SATHEE: Vectors 3 Question 4

Vectors 3 Question 4

4. 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

(2008, 3M)

(a) $\frac{1}{\sqrt{2}}$ cu unit

(b) $\frac{1}{2 \sqrt{2}}$ 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 $\hat{a}, \hat{b}, \hat{c}$ is given by $[\hat{a} \hat{b} \hat{c}] = \hat{a} \cdot (\hat{b} \times \hat{c})$

Now, $[\hat{a} \hat{b} \hat{c}]^{2} = | \begin{array}{lll} \hat{a} \cdot \hat{a} & \hat{a} \cdot \hat{b} & \hat{a} \cdot \hat{c} \\ \hat{b} \cdot \hat{a} & \hat{b} \cdot \hat{b} & \hat{b} \cdot \hat{c} \\ \hat{c} \cdot \hat{a} & \hat{c} \cdot \hat{b} & \hat{c} \cdot \hat{c} \end{array} | = | \begin{array}{ccc} 1 & 1 / 2 & 1 / 2 \\ 1 / 2 & 1 & 1 / 2 \\ 1 / 2 & 1 / 2 & 1 \end{array} |$

$\Rightarrow [\hat{a} \hat{b} \hat{c}]^{2} = 1 (1 - \frac{1}{4}) - \frac{1}{2} (\frac{1}{2} - \frac{1}{4}) + \frac{1}{2} (\frac{1}{4} - \frac{1}{2}) = \frac{1}{2}$

Thus, the required volume of the parallelopiped

$= \frac{1}{\sqrt{2}} cu unit$

Vectors 3 Question 4

4. 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

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Vectors 3 Question 4

4. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector a^,b^,c^ such that a^⋅b^=b^⋅c^=c^⋅a^=12. Then, the volume of the parallelopiped is

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4. 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