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 $\hat{a}.\hat{b}=\hat{b}.\hat{c}=\hat{c}.\hat{a}=\frac{1}{2}$. 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

Answer: (A)

$\frac{1}{\sqrt{2}}$

The volume of the parallelopiped with coterminus edges as $\hat{a},\hat{b},\hat{c}$ is given as

$\left[\hat{a}\hat{b}\hat{c}\right]=\hat{a}(\hat{b}\times \hat{c})$

Now,

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

$\left[\hat{a}\hat{b}\hat{c}\right]=\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|$

$\left[\hat{a}\hat{b}\hat{c}\right]=(1-\frac{1}{4})-\frac{1}{2}(\frac{1}{2}-\frac{1}{4})+\frac{1}{2}(\frac{1}{4}-\frac{1}{2})=\frac{1}{2}$

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