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- 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
- 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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