On 14 Aug 2009, at 22:53, dherring@ll.mit.edu wrote:
On Fri, 14 Aug 2009, Edward Grace wrote:
On 14 Aug 2009, at 18:43, DE wrote:
w = wedge(u,v);
w = cross_product(u,v) please :p
Err, that's only true in 3D (vectors of length 3). There's no such thing as a cross product between (say) vectors of length 2,4 or indeed anything else. The cross product is, in effect, a restricted version of the exterior (wedge) product which exists for higher dimensions.
Actually, it was developed the other way; the exterior product is one possible generalization of the cross product to higher dimensions.
Mea culpa - you're right of course. Maths history often seems to get 'renormalised' like this. Like the unit vectors being i,j,k when these characters originally (Hamilton) referred to the imaginary numbers of the quaternion. Then quaternions fell out of fashion and vector calculus took over so now everyone thinks i,j,k are and were always just the unit vectors of R3. Talk to an algebraist and they'll say - "oh, quaternions <insert dismissive sneer here> just a type of Clifford Algebra" - so the competition continues!
The cross product is unique in R3, but other generalizations are possible. IMO, cross_product(u,v) should still be provided for R3, even if it is simply an inline call to exterior_product(u,v).
Once one has gone and made things more complicated they can be simplified again and one can pretend it was always that way! That's how the game works no? ;-) -ed