Question

If $\overrightarrow{x}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$ and $\overrightarrow{x}\perp \overrightarrow{a},$ then $\overrightarrow{x}$ is equal to

• A. $\frac{(\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{a}}{\overrightarrow{b}.\overrightarrow{a}}$
• B. $\frac{\overrightarrow{b}\times (\overrightarrow{a}\times \overrightarrow{c})}{\overrightarrow{b}.\overrightarrow{c}}$
• C. $\frac{(\overrightarrow{c}\times \overrightarrow{b})\times \overrightarrow{a}}{\overrightarrow{a}.\overrightarrow{b}}$
• D. none of these.

Answer 1

Answer: (A)

$\vec{x}\times \vec{b}=\vec{c}\times \vec{b}$
$\Rightarrow \vec{a}\times \left(\vec{x}\times \vec{b}\right)=\vec{a}\times \left(\vec{c}\times \vec{b}\right)$
$\Rightarrow \left(\vec{a}\cdot \vec{b}\right)\vec{x}-\left(\vec{a}\cdot \vec{x}\right)\vec{b}$
$=-\left(\vec{c}\times \vec{b}\right)\times \vec{a}$
$=\left(\vec{b}\times \vec{c}\right)\times \vec{a}$
$\Rightarrow \left(\vec{a}\cdot \vec{b}\right)\vec{x}=\left(\vec{b}\times \vec{c}\right)\times \vec{a}[\because \vec{a}.\vec{x}$ as $=0\vec{x}\perp \vec{a}]$
$\Rightarrow \vec{x}=\frac{\left(\vec{b}\times \vec{c}\right)\times \vec{a}}{\vec{a}.\vec{b}}=\frac{\left(\vec{b}\times \vec{c}\right)\times \vec{a}}{\vec{b}.\vec{a}}$