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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}} $. 📰 Road To Perdition 1002848 📰 Unlock The Ultimate Moto X3M Experiencethis Bike Game Redefines Off Road Racing 7326896 📰 Film Julie Julia 1447149 📰 How Many Calories In A Chicken Breast 3722103 📰 Laura Kariuki 4524159 📰 Child Of Light 464247 📰 Why Is Meta Stock Down Today 9185081 📰 Panda Bear Pediatrics 679416 📰 This Num Row Oracle Changed How I Codewatch The Mind Blowing Results 7278097 📰 Stop This Program Instantlythese 3 Steps To Disable It Work Every Time 8810639 📰 Garden Royal Garden 2045003 📰 How To Connect A Ps5 Controller 5073792 📰 You Wont Believe What Happens When Mac Users Embrace Pure Solo Mode 5293023 📰 Redeem Robux Cards 2072300 📰 Tmgtmf Stock Hype Investors Are Losing It Gmgmf Just Got Nearly Unstoppable 8696208