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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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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.

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.

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 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

what is motion?
Abdulaziz Reply
where the solving of questions of this topic?
Bonifasi Reply
According to Nernst's distribution law there are about two solvents in which solutes undergo equilibria. But i don't understand how can you know which of two solvents goes bottom and one top? I real want to understand b'coz some books do say why they prefer one to top/bottom.
Elia Reply
I need chapter 25 last topic
Hafsa Reply
What is physics?
Abdulaziz
physics is the study of matter and energy in space and time and how they related to each other
Manzoor
interaction of matter and eneegy....
Abdullah
thanks for correcting me bro
Manzoor
What is electrostatics bassically?
Ehtisham Reply
study of charge at rest
wamis
A branch in physics that deals with statics electricity
Akona
what is PN junction?
Manzoor
please I don't understand the solution of the first example as in d working
habila Reply
what's the question? Write it here.
SABYASACHI
a cold body of 100°C and a hot body is of 100°F . Transfer heat = ?
jagan Reply
you are given two metal spheres mounted on portable insulating support. Find a way to give them equal and opposite charges. you may use a glass rod rubbed with silk but may not touch it to the spheres. Do the spheres have to be of equal size for your method to work?
Rai Reply
what is emotion?
Abdulaziz
in the 2nd example, for chapter 8.2 on page 3/3, I don't understand where the value 48uC comes from, I just couldn't get that value in my calculator.
Anita Reply
are you talking about the capacitance combination problem
sam
please write the problem or send a snap of th page....I don't have the book in my vicinity.
SABYASACHI
yes, the 2nd example called Network of Capacitors on page 3/3 of section 8.2.
Anita
12 V = (Q1/12uF)+(Q1/6uF). So, Q1 = 12x4 = 48 uC.
sam
ohhhh OK thanks so much!!!!!!!
Anita
hello guys,, I'm asking to know something about, How can i know which solvent goes down and which does up in determination of partion coefficient(Nernst's distribution law). Please Need help because i have seen many contradictions via few of text books even some videos on youtube they don't say
Elia
what is electromagnetic force. do electric and magnetic force happen differently
Short Reply
yes
Renugadevi
yes
Pranay
why
Godson
how?
Godson
when electric charge exert force on another electric charge then this force is known as electrostatic force and when a magnet exert force on another magnet then this force is known as magnetic force and when force exerted on magnet due to varying electric field then this electromagnetic force
Ilyas
Yes
Akona
derived the electric potential due to disk of charge
aron Reply
how can we derived potential electric due to the disk
aron
how can you derived electric potential of a disk
aron
how can you derived electric potential due to disk
aron
where is response?
aron
what is difference between heat and temperature?
Qasim Reply
temperature is the measure of degree of hotness or coldness. on the other hand, heat is the form of energy, which causes temperature. So we can safely say, heat is the reason and temperature is its consequence.
SABYASACHI
Heat is the reason and temperature is the consequences
Angela
how many liquid metals do we have
Jeffery Reply
do we have gasses as metals
Jeffery
who knows should please tell us
Sadiku
yes...gallium & cesium
Idris
Hg is liquid. No metal gasses at standard temp and pressure
Shane
I don't ever understand any of this formulae
isaac Reply
which formula
Sadiku
How to determine a temperature scale
Masia Reply
what is the formula for absolute error
Nyro
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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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