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

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


\[
\begin{aligned}
& \text { Now, }[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\left|\begin{array}{ccc}
\hat{\mathbf{a}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{c}} \\
\hat{\mathbf{b}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{c}} \\
\hat{\mathbf{c}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{c}}
\end{array}\right|=\left|\begin{array}{ccc}
1 & 1 / 2 & 1 / 2 \\
1 / 2 & 1 & 1 / 2 \\
1 / 2 & 1 / 2 & 1
\end{array}\right|=\frac{1}{2} \\
& \Rightarrow \quad[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\frac{1}{2} \\
&
\end{aligned}
\]
Thus, the required volume of the parallelopiped \(=\frac{1}{\sqrt{2}} \mathrm{cu}\) units