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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} $. 📰 Nutrition From Pears 1787183 📰 Ali Squid Game 7720490 📰 Cabo Stock Surge Is This Hidden Gem Worth Your Investment Today 5552 📰 Game Police Game 9027674 📰 Rblx Studio 9124064 📰 Ont Fios Ethernet And Coax Same Time 5280747 📰 Unlock Maximum Efficiency With These Must Try Windows Desktop Gadgets 1409305 📰 Why The Roth Conversion 5 Year Rule Could Double Your Retirement Savings Overnight 9964205 📰 The Ultimate Guide To Avoiding Cms Nppes Disasterdont Miss These Critical Tips 8740978 📰 Jurezs Heart On The Line As Club Americano Charges The Ringwill Honor Survive The Fire 6020605 📰 Whats Hiding In This Vi Beho Whitney Pistachio Cream Itll Change Your Desserts Forever 4536820 📰 Protecting Ai Relays Drastically Vibration Enclosure And Diagnostic Insights 4619178 📰 Osmosis Water System 1623985 📰 The Untold Secrets Of Ashley Manning Inside Her Hidden Career Shift 2513360 📰 52 Week Lows Are Blown Out Of The Waterwhy Todays Stocks Are A Hidden Goldmine 6854302