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📰 Solution: Since $\angle AOB = 90^\circ$, $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Compute $\|\overrightarrow{OC}\|^2 = 4\|\overrightarrow{OA}\|^2 + \|\overrightarrow{OB}\|^2 - 4\overrightarrow{OA} \cdot \overrightarrow{OB} = 4(4) + 9 - 0 = 25$. Thus, $\|\overrightarrow{OC}\| = 5$. 📰 Question: A plant biologist minimizes $(\tan x + \cot x)^2 + (\sin x + \csc x)^2$ for $0 < x < \frac{\pi}{2}$. Find the minimum value. 📰 Solution: Let $t = \sin x + \csc x$. Note $\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x}$. Let $y = \sin x \cos x = \frac{1}{2}\sin 2x$, so $\tan x + \cot x = \frac{2}{\sin 2x}$. The expression becomes $\left(\frac{2}{\sin 2x}\right)^2 + (\sin x + \csc x)^2$. Let $z = \sin x + \csc x \geq 2$ by AM-GM. Substitute $u = \sin 2x$, then minimize $\frac{4}{u^2} + \left(\frac{1}{\sin x} + \sin x\right)^2$. After calculus or substitution, the minimum occurs at $x = \frac{\pi}{4}$, yielding $(1 + 1)^2 + (\sqrt{2} + \frac{1}{\sqrt{2}})^2 = 4 + \left(\frac{3}{\sqrt{2}}\right)^2 = 4 + \frac{9}{2} = \frac{17}{2}$. However, correct simplification shows the minimum is $9$ when $x = \frac{\pi}{4}$. 📰 Apple Spreadsheet App 4254422 📰 Basketball Net That Cuts The Game In Halfyoure Seeing The Magic Now 5878134 📰 The Shocking Truth About Luke Skywalker You Didnt Know 7642937 📰 The Shocking Truth About Predalien Why Its Trending Hard Online Now 3269582 📰 Bank Of America Student Bank Account 6736074 📰 The Real Truth About Lena Anderson You Were Never Meant To Know 213589 📰 Lauren Graham Age 9016705 📰 Rambler Ru Exposed 7 Hidden Secrets You Cant Ignore 8083175 📰 Wells Fargo Check Reorder 4227415 📰 Free Apple Music Trial What Serious Features Do They Hide Behind That Spinnable Pop Up 8224374 📰 Crazy Games Bike Mystery Exposed Race Through Mind Blowing Challenges Surprises 8416607 📰 The One Internet Address Format Hack To Boost Your Online Credibility 432024 📰 Nintendo Us Twitter 8078425 📰 Osmolarity Vs Osmolality 2280967 📰 Patel Fbi 3176083