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houses for sale in versailles ohio - GetMeFoodie

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 = 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} $.

📌 Question: A regular tetrahedron has three of its vertices at $ (1,1,1) $, $ (1,-1,-1) $, and $ (-1,1,-1) $. Given that the fourth vertex has integer coordinates, find its location.

📌 Solution: Let the three given points be $ A = (1,1,1) $,

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