Vectors 3 Question 4
4. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector $\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}$ such that $\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}=\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}=\frac{1}{2}$. Then, the volume of the parallelopiped is
(2008, 3M)
(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
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Answer:
Correct Answer: 4. (a)
Solution:
- The volume of the parallelopiped with coterminus edges as $\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}$ is given by $[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]=\hat{\mathbf{a}} \cdot(\hat{\mathbf{b}} \times \hat{\mathbf{c}})$
Now, $[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^{2}=\left|\begin{array}{lll}\hat{\mathbf{a}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{a}} \cdot \hat{\mathbf{c}} \ \hat{\mathbf{b}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{b}} \cdot \hat{\mathbf{c}} \ \hat{\mathbf{c}} \cdot \hat{\mathbf{a}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{b}} & \hat{\mathbf{c}} \cdot \hat{\mathbf{c}}\end{array}\right|=\left|\begin{array}{ccc}1 & 1 / 2 & 1 / 2 \ 1 / 2 & 1 & 1 / 2 \ 1 / 2 & 1 / 2 & 1\end{array}\right|$
$\Rightarrow \quad[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^{2}=1\left(1-\frac{1}{4}\right)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}\right)+\frac{1}{2}\left(\frac{1}{4}-\frac{1}{2}\right)=\frac{1}{2}$
Thus, the required volume of the parallelopiped
$$ =\frac{1}{\sqrt{2}} \text { cu unit } $$
