If a→⋅b→=β and a→×b→=c→, then b→ is

a→×b→=c→or a→×(a→×b→)=a→×c→or (a→⋅b→)a→−|a→|2b→=a→×c→or b→=βa→−a→×c→|a→|2 (∵a→⋅b→=β)

If a→⋅b→=β and a→×b→=c→, then b→ is

a→×b→=c→or a→×(a→×b→)=a→×c→or (a→⋅b→)a→−|a→|2b→=a→×c→or b→=βa→−a→×c→|a→|2 (∵a→⋅b→=β)

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If $\vec{a} \cdot \vec{b} = β and \vec{a} \times \vec{b} = \vec{c}, then \vec{b}$ is

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$\frac{(β \vec{a} - \vec{a} \times \vec{c})}{| \vec{a} |^{2}}$

$\frac{(β \vec{a} + \vec{a} \times \vec{c})}{| \vec{a} |^{2}}$

$\frac{(β \vec{c} - \vec{a} \times \vec{c})}{| \vec{a} |^{2}}$

answer is A.

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Detailed Solution

$\vec{a} \times \vec{b} = \vec{c}$

$\begin{array}{l} or \vec{a} \times (\vec{a} \times \vec{b}) = \vec{a} \times \vec{c} \\ or (\vec{a} \cdot \vec{b}) \vec{a} - | \vec{a} |^{2} \vec{b} = \vec{a} \times \vec{c} \\ or \vec{b} = \frac{β \vec{a} - \vec{a} \times \vec{c}}{| \vec{a} |^{2}} (∵ \vec{a} \cdot \vec{b} = β) \end{array}$