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In Wikipedia, a cross product between two "vectors" is defined in terms of the angle between the vectors and their magnitudes.

  1. As I learned cross product in linear algebra, which I understand to be a topic about vector space, I now wonder if cross product is not defined on vector space, but instead can only be defined on an inner product space so that the angle between the vectors and their magnitudes can make sense?
  2. Or is there other definition of cross product on vector space?

Thanks and regards!

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3 Answers 3

up vote 11 down vote accepted

One point of view is that the cross product is the composition of the Hodge dual and the exterior product $V \times V \to \Lambda^2 V$ in three dimensions. The Hodge dual requires additional structure to define: you need not only an inner product, but an orientation. This reflects the fact that there is a choice of handedness in the definition of the cross product.

Another point of view is that the cross product is not an operation on spatial vectors in $\mathbb{R}^3$, but the Lie bracket on the Lie algebra of $\text{SO}(3)$. This is the definition, for example, which is relevant to the physics of angular momentum. Of course, if you insist on taking the cross product of spatial vectors you need a way to identify spatial vectors with elements of the Lie algebra of $\text{SO}(3)$. Writing $\text{SO}(3)$ as $\text{Aut}(V)$ where $V$ is a real oriented 3-dimensional inner product space, the Lie algebra of $\text{SO}(3)$ is naturally isomorphic to $\Lambda^2 V$, so this identification is again the Hodge dual.

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Thanks! (1) By the definition of cross product in terms of inner product as in Wikipedia I linked, it seems to me that cross product can be defined on an oriented inner product space with any dimension. Is this correct? Or as Eric mentioned, the definition in terms of inner product can only be applied to dim 1,3 and 7, because of uniqueness of perpendicular direction? (2) Assume the answer to the first question is yes. Is cross product still the composition of Hodge dual and exterior product? is it still the Lie bracket on some Lie algebra? –  Tim Jan 15 '11 at 16:43
    
The exterior product and Hodge dual are defined in any dimension, but only in 3 dimensions will their composition land you back in V. In dimension n, you can take the exterior product of n-1 vectors and the Hodge dual of that; that is an (n-1)-linear analogue of the cross product in any dimension. The 7-dimensional cross product is not a Lie bracket; see en.wikipedia.org/wiki/Seven-dimensional_cross_product . –  Qiaochu Yuan Jan 15 '11 at 17:46
    
Thanks! The link to 7-dim case is nice. Does cross product have to be defined on Euclidean space? Or can it be defined on finite-dimensional oriented inner product space, as the definition on the link to 7-dim case? –  Tim Jan 15 '11 at 20:43
    
@Tim: I'm not sure what you mean. –  Qiaochu Yuan Jan 15 '11 at 21:38
    
At en.wikipedia.org/wiki/…, the definition of cross product is defined on Euclidean space. But it only depends on inner product, so can cross product be defined on finite-dimensional oriented inner product space, not just Euclidean space? Euclidean space is finite-dimensional real inner product space (with or without orienetation), isn't it? –  Tim Jan 15 '11 at 21:42

The question was studied by Grassmann. At the time, people laughed at him, but now you can read all about it in the Wikipedia article on Exterior Algebra.

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You are correct in that you need an inner product to define the cross product. A geometric picture of it is as follows: take two vectors in three dimensions. They determine a parallelogram and the cross product is defined to be the vector perpendicular (with respect to the inner product) to the parallelogram and with magnitude equal to its area.

Note that there is something special in 3-dimensions that allows this definition to work, namely that to any plane there is a unique perpendicular direction. This is not the case in other dimensions so this definition does not generalize. Indeed, as the wikipedia article states, the only other dimension that has an analogous cross product is 7 (dimension 1 also has a cross product but it is trivial-- just negative of the regular product of real numbers). What is special about dimensions 1,3 and 7 is that there are division algebras only in dimensions 2,4, and 8 (the real numbers, the quaternions, and the octonions). The wikipedia article talks more about this construction.

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There are two vectors satisfying the description in your first paragraph. As I mentioned, you need an orientation to pick one. –  Qiaochu Yuan Jan 15 '11 at 14:23
    
Thanks! In your reply, is the definition of cross product between two vectors of 7-dim still the product of their magnitudes, sin of their angle and the perpendicular vector? –  Tim Jan 15 '11 at 17:28
    
@Qiaochu: thanks; i forgot to mention that. –  Eric O. Korman Jan 17 '11 at 22:44
    
@Tim: yes, the magnitude of the cross product is the product of their magnitudes and the sin of the angle between them. And it lies in the 5-dimensional subspace of vectors orthogonal to the plane spanned by the two vectors. –  Eric O. Korman Jan 17 '11 at 22:44

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