<< Chapter < Page Chapter >> Page >
This figure has two images. The first image has three vectors with the same initial point. Two of the vectors are labeled “u” and “v.” The angle between u and v is theta. The third vector is perpendicular to u and v. It is labeled “u cross v.” The second image has three vectors. The vectors are labeled “u, v, and u cross v.” “u cross v” is perpendicular to u and v. Also, on the image of these three vectors is a right hand. The fingers are in the direction of u. As the hand is closing, the direction of the closing fingers is the direction of v. The thumb is up and in the direction of “u cross v.”
The direction of u × v is determined by the right-hand rule.

Notice what this means for the direction of v × u . If we apply the right-hand rule to v × u , we start with our fingers pointed in the direction of v , then curl our fingers toward the vector u . In this case, the thumb points in the opposite direction of u × v . (Try it!)

Anticommutativity of the cross product

Let u = 0 , 2 , 1 and v = 3 , −1 , 0 . Calculate u × v and v × u and graph them.

This figure is the 3-dimensional coordinate system. It has two vectors in standard position. The first vector is labeled “u = <0, 2, 1>.” The second vector is labeled “v = <3, -1, 0>.”
Are the cross products u × v and v × u in the same direction?

We have

u × v = ( 0 + 1 ) , ( 0 3 ) , ( 0 6 ) = 1 , 3 , −6 v × u = ( −1 0 ) , ( 3 0 ) , ( 6 0 ) = −1 , −3 , 6 .

We see that, in this case, u × v = ( v × u ) ( [link] ). We prove this in general later in this section.

This figure is the 3-dimensional coordinate system. It has two vectors in standard position. The first vector is labeled “u = <0, 2, 1>.” The second vector is labeled “v = <3, -1, 0>.” It also has two vectors that are cross products. The first is “u x v = <1, 3, -6>.” The second is “v x u = <-1, -3, 6>.”
The cross products u × v and v × u are both orthogonal to u and v , but in opposite directions.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Suppose vectors u and v lie in the xy -plane (the z -component of each vector is zero). Now suppose the x - and y -components of u and the y -component of v are all positive, whereas the x -component of v is negative. Assuming the coordinate axes are oriented in the usual positions, in which direction does u × v point?

Up (the positive z -direction)

Got questions? Get instant answers now!

The cross products of the standard unit vectors i , j , and k can be useful for simplifying some calculations, so let’s consider these cross products. A straightforward application of the definition shows that

i × i = j × j = k × k = 0 .

(The cross product of two vectors is a vector, so each of these products results in the zero vector, not the scalar 0 . ) It’s up to you to verify the calculations on your own.

Furthermore, because the cross product of two vectors is orthogonal to each of these vectors, we know that the cross product of i and j is parallel to k . Similarly, the vector product of i and k is parallel to j , and the vector product of j and k is parallel to i . We can use the right-hand rule to determine the direction of each product. Then we have

i × j = k j × i = k j × k = i k × j = i k × i = j i × k = j .

These formulas come in handy later.

Cross product of standard unit vectors

Find i × ( j × k ) .

We know that j × k = i . Therefore, i × ( j × k ) = i × i = 0 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find ( i × j ) × ( k × i ) .


Got questions? Get instant answers now!

As we have seen, the dot product is often called the scalar product because it results in a scalar. The cross product results in a vector, so it is sometimes called the vector product    . These operations are both versions of vector multiplication, but they have very different properties and applications. Let’s explore some properties of the cross product. We prove only a few of them. Proofs of the other properties are left as exercises.

Properties of the cross product

Let u , v , and w be vectors in space, and let c be a scalar.

i. u × v = ( v × u ) Anticommutative property ii. u × ( v + w ) = u × v + u × w Distributive property iii. c ( u × v ) = ( c u ) × v = u × ( c v ) Multiplication by a constant iv. u × 0 = 0 × u = 0 Cross product of the zero vector v. v × v = 0 Cross product of a vector with itself vi. u · ( v × w ) = ( u × v ) · w Scalar triple product


For property i ., we want to show u × v = ( v × u ) . We have

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

Unlike most operations we’ve seen, the cross product is not commutative. This makes sense if we think about the right-hand rule.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?