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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} $. 📰 Unbelievable Aeo Ticker Spikes Single Dayis It The Next Big Thing 7320969 📰 Budget Sheet Template 6132017 📰 Rockhopper Penguin 6463402 📰 Final Trick To Filename Your Iphone Pictures Directly On Pcno Loss No Hassle 5778749 📰 Stop Thirsting At Nightcrystal Springs Water Glows With Health You Desperately Need 2068048 📰 Vc Meaning 3591526 📰 Step Inside This House Will Welcome You Like No Other Youll Wonder How You Missed It 9363503 📰 You Wont Believe What This Pst Changed About Your Breakfast Forever 1443045 📰 Microsoft Raw Image Extension 4283781 📰 How To Record A Call On Iphone 4339460 📰 The Ultimate Basmati Rice Water Ratio That Cooks Perfect Fluffy Rice Every Time 6948130 📰 Mazatlns Most Stunning Sights You Didnt Know Existed Until Now 9159677 📰 From Pixels To Perfection Simrealist Uses Ai To Create Unbelievable Art 6429765 📰 Morrigan Darkstalkers You Wont Believe How This Icon Killed Hundreds In Bloody Battle 6406264 📰 Jack Doherty Net Worth The True Financial Story Behind The Name You Know 8370555