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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} $. 📰 Cdt Time 3477009 📰 Grow A Garden Generator 3126744 📰 Blackberry Syndrome What This Tragic Trend Is Doing To Your Brain Stop It Now 3897996 📰 1V1 Lol Unblock 267848 📰 Hurrymicrosoft Finance Jobs Are Spike In Demand Land Your Dream Role Today 102410 📰 Currency Rates Of Today 8739656 📰 Future Endeavors 6822808 📰 Tortoise Shell Nails The Secret Behind Flawless Snap On Nails Everyones Demanding 3631632 📰 This Plaid Shirt Is So Unique Its Being Called A Hidden Trendmaster 9098932 📰 The Ultimate Summer Mens Parfum That Smells Like Vacationexcuse Me 328186 📰 5Question Let Gx2 2 X4 4X2 5 Find Gx2 2 309730 📰 Ashland Mo 2640343 📰 Filter For Kitchen Sink 918240 📰 Reese Witherspoon Book 735884 📰 Why Am I So Laggy On Roblox 1084650