## Projections

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.

## Definition

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 $\theta$ represents the angle between u and v , then, by properties of triangles, we know the length of ${\text{proj}}_{\text{u}}\text{v}$ is $‖{\text{proj}}_{\text{u}}\text{v}‖=‖\text{v}‖\text{cos}\phantom{\rule{0.2em}{0ex}}\theta .$ When expressing $\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ in terms of the dot product, this becomes

$\begin{array}{cc}\hfill ‖{\text{proj}}_{\text{u}}\text{v}‖& =‖\text{v}‖\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \hfill \\ & =‖\text{v}‖\left(\frac{\text{u}·\text{v}}{‖\text{u}‖‖\text{v}‖}\right)\hfill \\ & =\frac{\text{u}·\text{v}}{‖\text{u}‖}.\hfill \end{array}$

We now multiply by a unit vector in the direction of u to get ${\text{proj}}_{\text{u}}\text{v}\text{:}$

${\text{proj}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{‖\text{u}‖}\left(\frac{1}{‖\text{u}‖}\text{u}\right)=\frac{\text{u}·\text{v}}{{‖\text{u}‖}^{2}}\text{u}.$

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

$‖{\text{proj}}_{\text{u}}\text{v}‖={\text{comp}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{‖\text{u}‖}.$

## Finding projections

Find the projection of v onto u.

1. $\text{v}=⟨3,5,1⟩$ and $\text{u}=⟨-1,4,3⟩$
2. $\text{v}=3\text{i}-2\text{j}$ and $\text{u}=\text{i}+6\text{j}$
1. Substitute the components of v and u into the formula for the projection:
$\begin{array}{cc}\hfill {\text{proj}}_{\text{u}}\text{v}& =\frac{\text{u}·\text{v}}{{‖\text{u}‖}^{2}}\text{u}\hfill \\ & =\frac{⟨-1,4,3⟩·⟨3,5,1⟩}{{‖⟨-1,4,3⟩‖}^{2}}⟨-1,4,3⟩\hfill \\ & =\frac{-3+20+3}{{\left(-1\right)}^{2}+{4}^{2}+{3}^{2}}⟨-1,4,3⟩\hfill \\ & =\frac{20}{26}⟨-1,4,3⟩\hfill \\ & =⟨-\frac{10}{13},\frac{40}{13},\frac{30}{13}⟩.\hfill \end{array}$
2. To find the two-dimensional projection, simply adapt the formula to the two-dimensional case:
$\begin{array}{cc}\hfill {\text{proj}}_{\text{u}}\text{v}& =\frac{\text{u}·\text{v}}{{‖\text{u}‖}^{2}}\text{u}\hfill \\ & =\frac{\left(\text{i}+6\text{j}\right)·\left(3\text{i}-2\text{j}\right)}{{‖\text{i}+6\text{j}‖}^{2}}\left(\text{i}+6\text{j}\right)\hfill \\ & =\frac{1\left(3\right)+6\left(-2\right)}{{1}^{2}+{6}^{2}}\left(\text{i}+6\text{j}\right)\hfill \\ & =-\frac{9}{37}\left(\text{i}+6\text{j}\right)\hfill \\ & =-\frac{9}{37}\text{i}-\frac{54}{37}\text{j}.\hfill \end{array}$

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 $\text{v}=⟨6,-4⟩$ and let $\text{u}=⟨3,1⟩.$ 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 $\text{p}={\text{proj}}_{\text{u}}\text{v}.$ Then, we have

$\begin{array}{cc}\hfill \text{p}& =\frac{\text{u}·\text{v}}{{‖\text{u}‖}^{2}}\text{u}\hfill \\ & =\frac{18-4}{9+1}\text{u}\hfill \\ & =\frac{7}{5}\text{u}=\frac{7}{5}⟨3,1⟩=⟨\frac{21}{5},\frac{7}{5}⟩.\hfill \end{array}$

Now consider the vector $\text{q}=\text{v}-\text{p}.$ We have

$\begin{array}{cc}\hfill \text{q}& =\text{v}-\text{p}\hfill \\ & =⟨6,-4⟩-⟨\frac{21}{5},\frac{7}{5}⟩\hfill \\ & =⟨\frac{9}{5},-\frac{27}{5}⟩.\hfill \end{array}$

Clearly, by the way we defined q , we have $\text{v}=\text{q}+\text{p},$ and

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Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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Yasmin
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Cesar
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Uday
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
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Azam
Hello
Uday
I'm interested in Nanotube
Uday
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Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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