The edges of a parallelopiped are of unit length and are parallel to noncoplanar unit vectors $\displaystyle \hat{a}, \hat{b}, \hat{c}$ such that $\displaystyle \hat{a}\cdot \hat{b}= \hat{b}\cdot\hat{c}= \hat{c}\cdot \hat{a}= \frac{1}{2}$, then the volume of the parallelopiped is
Answer & explanation
Correct answer: option 1
Let V be the volume of the parallelepiped. Then
$V^2=|\hat a.(\hat b×\hat c)|^2=\begin{vmatrix}\hat a.\hat a&\hat a.\hat b&\hat a.\hat c\\\vec b.\vec a&\vec b.\vec b&\vec b.\vec c\\\vec c.\vec a&\vec c.\vec b&\vec c.\vec c\end{vmatrix}=\begin{vmatrix}1&1/2&1/2\\1/2&1&1/2\\1/2&1/2&1\end{vmatrix}=\frac{1}{2}$
Hence (A) is the correct answer.