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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, $x + 3x - 2 = 0 \Rightarrow 4x = 2 \Rightarrow x = \frac{1}{2}$. 📰 Kamala Harris Gun Owner 3942131 📰 Verizon In Riverdale Utah 783264 📰 3Fearless Trades Ge Stock Price Set To Break All Barriers In 2024Dont Miss Out 816500 📰 Other Words For Calm 5300528 📰 What Is Phagocytosis 7640441 📰 Hinata Nude 8136446 📰 Green Dotted Magic The Secret Tool Making Spaces Look Fresh Fresh 4838447 📰 What Cloisonne Reveals About Forgotten Crafts Finalest Hidden Message 7926287 📰 Fn Item Shop Today 4722499 📰 How Ehr Healthcare Cut Patient Wait Times By 70Heres How They Learned 3704413 📰 Fuw They Said This One Wordand Everything Changed 532399 📰 You Wont Believe What Happened When Oracle Opened These Critical Doors 2588172 📰 Best Water Distiller 5994374 📰 Kerri Kenney 7214585 📰 Travel Adapter Universal Plug 6364611