# 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.)

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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?