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📰 Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: $ x \cdot 3 + 2 \cdot (-x) = 3x - 2x = x $. Set $ x = 0 $. Thus, $ oxed{0} $ is the solution. 📰 Question: Determine the number of real solutions to the equation $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $. 📰 Solution: Let $ y = \sin(2x) $. The equation becomes $ y^2 + 3y + 2 = 0 $, which factors as $ (y + 1)(y + 2) = 0 $. Thus, $ y = -1 $ or $ y = -2 $. Since $ \sin(2x) $ ranges between $-1$ and $1$, $ y = -2 $ is invalid. For $ y = -1 $, $ \sin(2x) = -1 $ has infinitely many solutions (e.g., $ 2x = rac{3\pi}{2} + 2\pi k $, $ k \in \mathbb{Z} $). However, if restricted to a specific interval (not stated here), the count would depend on the domain. Assuming $ x \in \mathbb{R} $, there are infinitely many solutions. But if the question implies a general count, the answer is oxed{ ext{infinitely many}}. 📰 Move Whats Hidden The Untouched Green Coffee Mountain That Defies Discovery 5540401 📰 What Is The Best Iphone To Buy 400181 📰 Finance Yahoo Down 2602289 📰 Shenzhen Stock Exchange A Share Index Explodeswhy You Cant Afford To Miss This Surge 6730532 📰 Sp Global Stock Price Shock Heres What Shocked Wall Street Today 5931500 📰 Function Of The Frontal 1021344 📰 Unreal Skill Crazygames Ragdoll Archers Take Down Opponents Like Pros 3059816 📰 The Dark Side Of Zapaka Why This Trend Is Sickeningly Addictive And Dangerous 6647856 📰 Discover Rhythm Heaven The Ultimate Sanctuary Where Music Transforms Your Soul 4905208 📰 Tyrese Haliburton Contract 167287 📰 How Many Terms Can A Us President Serve 7762931 📰 Getting The Mmr Vaccine Right This Schedule Will Change Your Immunization Game 5089048 📰 How Many Brain Cells Do Humans Have 6380416 📰 Subtract 6 3X 180 1172942 📰 Bye In Russian 3977602