# 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. $h=1,$ what is variations in raman spectra for nanomaterials
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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
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Professor
I think
Professor
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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
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What is meant by 'nano scale'?
What is STMs full form?
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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
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 ?
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Kyle
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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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?  By   By  By  By By