# 2.4 The cross product  (Page 6/16)

 Page 6 / 16

Note that, as the name indicates, the triple scalar product produces a scalar. The volume formula just presented uses the absolute value of a scalar quantity.

## Proof

The area of the base of the parallelepiped is given by $‖\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}‖.$ The height of the figure is given by $‖{\text{proj}}_{\text{v×w}}\text{u}‖.$ The volume of the parallelepiped is the product of the height and the area of the base, so we have

$\begin{array}{cc}\hfill V& =‖{\text{proj}}_{\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}}\text{u}‖‖\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}‖\hfill \\ & =|\frac{\text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)}{‖\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}‖}|‖\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}‖\hfill \\ & =|\text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)|.\hfill \end{array}$

## Calculating the volume of a parallelepiped

Let $\text{u}=⟨-1,-2,1⟩,\text{v}=⟨4,3,2⟩,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}=⟨0,-5,-2⟩.$ Find the volume of the parallelepiped with adjacent edges $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ ( [link] ).

We have

$\begin{array}{cc}\hfill \text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)& =|\begin{array}{ccc}\hfill -1& \hfill -2& \hfill 1\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill -5& \hfill -2\end{array}|=\left(-1\right)|\begin{array}{cc}\hfill 3& \hfill 2\\ \hfill -5& \hfill -2\end{array}|+2|\begin{array}{cc}\hfill 4& \hfill 2\\ \hfill 0& \hfill -2\end{array}|+|\begin{array}{cc}\hfill 4& \hfill 3\\ \hfill 0& \hfill -5\end{array}|\hfill \\ & =\left(-1\right)\left(-6+10\right)+2\left(-8-0\right)+\left(-20-0\right)\hfill \\ & =-4-16-20\hfill \\ & =-40.\hfill \end{array}$

Thus, the volume of the parallelepiped is $|-40|=40$ units 3 .

Find the volume of the parallelepiped formed by the vectors $\text{a}=3\text{i}+4\text{j}-\text{k},$ $\text{b}=2\text{i}-\text{j}-\text{k},$ and $\text{c}=3\text{j}+\text{k}.$

$8$ units 3

## Applications of the cross product

The cross product appears in many practical applications in mathematics, physics, and engineering. Let’s examine some of these applications here, including the idea of torque, with which we began this section. Other applications show up in later chapters, particularly in our study of vector fields such as gravitational and electromagnetic fields ( Introduction to Vector Calculus ).

## Using the triple scalar product

Use the triple scalar product to show that vectors $\text{u}=⟨2,0,5⟩,\text{v}=⟨2,2,4⟩,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}=⟨1,-1,3⟩$ are coplanar—that is, show that these vectors lie in the same plane.

Start by calculating the triple scalar product to find the volume of the parallelepiped defined by $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}\text{:}$

$\begin{array}{cc}\hfill \text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)& =|\begin{array}{ccc}\hfill 2& \hfill 0& \hfill 5\\ \hfill 2& \hfill 2& \hfill 4\\ \hfill 1& \hfill -1& \hfill 3\end{array}|\hfill \\ & =\left[2\left(2\right)\left(3\right)+\left(0\right)\left(4\right)\left(1\right)+5\left(2\right)\left(-1\right)\right]-\left[5\left(2\right)\left(1\right)+\left(2\right)\left(4\right)\left(-1\right)+\left(0\right)\left(2\right)\left(3\right)\right]\hfill \\ & =2-2\hfill \\ & =0.\hfill \end{array}$

The volume of the parallelepiped is $0$ units 3 , so one of the dimensions must be zero. Therefore, the three vectors all lie in the same plane.

Are the vectors $\text{a}=\text{i}+\text{j}-\text{k},$ $\text{b}=\text{i}-\text{j}+\text{k},$ and $\text{c}=\text{i}+\text{j}+\text{k}$ coplanar?

No, the triple scalar product is $-4\ne 0,$ so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.

## Finding an orthogonal vector

Only a single plane can pass through any set of three noncolinear points. Find a vector orthogonal to the plane containing points $P=\left(9,-3,-2\right),Q=\left(1,3,0\right),$ and $R=\left(-2,5,0\right).$

The plane must contain vectors $\stackrel{\to }{PQ}$ and $\stackrel{\to }{QR}\text{:}$

$\begin{array}{c}\stackrel{\to }{PQ}=⟨1-9,3-\left(-3\right),0-\left(-2\right)⟩=⟨-8,6,2⟩\hfill \\ \stackrel{\to }{QR}=⟨-2-1,5-3,0-0⟩=⟨-3,2,0⟩.\hfill \end{array}$

The cross product $\stackrel{\to }{PQ}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{QR}$ produces a vector orthogonal to both $\stackrel{\to }{PQ}$ and $\stackrel{\to }{QR}.$ Therefore, the cross product is orthogonal to the plane that contains these two vectors:

$\begin{array}{cc}\hfill \stackrel{\to }{PQ}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{QR}& =|\begin{array}{ccc}\hfill \text{i}& \hfill \text{j}& \hfill \text{k}\\ \hfill -8& \hfill 6& \hfill 2\\ \hfill -3& \hfill 2& \hfill 0\end{array}|\hfill \\ & =0\text{i}-6\text{j}-16\text{k}-\left(-18\text{k}+4\text{i}+0\text{j}\right)\hfill \\ & =-4\text{i}-6\text{j}+2\text{k}.\hfill \end{array}$

We have seen how to use the triple scalar product and how to find a vector orthogonal to a plane. Now we apply the cross product to real-world situations.

Sometimes a force causes an object to rotate. For example, turning a screwdriver or a wrench creates this kind of rotational effect, called torque.

## Definition

Torque , $\tau$ (the Greek letter tau ), measures the tendency of a force to produce rotation about an axis of rotation. Let $\text{r}$ be a vector with an initial point located on the axis of rotation and with a terminal point located at the point where the force is applied, and let vector $\text{F}$ represent the force. Then torque is equal to the cross product of $\text{r}$ and $\text{F}\text{:}$

$\tau =\text{r}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{F}.$

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?