Cross Product in R^n: Defined or Undefined?

In summary, the cross product is not defined in R^n and does not lend itself well to generalizations in other spaces of different dimensions. However, an analogous product does exist in R^7. The easiest route to showing the equivalence of the algebraic and geometric definitions of the cross product is to start with the geometric definition and prove distributivity, which in turn can be used to derive the algebraic definition. While proving distributivity is not easy, it is doable. The name "cross product" refers to the simple Euclidean 3D case, but it can be generalized to p-forms on arbitrary manifolds through the use of Hodge dual and wedge product.
  • #1
quasar987
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Simple question: is the cross product defined in R^n ? In my linear algebra textbook, they talk about the dot product in length but don't even mention the cross product.
 
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  • #2
No, in general it is not (and cannot be) defined for [itex]\mathbb{R}^n[/itex]. An analogous product does exist in [itex]\mathbb{R}^7[/itex], though, constructed using the multiplication table for octonions.
 
  • #3
No the cross product is only defined in R^3.

It's defintion doesn't lend itself well to generalizations other spaces of different dimenion (especially if you want a binary operation). Of course thta's not to say that generalizations are impossible.
[tex]a\times b = \left|\begin{array}{ccc}\hat{x}&\hat{y}&\hat{z}\\a_x&a_y&a_z\\b_x&b_y&b_z\end{array}\right|[/tex]

Which matrix would you take the determinant of when n is not equal to 3?
 
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  • #4
What is the easiest route to showing the equivalence of the algebraic and geometric definitions of the cross product?
 
  • #5
That's a fairly difficult problem in general. Usually they start with the geometric definition, then show that the cross-product is distributive: A X (B+C)=(A X B)+(A X C)
Then you can derive the algebraic definition by writing the vectors out in components and use distributivity. Proving distributivity is not very easy, but certainly doable.
 
  • #6
The name "cross product" refers to the simple Euclidean 3D case.But since this "cross product" is nothing but a Hodge dual of a wedge product between two 1-forms,going to p-forms on arbitrary manifolds gives you the desired generalization...



Daniel.
 
  • #7
Galileo said:
That's a fairly difficult problem in general. Usually they start with the geometric definition, then show that the cross-product is distributive: A X (B+C)=(A X B)+(A X C)
Then you can derive the algebraic definition by writing the vectors out in components and use distributivity. Proving distributivity is not very easy, but certainly doable.
That's exactly what I was trying. And I think I'm on the right track to proving distributivity. :cool:
 
  • #8
quasar987 said:
That's exactly what I was trying. And I think I'm on the right track to proving distributivity. :cool:
Good luck :biggrin:
 

What is the cross product in R^n?

The cross product in R^n is a vector operation that takes two vectors as inputs and produces a new vector that is perpendicular to both of the original vectors. It is denoted as a x b and can only be performed on vectors in three-dimensional space (R^3).

How is the cross product defined?

The cross product is defined as the determinant of a 3x3 matrix, where the first row consists of the unit vectors i, j, and k, the second row consists of the components of the first vector, and the third row consists of the components of the second vector. The resulting vector is the cross product of the two original vectors.

Is the cross product defined for all vectors in R^n?

No, the cross product is only defined for vectors in three-dimensional space (R^3). This is because the determinant of a 3x3 matrix can only be calculated in R^3.

What is the geometric interpretation of the cross product?

The cross product has a geometric interpretation as the area of the parallelogram formed by the two original vectors. The direction of the resulting vector is perpendicular to the plane formed by the two original vectors, according to the right-hand rule.

When is the cross product undefined?

The cross product is undefined when the two original vectors are parallel or antiparallel. This is because the area of the parallelogram formed by parallel or antiparallel vectors is equal to 0, and therefore the cross product cannot be calculated.

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