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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.
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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)

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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 .

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Find ( i × j ) × ( k × i ) .


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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

Is there any normative that regulates the use of silver nanoparticles?
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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