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📰 \boxed{36.00} 📰 Question: In a science experiment, a drone flies along the line $ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix} $. Find the point on this line closest to $ (5, 7) $. 📰 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} $. 📰 Bank Heist Game Breakout Experience The Ultimate Heist Thrill On Your Device 6924731 📰 You Wont Believe What This Psp 3000 Does Better Than Any Gaming Console Ever 61829 📰 Dove Camerons Secrets Unfurl The Stunning Nude Moment Initially Hidden 2181361 📰 Why Adan Lake Is The Ultimate Escape Discover Its Breathtaking Beauty Before Its Too Late 4504751 📰 These 7 One Piece Characters Will Change How You See The Epic Adventure 6360951 📰 Kibosh Meaning 9703318 📰 2025 Roth Ira Income Limits Are You Just 10 Dollars Away From Limits 2770836 📰 Mount Azuria 6954660 📰 This Minor Character Hint In Brocoflix Exposed A Global Conspiracy No One Saw Coming 8545312 📰 Stuck Out Of Your Windows Account This Surprise Method Works Like Magic 5933264 📰 Waste Collection Schedules 2609374 📰 Justbuildlol How I Built My Ultimate Game In Minutes Youll Want To Try It Too 7980463 📰 Top Rated Ice Makers 8335343 📰 Pagaya Stock 1811583 📰 Dont Sleep Through This Hoodwinked Exposes The Biggest Consneal In Modern Cinema 7773583