Question:
Consider a tetrahedron with faces \( F_1, F_2, F_3, F_4 \). Let \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \vec{v_4} \) be area vectors perpendicular to these faces in the outward direction, then \( |\vec{v_1} + \vec{v_2} + \vec{v_3} + \vec{v_4}| \) equals
Consider a tetrahedron with faces \( F_1, F_2, F_3, F_4 \). Let \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \vec{v_4} \) be area vectors perpendicular to these faces in the outward direction, then \( |\vec{v_1} + \vec{v_2} + \vec{v_3} + \vec{v_4}| \) equals
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Consider a tetrahedron with faces $F1, F2, F3, F4$. Let $\vecv1, \vecV2, \vecv3, \vecv4$ be area vectors perpendicular to these faces in outward direction, then $|\vecv1+\vecV2+\vecV3+\vecV4|$ equals
Updated On: Apr 15, 2026
- 1
- 4
- $\vec{0}$
- None of these
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The Correct Option is C
Solution and Explanation
Step 1: Concept
The sum of outward area vectors for any closed surface is always zero.
Step 2: Analysis
Let the vertices be represented by vectors $\vec{a}, \vec{b}, \vec{c}$ from the fourth vertex. The area vectors are $\frac{1}{2}(\vec{a}\times\vec{b})$, $\frac{1}{2}(\vec{b}\times\vec{c})$, $\frac{1}{2}(\vec{c}\times\vec{a})$.
Step 3: Evaluation
The fourth area vector is $\frac{1}{2}\{(\vec{c}-\vec{a})\times(\vec{b}-\vec{a})\}$. Adding these results in the cancellation of all terms.
Step 4: Conclusion
The resultant vector is the zero vector, $\vec{0}$.
Final Answer: (c)
The sum of outward area vectors for any closed surface is always zero.
Step 2: Analysis
Let the vertices be represented by vectors $\vec{a}, \vec{b}, \vec{c}$ from the fourth vertex. The area vectors are $\frac{1}{2}(\vec{a}\times\vec{b})$, $\frac{1}{2}(\vec{b}\times\vec{c})$, $\frac{1}{2}(\vec{c}\times\vec{a})$.
Step 3: Evaluation
The fourth area vector is $\frac{1}{2}\{(\vec{c}-\vec{a})\times(\vec{b}-\vec{a})\}$. Adding these results in the cancellation of all terms.
Step 4: Conclusion
The resultant vector is the zero vector, $\vec{0}$.
Final Answer: (c)
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