Question

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three unit vectors $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{c}$ and angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is ${30}^o$, power that $\overrightarrow{a}=\pm 2(\overrightarrow{b}\times \overrightarrow{c})$.

Answer 1

Given,

$\overline{a},\overline{b},\overline{c}$ are unit vectors,

$\overline{a}$ is perpendicular to both $\overline{b}and\overline{c}$

$\overline{a}$ will be parallel to $\overline{b}\times \overline{c}$

so,$\overline{a}$= $\lambda (\overline{b}\times \overline{c})whereas,\lambda isascalar$

Also given,

angle between $\overline{b}\&\overline{c}$ is ${30}^0$

$\therefore \overline{a}=\lambda \frac{1}{2}$

Applying Modulus on both sides

$\left|\overline{a}\right|=\frac{\left|\lambda \right|}{2}$

$1=\frac{\left|\lambda \right|}{2}$

$\lambda =\pm 2$

therefor, $\overline{a}=\pm 2(\overline{b}\times \overline{c})$.