Question:
If \( V \) is the volume of the parallelepiped with edges \( \vec{a}, \vec{b}, \vec{c} \), then the volume of the parallelepiped with edges \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) (defined by dot products) is
If \( V \) is the volume of the parallelepiped with edges \( \vec{a}, \vec{b}, \vec{c} \), then the volume of the parallelepiped with edges \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) (defined by dot products) is
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If V is the volume of the parallelopiped with edges $\veca, \vecb, \vecc$, then the volume of the parallelopiped with edges $\vecα, \vecβ, \vecγ}$ (defined by dot products) is
Updated On: Apr 15, 2026
- $V^{3}$
- 3V
- $V^{2}$
- 2V
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The volume of a parallelopiped is given by the scalar triple product $[\vec{a}~\vec{b}~\vec{c}]$.
Step 2: Analysis
The new vectors $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are linear combinations of the original vectors.
Step 3: Evaluation
The new volume $V_{1} = |[\vec{\alpha}~\vec{\beta}~\vec{\gamma}]|$ is equivalent to the determinant formed by the dot products multiplied by the original scalar triple product.
Step 4: Conclusion
This simplifies to $[\vec{a}~\vec{b}~\vec{c}]^3 = V^3$.
Final Answer: (a)
The volume of a parallelopiped is given by the scalar triple product $[\vec{a}~\vec{b}~\vec{c}]$.
Step 2: Analysis
The new vectors $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are linear combinations of the original vectors.
Step 3: Evaluation
The new volume $V_{1} = |[\vec{\alpha}~\vec{\beta}~\vec{\gamma}]|$ is equivalent to the determinant formed by the dot products multiplied by the original scalar triple product.
Step 4: Conclusion
This simplifies to $[\vec{a}~\vec{b}~\vec{c}]^3 = V^3$.
Final Answer: (a)
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