Question:
Let \( \bar{a}, \bar{b}, \bar{c} \) be three vectors such that \( \bar{a} + \bar{b} + \bar{c} = \bar{0}, |\bar{a}| = 3, |\bar{b}| = 4, |\bar{c}| = 5 \), then \( \bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a} = \)}
Let \( \bar{a}, \bar{b}, \bar{c} \) be three vectors such that \( \bar{a} + \bar{b} + \bar{c} = \bar{0}, |\bar{a}| = 3, |\bar{b}| = 4, |\bar{c}| = 5 \), then \( \bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a} = \)}
Show Hint
If the sum of vectors is zero, the sum of their dot products is $-\frac{1}{2}\sum |\bar{v}|^2$.
Updated On: Apr 30, 2026
- 25
- -25
- 50
- -50
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Square the Sum
$|\bar{a} + \bar{b} + \bar{c}|^2 = 0^2$. $|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2 + 2(\bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a}) = 0$.
Step 2: Substitute Magnitudes
$3^2 + 4^2 + 5^2 + 2(\sum \bar{a} \cdot \bar{b}) = 0$. $9 + 16 + 25 + 2(\sum \bar{a} \cdot \bar{b}) = 0$.
Step 3: Calculation
$50 + 2(\sum \bar{a} \cdot \bar{b}) = 0 \implies \sum \bar{a} \cdot \bar{b} = -25$.
Step 4: Conclusion
The value is -25.
Final Answer:(B)
$|\bar{a} + \bar{b} + \bar{c}|^2 = 0^2$. $|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2 + 2(\bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a}) = 0$.
Step 2: Substitute Magnitudes
$3^2 + 4^2 + 5^2 + 2(\sum \bar{a} \cdot \bar{b}) = 0$. $9 + 16 + 25 + 2(\sum \bar{a} \cdot \bar{b}) = 0$.
Step 3: Calculation
$50 + 2(\sum \bar{a} \cdot \bar{b}) = 0 \implies \sum \bar{a} \cdot \bar{b} = -25$.
Step 4: Conclusion
The value is -25.
Final Answer:(B)
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