# vector product in general vector spaces

The vector product can be defined in any finite dimensional vector space $V$ with $\dim V=n$. Let $v_{1},\dots,v_{n}$ be a basis of $V$, we then define the vector product of the vectors $w_{1},\dots,w_{n-1}$ in the following way:

 $w_{1}\times\dots\times w_{n-1}=\sum_{j=1}^{n}v_{j}\det(w_{1},\dots,w_{n-1},v_{% j}).$

One can easily see that some of the properties of the vector product are the same as in $\mathbb{R}^{3}$:

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

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

• In a Euclidean vector space $w_{1}\times\dots\times w_{n-1}$ is perpendicular to all $w_{i}$.

Title vector product in general vector spaces VectorProductInGeneralVectorSpaces 2013-03-22 14:32:32 2013-03-22 14:32:32 mathwizard (128) mathwizard (128) 5 mathwizard (128) Definition msc 15A72 vector product