homeomorphic said:Presumably, you mean that a and b are vectors in R^3. Wedge product of vectors would be a bivector, so it couldn't possibly be equal to the left hand side.
If you take a wedge b and then the Hodge dual of that with respect to the Euclidean metric, you get the cross product.
Crash course in Hodge duals for this case:
Let e1, e2, e3 be a basis for R^3.
The Hodge dual of e1^ e2 can be written like (e1^e2)* and it will be e3.
And similarly, we have (e2^e3)* = e1 and (e3^e1)* = e2.
Then you can extend by linearity to the vector-space of all bivectors. You can see that this gives you the cross product.
dimension10 said:I have a question about the exterior product. Is it true that
[tex]\mathbf{a}\wedge\mathbf{b}=|| \mathbf{a}\times \mathbf{b}||[/tex]
If not, how does one relate the exterior product to the cross product?
Thanks in advance.
chiro said:Hey dimension10.
Are you familiar with the determinant form of the exterior product?
dimension10 said:The alternating exterior product?
chiro said:Just before I give an answer, I just want to be clear: is the wedge product and the exterior product the same thing? (I was under the impression it was).
dimension10 said:I think they are.
The exterior product, also known as the wedge product, is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both input vectors. The cross product, on the other hand, is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both input vectors and also has a magnitude equal to the area of the parallelogram formed by the two input vectors.
The cross product can be seen as a special case of the exterior product, where the input vectors are restricted to three dimensions and the output vector is further constrained to be perpendicular to both input vectors.
Exterior and cross products are used in various fields of science and engineering, such as physics, engineering, and computer graphics. They are particularly useful in calculating torque, determining the direction of magnetic fields, and creating 3D models in computer graphics.
Exterior and cross products are non-commutative operations, meaning that the order of the input vectors affects the output vector. They also follow the right-hand rule, where the direction of the output vector is determined by curling the fingers of the right hand from the first input vector to the second input vector.
One common misconception is that exterior and cross products are the same thing. As mentioned earlier, they are related, but not identical operations. Another misconception is that the output of the cross product is always a vector perpendicular to the input vectors. This is only true when the input vectors are perpendicular to each other.