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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

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Missy Reply
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Lale Reply
no can't
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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ya I also want to know the raman spectra
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Crow Reply
what about nanotechnology for water purification
RAW Reply
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yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
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can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 7

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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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