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:

  1. 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 } $



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