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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}}. 📰 Xle Stock Just Spiked 300Heres Why Investors Are Racing To Join The Ride 2381234 📰 Mac Key Command 2355301 📰 Eden Brolin 1731851 📰 Barry Gibbs 1919374 📰 Get Secret Data Free How To Remove Passwords From Excel Instantly No Crack Needed 941806 📰 Target Dei Lawsuit 617616 📰 Girafarig Evolution The Groundbreaking Discovery That Rewrote Animal Adaptation History 8282109 📰 Wade Wilson Murderer 6981707 📰 From Static To Sprint The Ultimate Story Of How Cars 2 Characters Changed Everything 6272353 📰 50S Clothes Dress Secrets That Will Make You Look Like A Timeless Beauty 329185 📰 Girl Dress Up Games 3990037 📰 This Secret Dirt Bike Fun Awaitsright Beyond Your Doorstep 5044364 📰 Ranks New Dti Codes Just Launched Boost Your Refund Tonight 4637763 📰 Max Amount To Contribute To Roth Ira 8364648 📰 Each Autonomous Vehicle Has 5 Choices 9068559