Question
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 :
A.
$$\frac{1}{{\sqrt 2 }}$$
B.
$$\frac{1}{{2\sqrt 2 }}$$
C.
$$\frac{{\sqrt 3 }}{2}$$
D.
$$\frac{1}{{\sqrt 3 }}$$
Answer :
$$\frac{1}{{\sqrt 2 }}$$
Solution :
We know that the volume of a parallelepiped with coterminous edges as the vectors $$\hat a,\,\hat b,\,\hat c$$ is given by
\[\begin{array}{l}
V = {\left[ {\vec a\,\vec b\,\vec c} \right]^2} = \left[ \begin{array}{l}
\vec a.\vec a\,\,\,\,\,\vec a.\vec b\,\,\,\,\,\vec a.\vec c\\
\vec b.\vec a\,\,\,\,\,\vec b.\vec b\,\,\,\,\,\vec b.\vec c\\
\vec c.\vec a\,\,\,\,\,\vec c.\vec b\,\,\,\,\,\vec c.\vec c
\end{array} \right]\\
{V^2} = \left| \begin{array}{l}
1\,\,\,\,\,\frac{1}{2}\,\,\,\,\,\frac{1}{2}\\
\frac{1}{2}\,\,\,\,\,1\,\,\,\,\,\frac{1}{2}\\
\frac{1}{2}\,\,\,\,\,\frac{1}{2}\,\,\,\,\,1
\end{array} \right| = \frac{1}{2}\\
\Rightarrow V = \frac{1}{{\sqrt 2 }}
\end{array}\]