# 2.4 The cross product  (Page 10/16)

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Find all vectors $\text{w}=⟨{w}_{1},{w}_{2},{w}_{3}⟩$ that satisfy the equation $⟨1,1,1⟩\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}=⟨-1,-1,2⟩.$

$\text{w}=⟨{w}_{3}-1,{w}_{3}+1,{w}_{3}⟩,$ where ${w}_{3}$ is any real number

Solve the equation $\text{w}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}⟨1,0,-1⟩=⟨3,0,3⟩,$ where $\text{w}=⟨{w}_{1},{w}_{2},{w}_{3}⟩$ is a nonzero vector with a magnitude of $3.$

[T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $30\text{°}$ angle with the horizontal. If the mechanic applies a vertical force of $10$ lb on the wrench handle, what is the magnitude of the torque at point $P$ (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.

8.66 ft-lb

[T] A boy applies the brakes on a bicycle by applying a downward force of $20$ lb on the pedal when the 6-in. crank makes a $40\text{°}$ angle with the horizontal (see the following figure). Find the torque at point $P.$ Express your answer in foot-pounds rounded to two decimal places.

[T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y -axis if the force is applied in the direction $⟨0,1,-2⟩$ and it produces a $100$ N·m torque to the bolt located at the origin.

250 N

[T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of $35\text{°}$ to produce a torque of $20$ N·m?

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{C}$ (in coulombs) moving in a magnetic field $\text{B}$ where the velocity vector $\text{v}$ is given by $\text{F}=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{B}}\right)$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ is in tesla [T], and $\text{F}$ is in newtons [N]). Find the force that acts on a proton that moves in the xy -plane at velocity $\text{v}={10}^{5}\text{i}+{10}^{5}\text{j}$ (in meters per second) in a magnetic field given by $\mathbf{\text{B}}=0.3\text{j}.$

$\text{F}=4.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-15}\text{k}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{N}}$

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{C}$ moving in a magnetic field $\text{B}$ where the velocity vector v is given by $\text{F}=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{B}}\right)$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ in $\text{T},$ and $\text{F}$ in $\text{N}\right).$ If the magnitude of force $\text{F}$ acting on a proton is $5.9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-17}$ N and the proton is moving at the speed of 300 m/sec in magnetic field $\text{B}$ of magnitude 2.4 T, find the angle between velocity vector $\text{v}$ of the proton and magnetic field $\text{B}.$ Express the answer in degrees rounded to the nearest integer.

[T] Consider $\mathbf{\text{r}}\left(t\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t,2t⟩$ the position vector of a particle at time $t\in \left[0,30\right],$ where the components of $\text{r}$ are expressed in centimeters and time in seconds. Let $\stackrel{\to }{OP}$ be the position vector of the particle after $1$ sec.

1. Determine unit vector $\mathbf{\text{B}}\left(t\right)$ (called the binormal unit vector ) that has the direction of cross product vector $\text{v}\left(t\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{a}}\left(t\right),$ where $\text{v}\left(t\right)$ and $\text{a}\left(t\right)$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $t$ seconds.
2. Use a CAS to visualize vectors $\text{v}\left(1\right),$ $\text{a}\left(1\right),$ and $\mathbf{\text{B}}\left(1\right)$ as vectors starting at point $P$ along with the path of the particle.

a. $\mathbf{\text{B}}\left(t\right)=⟨\frac{2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t}{\sqrt{5}},-\frac{2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t}{\sqrt{5}},\frac{1}{\sqrt{5}}⟩;$
b.

A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A\left(8,0,0\right),$ $B\left(8,18,0\right),$ $C\left(0,18,8\right),$ and $D\left(0,0,8\right)$ (see the following figure).

1. Find vector $\mathbf{\text{n}}=\stackrel{\to }{AB}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{AD}$ perpendicular to the surface of the solar panels. Express the answer using standard unit vectors.
2. Assume unit vector $\mathbf{\text{s}}=\frac{1}{\sqrt{3}}\text{i}+\frac{1}{\sqrt{3}}\text{j}+\frac{1}{\sqrt{3}}\text{k}$ points toward the Sun at a particular time of the day and the flow of solar energy is $\text{F}=900\mathbf{\text{s}}$ (in watts per square meter [ ${\text{W/m}}^{2}$ ]). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors $\text{F}$ and $\text{n}$ (expressed in watts).
3. Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. ( Hint : The angle between vectors $\text{n}$ and $\text{s}$ and the angle of elevation are complementary.)

What is deflation
Enumerate emotional intelligence to a manager
I need help in inflation graphs
Select inflation type, Demand pull, cost pull or anticipation 1- Select the set of data you intend on graphing i.e inflation rate of 2017, location (particular country) 2 - Select the type of measurement tool that best allows you to input the inflation data, Consumer price index is the most accurate
Jama
this is to make sure you have all the correct information, Also use should know 1- Cost pull is Aggregate Demand and Aggregate Supply AD - AS graphed 2- Demans pull is Aggregate Supply and Aggregate Demand AS - AD graphed
Jama
what is production
what is a monopolistic competition?
Enumerate emotional intelligence to a manager
Chinonso
Enumerate emotional intelligence to a manager
Chinonso
who is barter
exchange goods each other
Seven
what is economic
Bah
is the use of scares resources to satisfy our unlimited needs and wants
Desiderius
how many kinds of utility functions?
What is partnership?
Jackson
the legal association of two or more people as co-owners of a business for profit.
harmony
Would you expect the kinked demand curve to be more extreme (like a right angle) or less extreme (like a normal demand curve) if each firm in the cartel produces a near-identical product like OPEC and petroleum? What if each firm produces a somewhat different product?
no
what is supply
what is opportunity cost
Mizta
The opportunity gained interms of opportunity lost is known as opportunity cost Or The second best alternative use of resources
Mir
forgone alternative: like forgoing Something our of two to buy one
Tam-Waribo
what is macro economic s
macroeconomics is the study of economic as a whole level.
Gafar
meaning of positive science
positive science it is focused on facts and cause and effect and behavioural relationship and include developmental testing in economic theoreis.
Gafar
what is inflation
inflation is the general price increase of goods and services in an economy.
tesfie
Inflation is the persistent rise in the general price level
T-Max
inflation is characterized by increase in the general price of goods and services. when there is too much money in circulation. increase in demand of goods pursuing fewer goods. when purchasing power of money decreases .
Ejikeme
inflation is the persistent rise general price level
Habeeb
inflation is the persistent increase in price
Machall
hi
Rafiu
yes
boston
hi
Ayaan
how are you
Ayaan
increase in the general level of price...
what is deflation
Sele
is the gradual decrease of currency exchange in a country.
Gafar
why ecnomics important ? give answer plz
Saifullah
Because is a field of science study that reflects on our day to day activities with human behavior.
ANSU
why economic is a science
Imoro
Economics is referred to as a social science not a pure science. It's regarded as a social science because it makes use of the scientific method to solve problems. The scientific method refers to observation, asking questions, forming hypothesis, experimentation etc
Nkechi
Economics is a social science because it study human behavior how he relates with his daily activities with the available limited resources to satisfy his wants.
Ojo
what are the factors affecting the demand
Mohammed
yeah it uses the scientific method to study human behaviour.
Nkechi
inflation referes to the persistant increase in the general price of goods and services over a given period of time say a year.
Abdul
factors affacting Demand of good and services are 1.price of a commodity in question 2.price of related commodity 3.Income of a consumer 4.Population 5.tast and prefereance 6.Season or weather condition
Abdul
what is difference between perfect and non perfect market.
Saheed
what the difference between Trade off and Opportunity Cost?
Elzevery
Golda
In trade off, you increase the amount of something by decreasing the amount of something else. For example, you use 2 hours to study and 2 hours for leisure. if you increase study hour by 1 more hour, i.e 3 hours, leisure time will decrease by 1 hour, i.e 1 hour.
harmony
In all, you would have traded off 1 hour of leisure time for 3 hours of study time. But in opportunity cost, you let something go in order to obtain something else entirely.
harmony
Elzevery
please i want help on thid question given P=\$10 And TC=120+4Q2 1.find the profit maximizing level of price and quantity. 2.what will be the total profit?
Shemels
please how is substitutional effect affecting demand
Acha
if a price of a particular commodity is high people demand less ,they rather go for less one
FIDELIS
what is account
Wasif
account is an arrangement between a customer and a bank that allows the customer to play in and take out money (bank account)
Johnny
pay, not play sorry
Johnny
four effects of inflation in an economy
Acha
oil
TsendeTheRipper
decreases living standards .. decreases purchasing power.. Decreases internation competition . increases the cost of borrowing
MansoorAfghan
Inflation refers to persistent increase in the general prices of commodities.
Aluko
MansoorAfghan
Low standard of living, prices of commodities will be high, devaluation of currency in the economy
Aluko
Debtors gain while creditors loses
Ojo
Hi every one
Ani
Hello
Ansah
hi
kelvin
hi
Seven
Hii
raushan
how are you all
Seven
Fin and you
raushan
me also
Seven
Whare do you live
raushan
general tendency of rising price and continuous... .
Auqib
any one among you is going to aaper in net exam day after tomorrow
Auqib
Auqib
pakistan
Seven
Oo Pakistan I am India
raushan
really from Pakistan
Auqib
ok
Seven
You noo hindi
raushan
no
Seven
hi
ken
what is deference between the producer and firm?
ISMAILI
different between demand and quantity demand
No difference
MansoorAfghan
demand is the overall demand for it
MansoorAfghan
actually theres no difference
MansoorAfghan
quantity demanded is used in Equilibrium of d and s
MansoorAfghan
for evrything else u use deman
MansoorAfghan
the difference of it is that when demand simply denotes the willingness and a person's ability to purchase. And as against quantity demand represent the amount of an economic good or services desire by a consumer at a fixed price .☺
Gafar
how to calculate inflation
Explain the factors that have led to high quantity demanded
price of the product increase of price substitute product as people shift to cheap one
Black
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
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