Question

The edges of a parallelopiped are of unit length and are parallel to noncoplanar unit vectors $\hat{a},\hat{b},\hat{c}$ such that $\hat{a}\cdot \hat{b}=\hat{b}\cdot \hat{c}=\hat{c}\cdot \hat{a}=\frac{1}{2}$, 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 $\frac{1}{\sqrt{2}}$

The volume of the parallelopiped with coterminus edges as $\hat{a},\hat{b},\hat{c}$ is given by $\left[\hat{a}\hat{b}\hat{c}\right]=\hat{a}.(\hat{b}\times \hat{c})$

Now

$\left[\hat{a}\hat{b}\hat{c}\right]=\left|\begin{array}{ccc}\hat{a}.\hat{a}& \hat{a}.\hat{b}& \hat{a}.\hat{c}\\ \hat{b}.\hat{a}& \hat{b}.\hat{b}& \hat{b}.\hat{c}\\ \hat{c}.\hat{a}& \hat{c}.\hat{b}& \hat{c}.\hat{c}\end{array}\right|$

$=\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|$

$=1(1-\frac{1}{4})-\frac{1}{2}(\frac{1}{2}-\frac{1}{4})+\frac{1}{2}(\frac{1}{4}-\frac{1}{2})=\frac{1}{2}$

Thus, the required volume of the parallelopiped $=\frac{1}{\sqrt{2}}$