Understanding the Relationship Between Exterior and Cross Products

[dimension10]

I have a question about the exterior product. Is it true that

$$\mathbf{a}\wedge\mathbf{b}=|| \mathbf{a}\times \mathbf{b}||$$

If not, how does one relate the exterior product to the cross product?

Thanks in advance.

 

[homeomorphic]

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]

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.

Thanks.

I think I was confused by the picture in Wikipedia which makes it seem like the exterior product is the area of the parallelogram. This time, I noticed that there were 2 arrows around it, making it a bivector. So am I right to say that the magnitude of the exterior product is the same as the magnitude of the cross product whereas the exterior product itself is not the same as the magnitude?

Thanks.

 

[chiro]

I have a question about the exterior product. Is it true that

$$\mathbf{a}\wedge\mathbf{b}=|| \mathbf{a}\times \mathbf{b}||$$

If not, how does one relate the exterior product to the cross product?

Thanks in advance.

Hey dimension10.

Are you familiar with the determinant form of the exterior product?

 

[dimension10]

Hey dimension10.

Are you familiar with the determinant form of the exterior product?

The alternating exterior product?

 

[chiro]

The alternating exterior product?

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]

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).

I think they are.

 

[chiro]

I think they are.

In that case just use the definition of the wedge product (for R3 given two initial vectors) and the cross product and then use the property of norms (i.e. expand out the terms) to show they are equal.