vector product in general vector spaces
The vector product can be defined in any finite dimensional vector space^{} $V$ with $dimV=n$. Let ${v}_{1},\mathrm{\dots},{v}_{n}$ be a basis of $V$, we then define the vector product of the vectors ${w}_{1},\mathrm{\dots},{w}_{n1}$ in the following way:
$${w}_{1}\times \mathrm{\dots}\times {w}_{n1}=\sum _{j=1}^{n}{v}_{j}det({w}_{1},\mathrm{\dots},{w}_{n1},{v}_{j}).$$ 
One can easily see that some of the properties of the vector product are the same as in ${\mathbb{R}}^{3}$:

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If one of the ${w}_{i}$ is equal to $0$, then the vector product is $0$.

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If ${w}_{i}$ are linearly dependent, then the vector product is $0$.

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In a Euclidean vector space ${w}_{1}\times \mathrm{\dots}\times {w}_{n1}$ is perpendicular^{} to all ${w}_{i}$.
Title  vector product in general vector spaces 

Canonical name  VectorProductInGeneralVectorSpaces 
Date of creation  20130322 14:32:32 
Last modified on  20130322 14:32:32 
Owner  mathwizard (128) 
Last modified by  mathwizard (128) 
Numerical id  5 
Author  mathwizard (128) 
Entry type  Definition 
Classification  msc 15A72 
Synonym  vector product 