Vectors →A and →B satisfying the vector equation →A+→B=→a,→A×→B=→b and →A⋅→a=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|2−1)|→a|2
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C
→A=(→a×→b)+→a|→a|2
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D
→B=(→b×→a)−→a(|→a|2−1)|→a|2
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Solution
The correct options are B→B=(→b×→a)+→a(|→a|2−1)|→a|2 C→A=(→a×→b)+→a|→a|2 We have →A+→B=→a or →A⋅→a+→B⋅→a=→a⋅→a or 1+→B⋅→a=a2 (given →A⋅→a=1) or →B⋅→a=a2−1(i) Also, →A×→B=→b or →a×(→A×→B)=→a×→b or (→a⋅→B)→A−(→a⋅→A)→B=→a×→b or (a2−1)→A−→B=→a×→b(ii) (using (i) and →a⋅→A=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(a2−1)a2