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Question

If ¯¯¯a,¯¯b,¯¯c are coplanar, prove that
(¯¯¯a+¯¯b).[((¯¯b+¯¯c)ׯ¯¯a)+((¯¯b+¯¯¯a)ׯ¯b)]=0

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Solution

asa,b,carecoplanar,hencea(b×c)=0
nowLHS=(a+b)[(b×a)+(c×a)+(b×b)+(a×b)]
=(a+b)[(b×a)+(c×a)+(a×b)]=a.(b×a)+a.(c×a)+a.(a×b)+b.(b×a)+b.(c×a)+b.(a×b)
=0+0+0+0+0+0=0

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