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📰 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} $. 📰 Solution: Let $ \mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} $. The cross product is: 📰 Final Push Solo 401K Contribution Deadline This Weekclaim Your Tax Benefits 7033965 📰 Unlock Your Companys Potential The Ultimate Guide To Human Capital Management Solutions 8888048 📰 Southpaw Cast 2079178 📰 Shopee Cake 847978 📰 Final Black Friday Alert Prime Gaming Pc Dealsstock Run Begins Now 292817 📰 Optic Disc 8787958 📰 Best United Airlines Credit Card 9518227 📰 Tldr Bti Stock Price Jumps 200 Could This Be Your Next Big Win 3684094 📰 Kik 9890463 📰 What Time Is The Bills Vs Chiefs Game 1664699 📰 Alolan Persian Vs Other Languages The Hidden Dynamics Every Learner Needs To Know 2003651 📰 These Lyrics To Funk You Up Were So Funky They Broke My Infectionfind Out Why Now 5572692 📰 Your Upper Room Holds Secrets Worse Than Anything You Imagined 8443935 📰 Alert The Bubble You Cant Ignore Is Now Ready To Collapsewhat Will You Do 4276984 📰 Keith Papini 7686867