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📰 Solving $ 6 - 3y = 0 $ gives $ y = 2 $. Final answer: $ \boxed{2} $. 📰 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}} $. 📰 From Inches To Miles The Blast Radius Of An Atomic Bomb Explained In Clear Terms 8738151 📰 This Fake Id Front And Back Fooled Everyonesee How Fast It Works 7531162 📰 Pre Approved Home Loans 96694 📰 Huge Gains Or Risky Bet Discover Why The Rig Stock Price Is Spiking Now 4084760 📰 Atv Launcher 3222940 📰 Youll Learn The Ultimate Trick To Check Word Count On Word In 10 Seconds 8292179 📰 You Wont Believe Whats Inside Skibidi Gamesreload Now 5315444 📰 Why Investors Are Polarized Fidelity Etfs Tower Above The Restdont Miss Out 6294279 📰 He Who Saves His Country 632260 📰 Microsoft 5Th Ave 9976978 📰 You Wont Believe The Inside Secrets Of Turner Network Television Apps 8149251 📰 Haircare For Hard Water 1248443 📰 Claim Evidence Reasoning 3541860 📰 Functionalist Theory 9251779 📰 How To Calculate Rate Of Return 4301550