Question

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a},\hat{b},\hat{c}$ such that $\hat{a}.\hat{b}=\hat{b}\cdot \hat{c}=\hat{c}.\hat{a}=\frac{1}{2}$. Then the volume of the parallelopiped is

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

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

Volume $=|\hat{a}.(\hat{b}\times \hat{c}\left)\right|=\sqrt{\left|\begin{array}{ccc}\hat{a}.\hat{a}& \hat{a}.\hat{b}& \hat{a}.\hat{c}\\ \hat{b}.\hat{a}& \hat{b}.\hat{b}& \hat{b}.\hat{c}\\ \hat{c}.\hat{a}& \hat{c}.\hat{b}& \hat{c}.\hat{c}\end{array}\right|}=\sqrt{\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|}=\frac{1}{\sqrt{2}}$