The edges of a parallelepiped 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 parallelepiped is

$|\hat{a}| = |\hat{b}| = |\hat{c}| = 1$ $\hat{a}\cdot\hat{b} = \hat{b}\cdot\hat{c} = \hat{c}\cdot\hat{a} = \dfrac{1}{2}$ $\text{To find: } V = |\hat{a}\cdot(\hat{b}\times\hat{c})|, \quad V^2 = \det(G)$ $V^2 = \det(G), \quad G_{ij} = \hat{e}_i\cdot\hat{e}_j$ $G = \begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 1 \end{pmatrix}$ $\det G = 1\left(1-\tfrac14\right) - \tfrac12\left(\tfrac12-\tfrac14\right) + \tfrac12\left(\tfrac14-\tfrac12\right)$ $= \tfrac34 - \tfrac18 - \tfrac18 = \tfrac12$ $V^2 = \dfrac{1}{2} \Rightarrow V = \dfrac{1}{\sqrt2}$ $\boxed{V = \dfrac{1}{\sqrt2}} \quad \text{option (1)}$