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Question

If a+b+c=αd,b+c+d=βa and a, b, c are non-coplanar vectors, then show the a+b+c+d=0.

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Solution

Since,a+b+c=αdand,b+c+d=βaThisimplies,a+b+c+d=(α+1)d=(β+1)aSince,α±1d=(β+1α+1)aa+b+c=αd=α(β+1α+1)a(1α(β+1)α+1)a+b+c=aThisimpliesa+b+carecoplaner,WhichiscontradictionHence,α=1a+b+c+d=0

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