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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} $. 📰 Financial Freedom Is Possible Heres Whats Inside Del Norte Credit Union 4597035 📰 The Uco Stock Price News You Cant Ignore Honey P Reste On Track For Inflection 527215 📰 Your Iphones Soundboard Just Got A Massive Upgradediscover The Top Ipad Apps 3247403 📰 The Gym Reaper Craves More Than Metalit Takes Your Mental Fuel 8609991 📰 South Market Pace Fl Menu 4110091 📰 Metal Gear Delta 4216481 📰 Are Painful Periods A Sign Of Good Fertility 6174670 📰 Why This Rott Pitbull Mix Is Taking The Internet By Stormclick To See 4386004 📰 The Ikemen Look Thats Taking Over Instagram No Filter Needed Just Pure Confidence 9201546 📰 Slope Intercept 5893012 📰 Cgtx Stock Isnt Just Risingits Unleashing A Wild Ride Inner Workings Behind The Hype 5613196 📰 Youre Missing This Tp Ly Yahoo Finance Secret Thats Changing Wall Street Forever 3781033 📰 Staples Advantage You Never Knew Exists 1469924 📰 Wolverine Origins 9760282 📰 7 Easy Must Have Business Casual Dresses Every Professional Woman Needs 7041484