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📰 Solution: The closest point occurs where the vector from $ (5, 7) $ to the line is perpendicular to the direction vector $ \begin{pmatrix} 3 \\ 4 \end{pmatrix} $. Let the point on the line be $ (2 + 3t, -1 + 4t) $. The vector from $ (5, 7) $ to this point is $ \begin{pmatrix} 2 + 3t - 5 \\ -1 + 4t - 7 \end{pmatrix} = \begin{pmatrix} -3 + 3t \\ -8 + 4t \end{pmatrix} $. Dot product with $ \begin{pmatrix} 3 \\ 4 \end{pmatrix} $ must be zero: $ 3(-3 + 3t) + 4(-8 + 4t) = 0 $. Simplify: $ -9 + 9t - 32 + 16t = 0 \Rightarrow 25t - 41 = 0 \Rightarrow t = \frac{41}{25} $. Substitute back: $ x = 2 + 3 \cdot \frac{41}{25} = \frac{173}{25} $, $ y = -1 + 4 \cdot \frac{41}{25} = \frac{156}{25} $. 📰 \boxed{\left( \dfrac{173}{25}, \dfrac{156}{25} \right)} 📰 Question: A medical imaging algorithm uses vectors $ \begin{pmatrix} x \\ 2 \end{pmatrix} $ and $ \begin{pmatrix} 3 \\ -1 \end{pmatrix} $ to detect anomalies. Find $ x $ so the vectors are orthogonal. 📰 Air Plane Crash 7213608 📰 What Is A Woman Reviews 1109811 📰 Are Artichokes Good For You 1749545 📰 Greer Fire Sustainable Doesnt Mean Safethis Explosive Revelation Stands Alone 9218009 📰 This Bluegreen Owner Login Mistake Could Cost You Everythingact Now 8599036 📰 The Ultimate Water Bounce House Thatll Make Guests Squeal Shop Now Before Its Gone 9994361 📰 Cast Of The Movie Men In Black 3 6756980 📰 Verizon Going To Mexico 1167516 📰 You Wont Believe These Hidden Roth Ira Benefits You Need To Know 7228119 📰 Hipaa Coverage Explainedwhy Every Health Organization Must Stay Compliant Now 7322786 📰 The Ultimate Watercolor Background Guide Perfect For Instagram Blogs More 3718171 📰 Credit Cards With No Fee On Balance Transfers 3150419 📰 You Wont Burn In Hellsatan Hercules Mind Blowing Rise Exploded Online 5378151 📰 The Hidden Fault Lurking In Your Engine Code P0304 Will Shock You 7758939 📰 The Sp 500 Ticker Just Shocked Everyone Could Your Portfolio Crash Today 1873572