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Question

If r=αb×c+βc×a+γa×b and [abc]=2 , then α+β+γ=

A
r.(b×c+c×a+a×r)
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B
12r(a+b+c)
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C
2r(a+b+c)
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D
4
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Solution

The correct option is B 12r(a+b+c)
We have r=αb×c+βc×a+γa×b
r.a=αb×c.a+βc×a.a+γa.×b.a
r.a=αb×c.a+0+0
r.a=αb×c.a
=α[abc]
=2α from the question
We have r=αb×c+βc×a+γa+b
r.b=αb×c.b+βc×a.b+γa+b.b
r.b=0+βc×a.b+0
=β[abc]
=2β
We have r=αb×c+βc×a+γa×b
r.c=αb×c.c+βc×a.c+γa×b.c
r.c=0+0+γa×b.c
=γ[abc]
=2γ
Adding α+β+γ=12[r.a+r.b+r.c]
=12r(a+b+c)

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