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 $\hat a.\hat b=\hat b.\hat c=\hat c.\hat a=1/2$. Then the volume of the parallelopiped in cubic units, is

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

We have,

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

$∴[\vec a\,\,\vec b\,\,\vec c]^2=\begin{vmatrix}\hat a.\hat a&\hat a.\hat b&\hat a.\hat c\\\hat b.\hat a&\hat b.\hat b&\hat b.\hat c\\\hat c.\hat a&\hat c.\hat b&\hat c.\hat c\end{vmatrix}$

$⇒[\vec a\,\,\vec b\,\,\vec c]^2=\begin{vmatrix}1&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&1&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&1\end{vmatrix}=\frac{1}{2}$

$⇒[\vec a\,\,\vec b\,\,\vec c]=\frac{1}{\sqrt{2}}$ cubic units