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Question

If a+b+c=0, the prove that: a×b=b×c=c×a where a,b,c are non-zero vectors.

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Solution

Given a+b+c=0 ......(1)
Multiplying a in both the sides.
a×(a+b+c)=a×0
a×a+a×b+a×c=0 [a×a=0
a×bc×a=0 [a×c=c×a]
a×b=c×a .......(2)
Multiplying b in both the sides of equation (1) we have,
b×(a+b+c)=b×0
b×a+b×b+b×c=0
b×a+b×c=0 [b×b=0]
a×b+b×c=0 [b×a=a×b]
a×b=b×c .......(3)
By equation (2) and (3) we get,
a×b=b×c=c×a.

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