Question:
If \(\vec{a},\vec{b},\vec{c}\) are three vectors such that \(\vec{a}\cdot (\vec{b}+\vec{c})+\vec{b}\cdot (\vec{c}+\vec{a})+\vec{c}\cdot (\vec{a}+\vec{b})=0\) and \(|\vec{a}|=8,|\vec{b}|=1,|\vec{c}|=4\), then \(|\vec{a}+\vec{b}+\vec{c}|=...\)
If \(\vec{a},\vec{b},\vec{c}\) are three vectors such that \(\vec{a}\cdot (\vec{b}+\vec{c})+\vec{b}\cdot (\vec{c}+\vec{a})+\vec{c}\cdot (\vec{a}+\vec{b})=0\) and \(|\vec{a}|=8,|\vec{b}|=1,|\vec{c}|=4\), then \(|\vec{a}+\vec{b}+\vec{c}|=...\)
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Expand the given condition to get the sum of pairwise dot products, then use the square of the sum.
Updated On: Nov 12, 2026
- \(5\)
- \(9\)
- \(13\)
- \(18\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
Expanding the given condition: \(2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=0\).
Step 2: Square of the sum
\(|\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)\).
Step 3: Substitute
\(=64+1+16+0=81\).
Step 4: Take the root
\(|\vec a+\vec b+\vec c|=9\).
Final Answer:
The magnitude is 9. \[ \boxed{\text{(B) }9} \]
Expanding the given condition: \(2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)=0\).
Step 2: Square of the sum
\(|\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)\).
Step 3: Substitute
\(=64+1+16+0=81\).
Step 4: Take the root
\(|\vec a+\vec b+\vec c|=9\).
Final Answer:
The magnitude is 9. \[ \boxed{\text{(B) }9} \]
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