The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors…

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{2 \sqrt{2}}$
3. $\frac{\sqrt{3}}{2}$
4. $\frac{1}{\sqrt{3}}$

Solution

The volume of the parallelopiped with coterminus edges as $\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}$ is given by $[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]=\hat{\mathbf{a}} \cdot(\hat{\mathbf{b}} \times \hat{\mathbf{c}})$
$ \begin{aligned} & \text { Now, }[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\left|\begin{array}{ccc} \hat{\mathbf{a}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{c}} \\ \hat{\mathbf{b}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{c}} \\ \hat{\mathbf{c}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{c}} \end{array}\right|=\left|\begin{array}{ccc} 1 & 1 / 2 & 1 / 2 \\ 1 / 2 & 1 & 1 / 2 \\ 1 / 2 & 1 / 2 & 1 \end{array}\right|=\frac{1}{2} \\ & \Rightarrow \quad[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\frac{1}{2} \\ & \end{aligned} $ Thus, the required volume of the parallelopiped $=\frac{1}{\sqrt{2}} \mathrm{cu}$ units

Asked in: JEE Advanced 2008 (Paper 1)

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