Game 65, at Los Angeles Dodgers (June 10, 2023) - inBeat

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📰 Question: If $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are unit vectors with $ \mathbf{a} \cdot \mathbf{b} = \frac{1}{2} $, $ \mathbf{b} \cdot \mathbf{c} = \frac{\sqrt{3}}{2} $, find the maximum value of $ \mathbf{a} \cdot \mathbf{c} $. 📰 Solution: Let $ \theta $ be the angle between $ \mathbf{a} $ and $ \mathbf{b} $, so $ \cos\theta = \frac{1}{2} \Rightarrow \theta = 60^\circ $. Let $ \phi $ be the angle between $ \mathbf{b} $ and $ \mathbf{c} $, so $ \cos\phi = \frac{\sqrt{3}}{2} \Rightarrow \phi = 30^\circ $. To maximize $ \mathbf{a} \cdot \mathbf{c} = \cos(\alpha) $, where $ \alpha $ is the angle between $ \mathbf{a} $ and $ \mathbf{c} $, arrange $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ in a plane. The maximum occurs when $ \mathbf{a} $ and $ \mathbf{c} $ are aligned, but constrained by their angles relative to $ \mathbf{b} $. The minimum angle between $ \mathbf{a} $ and $ \mathbf{c} $ is $ 60^\circ - 30^\circ = 30^\circ $, so $ \cos(30^\circ) = \frac{\sqrt{3}}{2} $. However, if they are aligned, $ \alpha = 0^\circ $, but this requires $ \theta = \phi = 0^\circ $, which contradicts the given dot products. Instead, use the cosine law for angles: $ \cos\alpha \leq \cos(60^\circ - 30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} $. Thus, the maximum is $ \boxed{\frac{\sqrt{3}}{2}} $. 📰 Question: Find the vector $ \mathbf{v} $ such that $ \mathbf{v} \times \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} = \begin{pmatrix} -4 \\ 6 \\ 2 \end{pmatrix} $. 📰 Dr Now Diet 1638344 📰 Blonde Superhero 9574883 📰 The Changes Required To Run An Efficient Fleet Are Numerous Yet Decision Makers And Fleet Managers May Often Overlook One Of The Simplest And Most Cost Effective Ways To Improve Operational Efficiency Optimizing Tire Selection 6835273 📰 Wolfenstein Ii The New Colossus You Wont Believe How This Colossal Battle Redefines Survival 162828 📰 Experts Say Hotspot Hotspot Hotspot Is Fixing Internet Chaostry It Today 2901584 📰 How To Say N Word In Roblox 5969944 📰 Big Red Nyt 3570270 📰 Yellow Rice 3372220 📰 The Sweetheart With The Invisible Smile Thatll Make You Gasp 8867771 📰 Partial Balayage 3741463 📰 You Wont Believe What Happened When We Investigated Acworths Hidden Mystery 305625 📰 Mcflurry Calories 6043426 📰 Where To Watch Knicks Vs Charlotte Hornets 3212474 📰 Travel To Puerto Rico 1682679 📰 Never Miss A Beat How The Wisely Card Boosts Your Luck Instantly 7293777