Question

Given unit vectors $\overline{m},\overline{n}$ and $\overline{p}$ such that angle between $\overline{m}$ and $\overline{n}=$ angle between $\overline{p}$ and $(\overline{m}\times \overline{n})=\pi /6$ then $\left[\overline{n}\overline{p}\overline{m}\right]=$

• A. $\sqrt{3}/4$
• B. 3/4
• C. 1/4
• D. none

Answer 1

The correct option is A $\sqrt{3}/4$

As all vectors are unit vector so $|\overrightarrow{m}|=|\overrightarrow{n}|=|\overrightarrow{p}|=1$

Thus, $\left[\overrightarrow{n}\overrightarrow{p}\overrightarrow{m}\right]=\overrightarrow{n}.\overrightarrow{p}\times \overrightarrow{m}$

$=\overrightarrow{p}.(\overrightarrow{m}\times \overrightarrow{n})$

$=|\overrightarrow{p}||\overrightarrow{m}\times \overrightarrow{n}|cos\left(\pi /6\right)$

$=\frac{\sqrt{3}}{2}|\overrightarrow{m}\times \overrightarrow{n}|=\frac{\sqrt{3}}{2}|\overrightarrow{m}||\overrightarrow{n}|\sin \left(\pi /6\right)=\frac{\sqrt{3}}{2}\times \frac{1}{2}=\frac{\sqrt{3}}{4}$