Question

If u+v+w=0, show that u x v = v x w = w x u.

Answer 1

Initially I set u=(-1)(v+w), v=(-1)(u+w), w=(-1)(u+v) and then try to just insert those vectors into the thing I'm supposed to show; I don't know how to go from where I am though.

Answer 2

Currently at vxu+vxw+wxu

Answer 3

Quite certain you're supposed to use the fact that some of the vector products are orthogonal to each other and therefore 0

Answer 4

Just arrived at wxu; still haven't shown for vxw though, it seems very chaotic just inserting relations over and over, is there a better way?

Answer 5

So say we have
u=(u1,u2,u3)
v=(v1,v2,v3)
w=(w1,w2,w3)
,then
u1+v1+w1=0
u2+v2+w2=0
u3+v3+w3=0.
uxv=

=i(u2v3-u3v2)-j(u1v3-v1u3)+k(u1v2-u2v2)
SO I guess you could use this to show wxu by writing what I just wrote but your ui's as wi's and your vi's as ui's.
(Then do the same for vxw)
But this seams pretty long to me.
I will have to see if I can find a shorter way.