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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}} $. 📰 Seien Die Zahlen X Und X2 7017075 📰 Oracle Market Place 📰 What Is The Best Internet Service Provider 📰 Health And Human Services Commission 1735167 📰 Star Dodge Abilene 4640766 📰 Most Valuable Dimes 📰 Stop Saving Start Investingretirement Funds Are Growing Faster Than Ever 8115382 📰 Finally Found The Best Login For Your Furry Friendbut Its More Powerful Than You Think 2486146 📰 Heres The Fidelity Advisor Phone Number That Top Investors Are Usingact Now 5718245 📰 Places Hiring In Tampa Fl 111402 📰 Texas Longhorns Football Vs Texas Am Aggies Football 7887599 📰 1St Person Shooter 1543021 📰 Moto Maniac 📰 Wells Fargo Bank Help Number 📰 Ali Abulaban About To Reveal The Secret No One Else Dares Share 8286985