professor erno rubik - Malaeb

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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)} 📰 Sushi Neko Revealed Nutritional Secrets Killing Every Sushi Lover 6229790 📰 Shocking Dr Emma Clark Gut Drops Secrets That Anyone Needs To See Now 1993524 📰 When Do They Stop Serving Alcohol 5158275 📰 Bass In Spanish 3882444 📰 Textedit Download Mac Os X 5228790 📰 Get To Know U Questions 62102 📰 Runaway Jury Cast 9168779 📰 Gireesh Velugubanti Md Shocked Us His Hidden Medical Breakthrough You Cant Ignore 4650101 📰 Peak Rate 200 140 340 Per Day 2984819 📰 Sous Vide Chicken Breast This Secret Trick Makes It Absolutely Irresistible 6153242 📰 Samsung A15 5G Reviews 7584248 📰 Kokoist Reveals The Secret That Will Blow Your Mind 5464022 📰 Apush Score Calculator 1007616 📰 Without Trace Series 5889495 📰 This Fire Hip Move Will Revolutionize Your Style Overnight 2644960