What is the relationship between wedge product and cross product?

  • Thread starter Thread starter closet mathemetician
  • Start date Start date
  • Tags Tags
    Product Wedge
Click For Summary
The wedge product and cross product are both antisymmetric operations on vectors, but they yield different mathematical objects: the wedge product results in a bivector, while the cross product produces a pseudo vector. The wedge product is defined for any vector space, existing in the space of 2-forms, denoted /\^2(V). In three-dimensional space, the bivector space is isomorphic to the vector space, leading to the identification of the wedge product with the cross product. This identification is non-canonical, meaning it depends on the specific context of three dimensions. Understanding these distinctions is crucial for grasping their applications in geometry and physics.
closet mathemetician
Messages
44
Reaction score
0
What's the difference between a wedge product and a cross product?
 
Mathematics news on Phys.org
Although they are both antisymmetric in their arguments,
the wedge product of two vectors is a bivector (a 2-index tensor);
the cross product of two vectors is another [psuedo] vector.
 
Pretty much it's just down to how you view these things.

x/\y is always defined for all x,y in any vector space, they just live in the space /\^2(V). It so happens that in the case when dim(V)=3, then /\^2(V) is (non-canonically) isomorphic to V, so people identify them, and call the resulting thing the cross product.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
View full post »

Similar threads

I Wedge product of basis vectors
Replies
4
Views
2K
I What criteria identifies a math operation as a "product"?
Replies
14
Views
2K
B Confusion about the angle between two vectors in a cross product
Replies
7
Views
3K
I Vector fields wedge product vs covector field
Replies
11
Views
2K
B Orientation of double integrals
Replies
13
Views
2K
I Help with Differential Forms - self wedge product terms
Replies
13
Views
4K
I Dot product, inner product, and projections
Replies
7
Views
2K
I Question about the vector cross product in spherical or cylindrical coordinates
Replies
5
Views
5K
I Question regarding the 3-form ##dx^i \wedge dx^j \wedge dx^k##
Replies
13
Views
3K
No Cross Product in higher dimensions?
Replies
5
Views
3K