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Question

Given $$A+B = a, A\times B = b, A\cdot a =1$$ Find the vector $$B$$.


A
(b×a)+b(a21)a2
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B
(b×a)+b(a21)b2
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C
(b×a)+a(a21)a2
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D
(b×a)+a(a21)b2
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Solution

The correct option is D $$\dfrac{(b\times a)+a(a^2-1)}{a^2}$$
Given 
$$A+B=a$$-------(1)
$$A\times B=b$$------(2)
$$A\cdot a=1$$------(3)
cross product w.r to B
$$A\times B+B\times B=a\times B$$
$$b=a\times B$$    from eq (2)
cross product w.r to a
$$b\times a=(a\times B)\times a$$
$$b\times a=(a\cdot a)B-(a\cdot B)a$$
$$b\times a+(a\cdot B)a=a^2B$$
putting value of B from eq (1)
$$b\times a+(a\cdot (a-A))a=a^2B$$
$$b\times a+(a^2-A\cdot a)a=a^2B$$
$$b\times a+(a^2-1)a=a^2B$$
$$a^2B=b\times a+a(a^2-1)$$
$$B=\dfrac{b\times a+a(a^2-1)}{a^2}$$



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