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Question

Vectors A and B satisfying the vector equation A+B=a, A×B=b and Aa=1, where a and b are given vectors, are

A
A=(a×b)a|a|2
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B
B=(b×a)+a(|a|21)|a|2
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C
A=(a×b)+a|a|2
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D
B=(b×a)a(|a|21)|a|2
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Solution

The correct options are
B B=(b×a)+a(|a|21)|a|2
C A=(a×b)+a|a|2
We have A+B=a
or Aa+Ba=aa
or 1+Ba=a2 (given Aa=1)
or Ba=a21 (i)
Also, A×B=b
or a×(A×B)=a×b
or (aB)A(aA)B=a×b
or (a21)AB=a×b (ii)
(using (i) and aA=1)
and A+B=a (iii)
From (ii) and (iii), we have
A=(a×b)+aa2
B=a(a×b)+aa2
B=(b×a)+a(a21)a2

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