What is the relationship between wedge product and cross product?

  • Thread starter Thread starter closet mathemetician
  • Start date Start date
  • Tags Tags
    Product Wedge
AI Thread 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 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Back
Top