Question

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\bar{a}| = 3, |\bar{b}| = 5, |\bar{c}| = 7$ then $|\bar{a} - \bar{b}|^2 + |\bar{b} - \bar{c}|^2 + |\bar{c} - \bar{a}|^2$ does not exceed}

• A. 83
• B. 249
• C. 166
• D. 105

Hint:

The sum of squared distances between endpoints of vectors is maximized when the vectors sum to zero.

Answer 1

The Correct Option is B

Step 1: Concept
Expand the squared magnitudes: $|\bar{a} - \bar{b}|^2 = |\bar{a}|^2 + |\bar{b}|^2 - 2\bar{a} \cdot \bar{b}$.

Step 2: Meaning
Sum $= 2(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2) - 2(\bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a})$.

Step 3: Analysis
From $|\bar{a} + \bar{b} + \bar{c}|^2 \ge 0$, we get $2(\bar{a} \cdot \bar{b} + \bar{b} \cdot \bar{c} + \bar{c} \cdot \bar{a}) \ge -(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2)$. Maximum value occurs when $\bar{a} + \bar{b} + \bar{c} = 0$, giving Sum $= 3(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2)$.

Step 4: Conclusion
Sum $\le 3(3^2 + 5^2 + 7^2) = 3(9 + 25 + 49) = 3(83) = 249$. Final Answer: (B)