Q. The edges of a parallelopiped are unit length and are parallel to non-coplanar unit vectors $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ such that $\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} = \frac{1}{2}\ \ \ Then,\ the\ volume\ of\ the\ parallelopiped\ is$
Explanation
(a)
The volume of the parallelepiped with coterminous edges as $\widehat{\mathbf{a}},\widehat{\mathbf{b}},\widehat{\mathbf{c}}$ is given by $\left\lbrack \widehat{\mathbf{a}},\widehat{\mathbf{b}},\widehat{\mathbf{c}} \right\rbrack = \widehat{\mathbf{a}} \bullet (\widehat{\mathbf{b}} \times \widehat{\mathbf{c}})$
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$Now,\ \left\lbrack \widehat{\mathbf{a}},\widehat{\mathbf{b}},\widehat{\mathbf{c}} \right\rbrack^{2} = \left| \begin{matrix}
\widehat{\mathbf{a}} \bullet \widehat{\mathbf{a}} & \widehat{\mathbf{a}} \bullet \widehat{\mathbf{b}} & \widehat{\mathbf{a}} \bullet \widehat{\mathbf{c}} \\
\widehat{\mathbf{b}} \bullet \widehat{\mathbf{a}} & \widehat{\mathbf{b}} \bullet \widehat{\mathbf{b}} & \widehat{\mathbf{b}} \bullet \widehat{\mathbf{c}} \\
\widehat{\mathbf{c}} \bullet \widehat{\mathbf{a}} & \widehat{\mathbf{c}} \bullet \widehat{\mathbf{b}} & \widehat{\mathbf{c}} \bullet \widehat{\mathbf{c}}
\end{matrix} \right|$
$= \left| \begin{matrix}
1 & 1/2 & 1/2 \\
1/2 & 1 & 1/2 \\
1/2 & 1/2 & 1
\end{matrix} \right| = \frac{1}{2}$
$\left\lbrack \because\left| \overrightarrow{\mathbf{a}} \right| = |\overrightarrow{\mathbf{b}}| = |\overrightarrow{\mathbf{c}}| = 1 \right\rbrack$
$\Longrightarrow \ \left\lbrack \widehat{\mathbf{a}},\widehat{\mathbf{b}},\widehat{\mathbf{c}} \right\rbrack^{2} = \frac{1}{2}$
Thus, the required volume of the parallelopiped
$= \frac{1}{\sqrt{2}}\ cu\ unit$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.