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📰 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} $. 📰 \boxed{\left( \dfrac{173}{25}, \dfrac{156}{25} \right)} 📰 Master Hacking Nitro Type With These Pro Tricks Delivering Instant Kills 3198349 📰 Unleash Your Inner Artist Mario Cow Coloring Pages For Endless Creativity 9192536 📰 No Oracle But This Answer Will Change Everythingyes Or No 4909940 📰 Tatget Hacks That Saved Me 500This Secret Stop Everything Changed 6018090 📰 Unlock Hidden Gems With Beau Cam The Ultimate Hidden Beauty Stream Waiting For You 3515632 📰 Stadium Centre 8964709 📰 Stunt Madalin Cars 2 Unbelievable Car Tricks You Wont Believe Are Real 430368 📰 Colette Momsen 9462507 📰 Jepq Yield Is Soaringheres How To Tap Into Its Massive Returns 6473103 📰 Hypocorism 597453 📰 Why Is Steam Down 3218405 📰 Youll Never Guess What A Socket Is In Computer Networkslearn It Now 628448 📰 Aire De Olive 1883359 📰 Gg Hoopjourney 8114944 📰 Whats Really In The Diagnostic Criteria For Autistic Disorder Learn Before Its Too Late 898621