# 2.4 The cross product  (Page 8/16)

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$\text{u}=⟨3,-1,2⟩,$ $\text{v}=⟨-2,0,1⟩$

$\text{w}=-\frac{1}{3\sqrt{6}}\text{i}-\frac{7}{3\sqrt{6}}\text{j}-\frac{2}{3\sqrt{6}}\text{k}$

$\text{u}=⟨2,6,1⟩,$ $\text{v}=⟨3,0,1⟩$

$\text{u}=\stackrel{\to }{AB},$ $\text{v}=\stackrel{\to }{AC},$ where $A\left(1,0,1\right),$ $B\left(1,-1,3\right),$ and $C\left(0,0,5\right)$

$\text{w}=-\frac{4}{\sqrt{21}}\text{i}-\frac{2}{\sqrt{21}}\text{j}-\frac{1}{\sqrt{21}}\text{k}$

$\text{u}=\stackrel{\to }{OP},$ $\text{v}=\stackrel{\to }{PQ},$ where $P\left(-1,1,0\right)$ and $Q\left(0,2,1\right)$

Determine the real number $\alpha$ such that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ and $\text{i}$ are orthogonal, where $\text{u}=3\text{i}+\text{j}-5\text{k}$ and $\text{v}=4\text{i}-2\text{j}+\alpha \text{k}.$

$\alpha =10$

Show that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ and $2\text{i}-14\text{j}+2\text{k}$ cannot be orthogonal for any $\alpha$ real number, where $\text{u}=\text{i}+7\text{j}-\text{k}$ and $\text{v}=\alpha \text{i}+5\text{j}+\text{k}.$

Show that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ is orthogonal to $\text{u}+\text{v}$ and $\text{u}-\text{v},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Show that $\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}$ is orthogonal to $\left(\text{u}·\text{v}\right)\left(\text{u}+\text{v}\right)+\mathbf{\text{u}},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Calculate the determinant $|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 1\hfill & \hfill -1& \hfill 7\\ 2\hfill & \hfill 0& \hfill 3\end{array}|.$

$-3\text{i}+11\text{j}+2\text{k}$

Calculate the determinant $|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 0\hfill & \hfill 3& \hfill -4\\ 1\hfill & \hfill 6& \hfill -1\end{array}|.$

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given. Use determinant notation to find vector $\text{w}$ orthogonal to vectors $\text{u}$ and $\text{v}.$

$\text{u}=⟨-1,0,{e}^{t}⟩,$ $\text{v}=⟨1,{e}^{\text{−}t},0⟩,$ where $t$ is a real number

$\text{w}=⟨-1,{e}^{t},\text{−}{e}^{\text{−}t}⟩$

$\text{u}=⟨1,0,x⟩,$ $\text{v}=⟨\frac{2}{x},1,0⟩,$ where $x$ is a nonzero real number

Find vector $\left(\text{a}-2\mathbf{\text{b}}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{c}},$ where $\text{a}=|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 2\hfill & \hfill -1& \hfill 5\\ 0\hfill & \hfill 1& \hfill 8\end{array}|,$ $\mathbf{\text{b}}=|\begin{array}{ccc}\mathbf{\text{i}}\hfill & \hfill \mathbf{\text{j}}& \hfill \mathbf{\text{k}}\\ 0\hfill & \hfill 1& \hfill 1\\ 2\hfill & \hfill -1& \hfill -2\end{array}|,$ and $\mathbf{\text{c}}=\text{i}+\text{j}+\mathbf{\text{k}}.$

$-26\text{i}+17\text{j}+9\text{k}$

Find vector $\mathbf{\text{c}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{a}+3\mathbf{\text{b}}\right),$ where $\text{a}=|\begin{array}{ccc}\text{i}\hfill & \text{j}\hfill & \text{k}\hfill \\ 5\hfill & 0\hfill & 9\hfill \\ 0\hfill & 1\hfill & 0\hfill \end{array}|,$ $\mathbf{\text{b}}=|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 0\hfill & \hfill -1& \hfill 1\\ 7\hfill & \hfill 1& \hfill -1\end{array}|,$ and $\mathbf{\text{c}}=\text{i}-\text{k}.$

[T] Use the cross product $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ to find the acute angle between vectors $\text{u}$ and $\text{v},$ where $\text{u}=\text{i}+2\text{j}$ and $\text{v}=\text{i}+\text{k}.$ Express the answer in degrees rounded to the nearest integer.

$72\text{°}$

[T] Use the cross product $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ to find the obtuse angle between vectors $\text{u}$ and $\text{v},$ where $\text{u}=\text{−}\text{i}+3\text{j}+\text{k}$ and $\text{v}=\text{i}-2\text{j}.$ Express the answer in degrees rounded to the nearest integer.

Use the sine and cosine of the angle between two nonzero vectors $\text{u}$ and $\text{v}$ to prove Lagrange’s identity: ${‖\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}‖}^{2}={‖\text{u}‖}^{2}{‖\text{v}‖}^{2}-{\left(\text{u}·\text{v}\right)}^{2}.$

Verify Lagrange’s identity ${‖\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}‖}^{2}={‖\text{u}‖}^{2}{‖\text{v}‖}^{2}-{\left(\text{u}·\text{v}\right)}^{2}$ for vectors $\text{u}=\text{−}\text{i}+\text{j}-2\text{k}$ and $\text{v}=2\text{i}-\text{j}.$

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that $\text{u}$ and $\text{v}$ are collinear if and only if $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}=0.$

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that vectors $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}$ are collinear, where $A\left(4,1,0\right),$ $B\left(6,5,-2\right),$ and $C\left(5,3,-1\right).$

Find the area of the parallelogram with adjacent sides $\text{u}=⟨3,2,0⟩$ and $\text{v}=⟨0,2,1⟩.$

$7$

Find the area of the parallelogram with adjacent sides $\text{u}=\text{i}+\text{j}$ and $\text{v}=\text{i}+\text{k}.$

Consider points $A\left(3,-1,2\right),B\left(2,1,5\right),$ and $C\left(1,-2,-2\right).$

1. Find the area of parallelogram $ABCD$ with adjacent sides $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}.$
2. Find the area of triangle $ABC.$
3. Find the distance from point $A$ to line $BC.$

a. $5\sqrt{6};$ b. $\frac{5\sqrt{6}}{2};$ c. $\frac{5\sqrt{6}}{\sqrt{59}}$

Consider points $A\left(2,-3,4\right),B\left(0,1,2\right),$ and $C\left(-1,2,0\right).$

1. Find the area of parallelogram $ABCD$ with adjacent sides $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}.$
2. Find the area of triangle $ABC.$
3. Find the distance from point $B$ to line $AC.$

In the following exercises, vectors $\text{u},\text{v},\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are given.

1. Find the triple scalar product $\text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{w}}\right).$
2. Find the volume of the parallelepiped with the adjacent edges $\text{u},\text{v},\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}.$

$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}+\text{k},$ and $\text{w}=\text{i}+\text{k}$

a. $2;$ b. $2$

$\text{u}=⟨-3,5,-1⟩,$ $\text{v}=⟨0,2,-2⟩,$ and $\text{w}=⟨3,1,1⟩$

Calculate the triple scalar products $\text{v}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{w}}\right)$ and $\text{w}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right),$ where $\text{u}=⟨1,1,1⟩,$ $\text{v}=⟨7,6,9⟩,$ and $\text{w}=⟨4,2,7⟩.$

$\text{v}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)=-1,$ $\text{w}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=1$

Calculate the triple scalar products $\text{w}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}\right)$ and $\text{u}·\left(\mathbf{\text{w}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right),$ where $\text{u}=⟨4,2,-1⟩,$ $\text{v}=⟨2,5,-3⟩,$ and $\text{w}=⟨9,5,-10⟩.$

Find vectors $\text{a},\text{b},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{c}$ with a triple scalar product given by the determinant

$|\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 0\hfill & 2\hfill & 5\hfill \\ 8\hfill & 9\hfill & 2\hfill \end{array}|.$ Determine their triple scalar product.

$\text{a}=⟨1,2,3⟩,$ $\mathbf{\text{b}}=⟨0,2,5⟩,$ $\mathbf{\text{c}}=⟨8,9,2⟩;$ $\text{a}·\left(\text{b}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{c}\right)=-9$

The triple scalar product of vectors $\text{a},\text{b},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{c}$ is given by the determinant

$|\begin{array}{ccc}\hfill 0& \hfill -2& \hfill 1\\ \hfill 0& \hfill 1& \hfill 4\\ \hfill 1& \hfill -3& \hfill 7\end{array}|.$ Find vector $\text{a}-\mathbf{\text{b}}+\mathbf{\text{c}}.$

Consider the parallelepiped with edges $OA,OB,$ and $OC,$ where $A\left(2,1,0\right),B\left(1,2,0\right),$ and $C\left(0,1,\alpha \right).$

1. Find the real number $\alpha >0$ such that the volume of the parallelepiped is $3$ units 3 .
2. For $\alpha =1,$ find the height $h$ from vertex $C$ of the parallelepiped. Sketch the parallelepiped.

a. $\alpha =1;$ b. $h=1,$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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?