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📰 Question: An environmental consultant models a river's flow as the line $y = -\frac{1}{2}x + 5$. Find the point on this line closest to the pollution source at $(4, 3)$. 📰 Solution: The closest point is the projection of $(4, 3)$ onto the line. The formula for the projection of a point $(x_0, y_0)$ onto $ax + by + c = 0$ is used. Rewriting the line as $\frac{1}{2}x + y - 5 = 0$, we compute the projection. Alternatively, parametrize the line and minimize distance. Let $x = t$, then $y = -\frac{1}{2}t + 5$. The squared distance to $(4, 3)$ is $(t - 4)^2 + \left(-\frac{1}{2}t + 5 - 3\right)^2 = (t - 4)^2 + \left(-\frac{1}{2}t + 2\right)^2$. Expanding: $t^2 - 8t + 16 + \frac{1}{4}t^2 - 2t + 4 = \frac{5}{4}t^2 - 10t + 20$. Taking derivative and setting to zero: $\frac{5}{2}t - 10 = 0 \Rightarrow t = 4$. Substituting back, $y = -\frac{1}{2}(4) + 5 = 3$. Thus, the closest point is $(4, 3)$, which lies on the line. $\boxed{(4, 3)}$ 📰 Question: A hydrologist models groundwater flow with vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find the angle between these flow directions. 📰 You Wont Believe How To Secretly Encrypt Outlook Emails In 2025 Finally Secure Your Mail 7608901 📰 Setsuna Tokage Shocking Powers What Sorcerer Harnesses The Storms Full Fury Dont Miss 2237657 📰 Things To Do When Bored 📰 Win Big Effort Free File Transfers With This Revolutionary Windows Usb Stick Setup 8439597 📰 What Time Is The Nascar Race 5088277 📰 Roblox Free Models 2930339 📰 Crazy Gsmes 📰 Sop Uncovered The Ultimate Definition That Will Revolutionize Your Revenue Strategy 75859 📰 Role Playing Game Games 📰 Why Atomic Mass Matters Unlocking The Mystery Of Atomic Number And Mass Number 5774545 📰 Sp 100 Explosion Why This 100 Stock Play Could Rewrite Your Portfolio 8491605 📰 Dd Yyyy Mm Revealed These 7 Days Changed Everything You Thought You Knew 6058360 📰 Lexmark Xm1246 Driver 📰 Zatima Episodes 3439146 📰 Mercenary Kings