Question

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

• A. $2\sin \theta .\cos \left(\frac{\theta }{2}\right)$
• B. $2\cos \theta .\sin \left(\frac{\theta }{2}\right)$
• C. $2{\cos }^2\theta .\sin \theta$
• D. $2\cos \left(\frac{\theta }{2}\right).{\sin }^2\theta$

Answer 1

The correct option is B $2\cos \theta .\sin \left(\frac{\theta }{2}\right)$

$|(\overrightarrow{p}.\overrightarrow{r})\overrightarrow{q}-(\overrightarrow{p}.\overrightarrow{q})\overrightarrow{r}|=\cos \theta |\overrightarrow{q}-\overrightarrow{r}|$

$=\cos \theta \times \sqrt{1+1-2\cos \theta }$

$=(2\cos \theta .\sin \frac{\theta }{2})$