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Home » Square and add the equations: - Londonproperty

Square and add the equations: - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 Solution: Since $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vectors, we use the identity:

📌 \|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 + \|\mathbf{c}\|^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})

📌 = 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})

📌 Each dot product $ \leq 1 $, and maximum occurs when $ \mathbf{a} = \mathbf{b} = \mathbf{c} $, then all dot products are 1:

📌 \|\mathbf{a} + \mathbf{b} + \mathbf{c}\|^2 = 3 + 2(1 + 1 + 1) = 3 + 6 = 9

📌 \Rightarrow \|\mathbf{a} + \mathbf{b} + \mathbf{c}\| = 3

📌 This is achievable when all vectors are equal.

📌 Thus, the maximum value is $ \boxed{3} $.

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