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📰 Solution: Let $\theta$ be the angle between any pair of vectors. Since $\mathbf{u} \cdot \mathbf{v} = \cos\theta = \frac{1}{2}$, $\theta = 60^\circ$. The magnitude squared is $|\mathbf{u} + \mathbf{v} + \mathbf{w}|^2 = 3 + 2(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}) = 3 + 2(3 \cdot \frac{1}{2}) = 3 + 3 = 6$. Thus, the maximum value is $\sqrt{6}$. 📰 Question: A plant biologist studies stress responses using vectors $\begin{pmatrix} 1 \\ x \\ 2 \end{pmatrix}$ and $\begin{pmatrix} x \\ 3 \\ -1 \end{pmatrix}$. Find $x$ such that the vectors are orthogonal. 📰 Solution: For orthogonality, their dot product must be zero: $1 \cdot x + x \cdot 3 + 2 \cdot (-1) = 0$. Simplifying, $x + 3x - 2 = 0 \Rightarrow 4x = 2 \Rightarrow x = \frac{1}{2}$. 📰 Commercial Chef Approved Wegmans Catering Delivers Restaurants Quality Meals Fast 419204 📰 From Casual To Pro Discover The Wild World Of Car Games Oyoull Regret Not Playing 1474738 📰 Unlimited Wings 5034980 📰 Best Chatgpt Prompt 9330805 📰 Grab These Outlook Email Templates And Impress Clients In Seconds 1570297 📰 Business Savings 4291547 📰 Sakura Anime 5149056 📰 Get The Roku Remote Downloadyour Perfect Tv Guide Starts Here 8069511 📰 Arctovish 1312574 📰 Unlock Epic 1940S Tank Fightsthe Best Online Multiplayer Titles Hacking 2024 1256524 📰 J 20 Unveiled The Shocking Truth About This Simple Musical Note 3982434 📰 S 2Pi R2 2Pi Rh 6404966 📰 Essie Davis 6594821 📰 Sats Stock 3959399 📰 Airsupra Dosage 4936468