Exploring Wedge Products of Vectors

In summary, the wedge product is a product operation on two alternating tensors that yields another alternating tensor. However, it is also often used for two vectors because vectors are isomorphic to first-order alternating tensors. This is similar to how we call vectors vectors even though they are really first-order tensors. A first-order tensor can act as a map from the 0-order tensor space to a vector space or vice versa, allowing us to use the wedge product for vectors.
  • #1
JG89
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I've learned that the wedge product is a product operation on two alternating tensors that yields another alternating tensor, but sometimes while surfing the net I see people using the wedge product for two vectors. For example, on the wiki page titled "Exterior algebra" it says that "using the standard basis [itex] \{ e_1, e_2, e_3 \} [/itex], the wedge product of a pair of vectors u and v is ..." (the result is an alternating tensor, which seems correct)

I didn't write out the formula because it's irrelevant. My question is, why are they computing the wedge product of two vectors if the wedge product is defined on the set of alternating tensors?
 
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  • #2
Hi JG89! :wink:
JG89 said:
… why are they computing the wedge product of two vectors if the wedge product is defined on the set of alternating tensors?

Same as "why do we call vectors vectors when they're really first-order tensors" and "why do we call angular momentum a vector when it's really a pseuodvector".

Vectors are isomorphic to first-order alternating tensors …

so if it looks like a vector, quacks like a vector, and is isomorphic to a vector,

then let''s call it a vector! :smile:
 
  • #3
tiny-tim said:
so if it looks like a vector, quacks like a vector, and is isomorphic to a vector,

then let''s call it a vector! :smile:
Brilliant line! :rofl: :approve:
 
  • #4
A first-order tensor over a vector space V would be a linear transformation from V to the set of real numbers. Suppose [itex] V = \mathbb{R}^n [/itex]. I don't see how a vector in [itex] \mathbb{R}^n [/itex] would be associated in any natural way with a linear transformation from [itex] \mathbb{R}^n [/itex] to [itex] \mathbb{R} [/itex]
 
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  • #5
JG89 said:
A first-order tensor over a vector space V would be a linear transformation from V to the set of real numbers. Suppose [itex] V = \mathbb{R}^n [/itex]. I don't see how a vector in [itex] \mathbb{R}^n [/itex] would be associated with a linear transformation from [itex] \mathbb{R}^n [/itex] to [itex] \mathbb{R} [/itex]
In actuality, a first order tensor can act as a map from the 0-order tensor space to a vector space OR as a map from a vector space to the 0-order tensor space. For example, let [itex]\boldsymbol{a}\in\mathbb{R}^n[/itex] (i.e. a first order tensor). Then if [itex]b\in\mathbb{R}[/itex] (i.e. 0-order tensor), then

[tex]\begin{aligned}\boldsymbol{a} & :\mathbb{R}\mapsto\mathbb{R}^n\\&:b\mapsto b\boldsymbol{a}\;.\end{aligned}[/tex]

Equally, let [itex]\boldsymbol{a}\in\mathbb{R}^n[/itex] (i.e. a first order tensor). Then if [itex]\mathbb{b}\in\mathbb{R}^n[/itex] (i.e. 0-order tensor), then

[tex]\begin{aligned}\boldsymbol{a} & :\mathbb{R}^n\mapsto\mathbb{R}\\&:b\mapsto \boldsymbol{a}\cdot\boldsymbol{b}\;.\end{aligned}[/tex]
 
  • #6
So suppose u and v are two vectors in [itex] \mathbb{R}^n [/itex], then we can think of both u and v as the 1-tensors defined by u(x) = xu and v(x) = xv where x is a real number. So when we talk about the wedge product of the vectors u and v, then I can use my good old definition given in my book for the wedge product of the 1-tensors u and v, correct?
 

1. What is the wedge product of vectors?

The wedge product, also known as the exterior product, is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both of the original vectors. It is used in geometric algebra to represent the area spanned by two vectors in a higher-dimensional space.

2. How is the wedge product different from the dot product?

The dot product is a scalar operation that produces a single number by multiplying the lengths of two vectors and the cosine of the angle between them. The wedge product, on the other hand, is a vector operation that produces a new vector. It is more useful for representing areas and volumes, while the dot product is used for finding angles and projections.

3. What are some applications of the wedge product in science and engineering?

The wedge product is used in physics for calculating the magnetic field and in engineering for calculating moments of inertia and cross-sectional areas. It is also used in computer graphics for calculating lighting and shading effects.

4. How is the wedge product calculated?

The wedge product can be calculated by taking the cross product of two vectors and then multiplying it by the sine of the angle between them. This can also be represented by a determinant or matrix multiplication. The result is a new vector that is perpendicular to both of the original vectors.

5. Are there any limitations to using the wedge product?

One limitation is that the wedge product is only defined in three-dimensional space or higher. It cannot be used in two-dimensional space. Additionally, the wedge product is not commutative, meaning that the order of the vectors matters. Finally, the wedge product can only be used with vectors, not with scalars or other types of mathematical objects.

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