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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``A (better?) definition''
by stevecheng on 2005-07-13 18:29:38 |
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| Hello,
I think the following definition is slightly better than the current one (though equivalent for orthnormal {v_i})
For a n-dimensional inner product space,
the cross product $z = v_1 \times \dotm \times v_{n-1}$
is the vector such that
$\langle z, w \rangle = \det(v_1, \dotsb, v_{n-1}, w$
for all vectors $w$.
The angle $z$ exists and is unique by the (finite-dimensional) Riesz Representation Theorem. Or it could be constructed simply by setting $w = e_1, e_2, \dotsc$
(i.e. the basis vectors) successively.
By the way, this comes from Spivak's Calculus on Manifolds book.
I think this definition is better because it avoids the use of a basis representation,
and most of the properties for the cross product that have to be painstakingly
proven for R^3 become almost trivial with this definition.
(e.g. the triple product formula, invariance under rotations and under change
of bases of the same orientation, etc.)
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