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  • Calculate the cross product of two given vectors.
  • Use determinants to calculate a cross product.
  • Find a vector orthogonal to two given vectors.
  • Determine areas and volumes by using the cross product.
  • Calculate the torque of a given force and position vector.

Imagine a mechanic turning a wrench to tighten a bolt. The mechanic applies a force at the end of the wrench. This creates rotation, or torque, which tightens the bolt. We can use vectors to represent the force applied by the mechanic, and the distance (radius) from the bolt to the end of the wrench. Then, we can represent torque by a vector oriented along the axis of rotation. Note that the torque vector is orthogonal to both the force vector and the radius vector.

In this section, we develop an operation called the cross product, which allows us to find a vector orthogonal to two given vectors. Calculating torque is an important application of cross products, and we examine torque in more detail later in the section.

The cross product and its properties

The dot product is a multiplication of two vectors that results in a scalar. In this section, we introduce a product of two vectors that generates a third vector orthogonal to the first two. Consider how we might find such a vector. Let u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 be nonzero vectors. We want to find a vector w = w 1 , w 2 , w 3 orthogonal to both u and v —that is, we want to find w such that u · w = 0 and v · w = 0 . Therefore, w 1 , w 2 , and w 3 must satisfy

u 1 w 1 + u 2 w 2 + u 3 w 3 = 0 v 1 w 1 + v 2 w 2 + v 3 w 3 = 0.

If we multiply the top equation by v 3 and the bottom equation by u 3 and subtract, we can eliminate the variable w 3 , which gives

( u 1 v 3 v 1 u 3 ) w 1 + ( u 2 v 3 v 2 u 3 ) w 2 = 0 .

If we select

w 1 = u 2 v 3 u 3 v 2 w 2 = ( u 1 v 3 u 3 v 1 ) ,

we get a possible solution vector. Substituting these values back into the original equations gives

w 3 = u 1 v 2 u 2 v 1 .

That is, vector

w = u 2 v 3 u 3 v 2 , ( u 1 v 3 u 3 v 1 ) , u 1 v 2 u 2 v 1

is orthogonal to both u and v , which leads us to define the following operation, called the cross product.

Definition

Let u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 . Then, the cross product     u × v is vector

u × v = ( u 2 v 3 u 3 v 2 ) i ( u 1 v 3 u 3 v 1 ) j + ( u 1 v 2 u 2 v 1 ) k = u 2 v 3 u 3 v 2 , ( u 1 v 3 u 3 v 1 ) , u 1 v 2 u 2 v 1 .

From the way we have developed u × v , it should be clear that the cross product is orthogonal to both u and v . However, it never hurts to check. To show that u × v is orthogonal to u , we calculate the dot product of u and u × v .

u · ( u × v ) = u 1 , u 2 , u 3 · u 2 v 3 u 3 v 2 , u 1 v 3 + u 3 v 1 , u 1 v 2 u 2 v 1 = u 1 ( u 2 v 3 u 3 v 2 ) + u 2 ( u 1 v 3 + u 3 v 1 ) + u 3 ( u 1 v 2 u 2 v 1 ) = u 1 u 2 v 3 u 1 u 3 v 2 u 1 u 2 v 3 + u 2 u 3 v 1 + u 1 u 3 v 2 u 2 u 3 v 1 = ( u 1 u 2 v 3 u 1 u 2 v 3 ) + ( u 1 u 3 v 2 + u 1 u 3 v 2 ) + ( u 2 u 3 v 1 u 2 u 3 v 1 ) = 0

In a similar manner, we can show that the cross product is also orthogonal to v .

Finding a cross product

Let p = −1 , 2 , 5 and q = 4 , 0 , −3 ( [link] ). Find p × q .

This figure is the 3-dimensional coordinate system. It has two vectors in standard position. The first vector is labeled “p = <-1, 2, 5>.” The second vector is labeled “q = <4, 0, -3>.”
Finding a cross product to two given vectors.

Substitute the components of the vectors into [link] :

p × q = −1 , 2 , 5 × 4 , 0 , −3 = p 2 q 3 p 3 q 2 , p 1 q 3 p 3 q 1 , p 1 q 2 p 2 q 1 = 2 ( −3 ) 5 ( 0 ) , ( −1 ) ( −3 ) + 5 ( 4 ) , ( −1 ) ( 0 ) 2 ( 4 ) = −6 , 17 , −8 .
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Find p × q for p = 5 , 1 , 2 and q = −2 , 0 , 1 . Express the answer using standard unit vectors.

i 9 j + 2 k

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Although it may not be obvious from [link] , the direction of u × v is given by the right-hand rule. If we hold the right hand out with the fingers pointing in the direction of u , then curl the fingers toward vector v , the thumb points in the direction of the cross product, as shown.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Radek Reply
Practice Key Terms 6

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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