Question:
$\vec{a}, \vec{b}, \vec{c}$ are vectors such that $|\vec{a}| = 5, |\vec{b}| = 4, |\vec{c}| = 3$ and each is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|^2 = $
$\vec{a}, \vec{b}, \vec{c}$ are vectors such that $|\vec{a}| = 5, |\vec{b}| = 4, |\vec{c}| = 3$ and each is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|^2 = $
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When each vector in a set of three is perpendicular to the sum of the other two, all the cross-terms in the expansion cancel out perfectly. The squared magnitude of their sum is simply the sum of their individual squared magnitudes!
Updated On: Jun 4, 2026
- $60$
- $12$
- $47$
- $50$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are given the magnitudes of three vectors and told that each vector is orthogonal (perpendicular) to the vector sum of the remaining two. We need to find the square of the magnitude of their total sum.
Step 2: Key Formula or Approach:
Two vectors are perpendicular if their dot product equals zero. We will set up three dot product equations based on the conditions given. Also, we will expand the squared magnitude 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: Detailed Explanation:
Translate the perpendicularity conditions into dot products:
1. $\vec{a} \perp (\vec{b} + \vec{c}) \implies \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \implies \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0$
2. $\vec{b} \perp (\vec{a} + \vec{c}) \implies \vec{b} \cdot (\vec{a} + \vec{c}) = 0 \implies \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0$
3. $\vec{c} \perp (\vec{a} + \vec{b}) \implies \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \implies \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0$
Adding all three equations together gives:
$$2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0$$ Now, substitute the given magnitudes ($|\vec{a}| = 5, |\vec{b}| = 4, |\vec{c}| = 3$) into the expansion formula:
$$|\vec{a} + \vec{b} + \vec{c}|^2 = (5)^2 + (4)^2 + (3)^2 + 0$$ $$|\vec{a} + \vec{b} + \vec{c}|^2 = 25 + 16 + 9 = 50$$
Step 4: Final Answer:
The value is $50$, matching option (D).
We are given the magnitudes of three vectors and told that each vector is orthogonal (perpendicular) to the vector sum of the remaining two. We need to find the square of the magnitude of their total sum.
Step 2: Key Formula or Approach:
Two vectors are perpendicular if their dot product equals zero. We will set up three dot product equations based on the conditions given. Also, we will expand the squared magnitude 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: Detailed Explanation:
Translate the perpendicularity conditions into dot products:
1. $\vec{a} \perp (\vec{b} + \vec{c}) \implies \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \implies \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0$
2. $\vec{b} \perp (\vec{a} + \vec{c}) \implies \vec{b} \cdot (\vec{a} + \vec{c}) = 0 \implies \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0$
3. $\vec{c} \perp (\vec{a} + \vec{b}) \implies \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \implies \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0$
Adding all three equations together gives:
$$2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0$$ Now, substitute the given magnitudes ($|\vec{a}| = 5, |\vec{b}| = 4, |\vec{c}| = 3$) into the expansion formula:
$$|\vec{a} + \vec{b} + \vec{c}|^2 = (5)^2 + (4)^2 + (3)^2 + 0$$ $$|\vec{a} + \vec{b} + \vec{c}|^2 = 25 + 16 + 9 = 50$$
Step 4: Final Answer:
The value is $50$, matching option (D).
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