Daily Archives: June 13, 2014

Solution: 2014-13 Unit vectors

Prove that, for any unit vectors \( v_1, v_2, \cdots, v_n \) in \( \mathbb{R}^n \), there exists a unit vector \( w \) in \( \mathbb{R}^n \) such that \( \langle w, v_i \rangle \leq n^{-1/2} \) for all \( i = 1, 2, \cdots, n \). (Here, \( \langle \cdot, \cdot \rangle \) is a usual scalar product in \( \mathbb{R}^n \).)

The best solution was submitted by 어수강. Congratulations!

Alternative solutions were submitted by 이종원 (+3), 장기정 (+3), 정성진 (+3), 채석주 (+1), 황성호 (+1). Thank you for your participation.

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