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| ``Re: A (better?) definition''
by stevecheng on 2005-07-15 12:10:15 |
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| Reading your insightful comment and after thinking hard about it for a while,
I managed to devise a definition
(although I haven't seen a reference source on such "cross products",
so I would appreciate it if you or anyone else can point me to one or
verify what I have is correct):
Let $V$ be a $n$-dimensional real inner product space with a given orientation,
and $\omega$ denote the associated volume form (a $n$-form).
There is a natural embedding $\hat{\cdot}\colon V \to V^*$ (from vectors to $1$-forms) given by $\hat{v}(w) = \langle v, w \rangle$.
The ``cross product'' of $x_1, \dotsc, x_k \in V$ is defined
to be the $n-k$-form $\theta$ such that
$$
\theta(y_1, \dotsc, y_{n-k}) = \omega(x_1, \dotsc, x_k, y_1, \dotsc, y_{n-k})
$$
for all $y_1, \dotsc, y_{n-k} \in V$.
Then Theorem:
There exists $x_{k+1}, \dotsc, x_n \in V$ mutually orthogonal and orthogonal to
$x_1, \dotsc, x_k$, unique except for ordering and scaling, such that
$$
\theta = \lambda \hat{x_{k+1}} \wedge \dotsb \wedge \hat{x_n}
$$
for some scalar $\lambda$.
So this constructs the orthogonal spaces as you hinted.
In the case of $k = n-1$, the wedge product drops out,
and $\lambda x_n$ simply becomes the cross product definition
I gave at first. (Good, because I rather avoid wedge products for
such an elementary thing :)
Should we add a definition like this?
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