# 2.3 The dot product  (Page 5/16)

 Page 5 / 16

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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where the solving of questions of this topic?
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.
I need chapter 25 last topic
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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
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study of charge at rest
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A branch in physics that deals with statics electricity
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please I don't understand the solution of the first example as in d working
what's the question? Write it here.
SABYASACHI
a cold body of 100°C and a hot body is of 100°F . Transfer heat = ?
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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.
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.
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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
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yes
yes
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why
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how?
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Yes
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how can you derived electric potential of a disk
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how can you derived electric potential due to disk
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what is difference between heat and temperature?
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
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who knows should please tell us
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Hg is liquid. No metal gasses at standard temp and pressure
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