Question

Let $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ be three non-coplanar unit vectors equally inclined to each other at an angle of $\frac{\pi }{3}$. Then the value of $|\overrightarrow{p}\times (\overrightarrow{q}\times \overrightarrow{r})|$ is

• A. $3$
• B. $\frac{1}{2}$
• C. $\frac{\sqrt{3}}{4}$
• D. $\frac{3\sqrt{3}}{4}$

Answer 1

The correct option is B $\frac{1}{2}$

$|\overrightarrow{p}|=|\overrightarrow{q}|=|\overrightarrow{r}|=1$

$|\overrightarrow{p}\times (\overrightarrow{q}\times \overrightarrow{r})|$
$=|(\overrightarrow{p}.\overrightarrow{r})\overrightarrow{q}-(\overrightarrow{p}.\overrightarrow{q})\overrightarrow{r}|$
$=||\overrightarrow{p}||\overrightarrow{r}|\cos \left(\frac{\pi }{3}\right)\overrightarrow{q}-|\overrightarrow{p}||\overrightarrow{q}|\cos \left(\frac{\pi }{3}\right)\overrightarrow{r}|$
$=\cos \left(\frac{\pi }{3}\right)|\overrightarrow{q}-\overrightarrow{r}|=\cos \left(\frac{\pi }{3}\right)\times \sqrt{(\overrightarrow{q}-\overrightarrow{r})\cdot (\overrightarrow{q}-\overrightarrow{r})}=\frac{1}{2}\times \sqrt{1+1-2\cos \left(\frac{\pi }{3}\right)}=\frac{1}{2}\times \sqrt{1+1-1}=\frac{1}{2}$