If a + b + c = 0 then a cross b is
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a+b+c = 0
a+b = -c
cross product by a on both sides
a×a+b×a=-c×a
b×a = a×c by vector property
now
a+c = -b ........(b)
cross product by a on both sides
a×b+b×b=-c×b
a×b=b×c .........(a) by vector property
henceby comparing equations (a) and (b)
a×b=b×c=c×a
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