Question

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector $\hat{ a }, \hat{ b }, \hat{ c }$ such that $\hat{ a } \cdot \hat{ b }=\hat{ b } \cdot \hat{ c }=\hat{ c } \cdot \hat{ a }=\frac{1}{2}$. Then, the volume of the parallelopiped is

• A. $\frac{1}{\sqrt{2}}$ cu unit
• B. $\frac{1}{2\sqrt{2}} $ cu unit
• C. $\frac{\sqrt{3}}{2}$ cu unit
• D. $\frac{1}{\sqrt{3}}$ cu unit

Answer 1

Answer: (A)

The volume of the parallelopiped with coterminus edges as $\hat{ a }, \hat{ b }, \hat{ c }$ is given by $[\hat{ a } \hat{ b } \hat{ c }]=\hat{ a } \cdot(\hat{ b } \times \hat{ c })$

Now, $[\hat{ a } \hat{ b } \hat{ c }]^{2}=\begin{vmatrix}\hat{ a } \cdot \hat{ a } & \hat{ a } \cdot \hat{ b } & \hat{ a } \cdot \hat{ c } \\ \hat{ b } \cdot \hat{ a } & \hat{ b } \cdot \hat{ b } & \hat{ b } \cdot \hat{ c } \\ \hat{ c } \cdot \hat{ a } & \hat{ c } \cdot \hat{ b } & \hat{ c } \cdot \hat{ c }\end{vmatrix}=\begin{vmatrix}1 & 1 / 2 & 1 / 2 \\ 1 / 2 & 1 & 1 / 2 \\ 1 / 2 & 1 / 2 & 1\end{vmatrix}$
$\Rightarrow [\hat{ a } \hat{ b } \hat{ c }]^{2}=1\left(1-\frac{1}{4}\right)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}\right)+\frac{1}{2}\left(\frac{1}{4}-\frac{1}{2}\right)=\frac{1}{2}$
Thus, the required volume of the parallelopiped
$=\frac{1}{\sqrt{2}}$ cu unit