Question:
If
\[
| \mathbf{a} \times \mathbf{b} \times \mathbf{c} | = 4,
\]
then the volume of the parallelepiped with coterminous edges
\[
\mathbf{a} + 2\mathbf{b}, \quad \mathbf{b} + 2\mathbf{c}, \quad \mathbf{c} + 2\mathbf{a}
\]
is
If
\[
| \mathbf{a} \times \mathbf{b} \times \mathbf{c} | = 4,
\]
then the volume of the parallelepiped with coterminous edges
\[
\mathbf{a} + 2\mathbf{b}, \quad \mathbf{b} + 2\mathbf{c}, \quad \mathbf{c} + 2\mathbf{a}
\]
is
Show Hint
For problems involving the volume of a parallelepiped, always use the scalar triple product and apply distributive properties as needed.
Updated On: Jan 30, 2026
- 36 units\(^3\)
- 32 units\(^3\)
- 20 units\(^3\)
- 40 units\(^3\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the volume of the parallelepiped.
The volume \( V \) of a parallelepiped with coterminous edges \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is given by the scalar triple product: \[ V = | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) |. \] The volume of the new parallelepiped formed by the vectors \( \mathbf{a} + 2\mathbf{b}, \mathbf{b} + 2\mathbf{c}, \mathbf{c} + 2\mathbf{a} \) can be found by using the properties of the scalar triple product.
Step 2: Applying the identity.
Using the distributive property of the scalar triple product, we calculate the volume, which turns out to be 36 units\(^3\). Thus, the correct answer is option (A).
The volume \( V \) of a parallelepiped with coterminous edges \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is given by the scalar triple product: \[ V = | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) |. \] The volume of the new parallelepiped formed by the vectors \( \mathbf{a} + 2\mathbf{b}, \mathbf{b} + 2\mathbf{c}, \mathbf{c} + 2\mathbf{a} \) can be found by using the properties of the scalar triple product.
Step 2: Applying the identity.
Using the distributive property of the scalar triple product, we calculate the volume, which turns out to be 36 units\(^3\). Thus, the correct answer is option (A).
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