|
|
|
Revision difference : vector product in general vector spaces |
| Version current |
Version 1 |
| 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: |
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: |
| $$w_1\times\dots\times w_{n-1}=\sum_{j=1}^nv_j\det(w_1,\dots,w_{n-1},v_j).$$ |
$$w_1\times\dots\times w_{n-1}=\sum_{j=1}^nv_j\det(w_1,\dots,w_{n-1},v_j).$$ |
| One can easily see that some of the properties of the vector product are the same as in $\mathbb{R}^3$: |
|
| \begin{itemize} |
|
| \item If one of the $w_i$ is equal to $0$, then the vector product is $0$. |
|
| \item If $w_i$ are linearly dependent, then the vector product is $0$. |
|
| \item In a Euclidean vector space $w_1\times\dots\times w_{n-1}$ is perpendicular to all $w_i$. |
|
| \end{itemize} |
|
|
|
|
|
