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📰 Solution: Two vectors are orthogonal if their dot product equals zero. Compute the dot product: $ x \cdot 2 + 3 \cdot (-x) = 2x - 3x = -x $. Set this equal to zero: $ -x = 0 $. Solving gives $ x = 0 $. \boxed{0} 📰 Question: Determine the minimum value of $ (\tan x + \cot x)^2 $ for $ x $ in the domain where $ \tan x $ and $ \cot x $ are defined. 📰 Solution: Expand $ (\tan x + \cot x)^2 = \tan^2 x + 2 \tan x \cot x + \cot^2 x $. Simplify using $ \tan x \cot x = 1 $: $ \tan^2 x + 2 + \cot^2 x $. Use identity $ \tan^2 x + \cot^2 x = (\tan x - \cot x)^2 + 2 $, so expression becomes $ (\tan x - \cot x)^2 + 4 $. The minimum occurs when $ \tan x = \cot x $, i.e., $ x = \frac{\pi}{4} + k\frac{\pi}{2} $, giving $ 0 + 4 = 4 $. \boxed{4} 📰 Kentucky Baptist Convention 606303 📰 Wes Anderson Hotel Chevalier 72241 📰 Half Dome Hike 603598 📰 Breaking Elizabeth Williams Reveals The Hidden Motivation Behind Her Breakout Role 9129792 📰 Unitedhealth Dividend Approvedstock Surges Over 15 In Viral Momentum 4757200 📰 How Many Ounces Of Water Should I Drink 372373 📰 This Netbenefits Hack On Wwwfidelitycom Is Changing How People Save Forever 3431978 📰 Dominantly Synonyms 9157114 📰 The Langoliers 3839402 📰 2021 International Plumbing Code 849495 📰 Kenyon 5400696 📰 Abandoned Archive 4878800 📰 Datetimeadd Sql Secrets Automate Date Math Like A Pro Watch Now 5818201 📰 Youll Never Guess What Chaotic Expression This Guy Made 8584187 📰 Rockstar Dept 3514500