Question

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector $\hat{a},\hat{b},\hat{c}$ 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. $\frac{1}{\sqrt{2}}$ cu unit
• B. $\frac{1}{2\sqrt{2}}$ cu unit
• C. $\frac{\sqrt{3}}{2}$ cu unit
• D. $\frac{1}{\sqrt{3}}$ cu unit

Answer 1

Answer: (A)

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

Now, $[\hat{a}\hat{b}\hat{c}{]}^2=\left|\begin{array}{ccc}\hat{a}\cdot \hat{a}& \hat{a}\cdot \hat{b}& \hat{a}\cdot \hat{c}\\ \hat{b}\cdot \hat{a}& \hat{b}\cdot \hat{b}& \hat{b}\cdot \hat{c}\\ \hat{c}\cdot \hat{a}& \hat{c}\cdot \hat{b}& \hat{c}\cdot \hat{c}\end{array}\right|=\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|$
$\Rightarrow [\hat{a}\hat{b}\hat{c}{]}^2=1\left(1-\frac{1}{4}\right)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}\right)+\frac{1}{2}\left(\frac{1}{4}-\frac{1}{2}\right)=\frac{1}{2}$
Thus, the required volume of the parallelopiped
$=\frac{1}{\sqrt{2}}$ cu unit