Question

Let $\overline{v}$ be a unit vector which follows the equation, $\overline{v}\times \overline{b}=\overline{c}$. Also, $\left|\overline{b}\right|=2$ and $\left|\overline{c}\right|=\sqrt{3}$ then

• A. $\overline{v}=-\overline{b}+\overline{b}\times \overline{c}$
• B. $\overline{v}=\frac{3}{4}(\overline{b}+2\overline{b}\times \overline{c})$
• C. $\overline{v}=\frac{1}{4}(\overline{b}+\overline{b}\times \overline{c})$
• D. $\overline{v}=\frac{3}{4}(\overline{b}+\overline{b}\times \overline{c})$

Answer 1

The correct option is C $\overline{v}=\frac{1}{4}(\overline{b}+\overline{b}\times \overline{c})$

$\overline{b}\times (\overline{v}\times \overline{b})=\overline{b}\times \overline{c}=(\overline{b}.\overline{b})\overline{v}-(\overline{b}.\overline{v})\overline{b}=\overline{b}\times \overline{c}$
$4\overline{v}-(\overline{b}.\overline{v})\overline{b}=\overline{b}\times \overline{c}$
$|\overline{v}\times \overline{b}|=\left|\overline{c}\right|$
$\Rightarrow \left|\overline{v}\right|\left|\overline{b}\right|sin\theta =\left|\overline{c}\right|$
$2\left|\overline{v}\right|sin\theta =\sqrt{3}$
$sin\theta =\frac{\sqrt{3}}{2}$
$cos\theta =\frac{1}{2}$
$\Rightarrow 4\overline{v}-(2\times 1\times \frac{1}{2})\overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}$
$\overline{v}=\frac{1}{4}(\overline{b}+\overline{b}\times \overline{c})$