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Question

For any three vectors a,b and c, prove that vectors ab,bc and ca are coplanar.

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Solution

If 3 vectors say A,B,C are coplaner.
[ABC]=0
=[abbcca]
=(ab).[(bc)×(ca)]
=(ab).[b×cb×ac×c+c×a]
=(ab).[b×ca×b0+c×a]
=(a.b×c)+a.a×b+a.a×cb.b×cb.a×bb.c×a
=[abc]+[aab]+[aca][bbc][bab][bca]
=[abc]=[bca]=[cab]
=[abc][abc]=0
ab,bc,ca are co planer.

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