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📰 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. 📰 Solution: The angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $2(1) + (-3)(4) = 2 - 12 = -10$. Compute magnitudes: $\|\mathbf{a}\| = \sqrt{2^2 + (-3)^2} = \sqrt{13}$, $\|\mathbf{b}\| = \sqrt{1^2 + 4^2} = \sqrt{17}$. Thus, $\cos\theta = \frac{-10}{\sqrt{13}\sqrt{17}}$. Rationalizing, $\theta = \arccos\left(-\frac{10}{\sqrt{221}}\right)$. $\boxed{\arccos\left(-\dfrac{10}{\sqrt{221}}\right)}$ 📰 This Secret Tool In Litcharts Changes College Grades Forever 1840251 📰 Switch 2 Vs Nintendo Switch The Ultimate Showdown You Didnt Expect 5421340 📰 From Speedster To Legend The Untold Story Of Barry Allen That Defies Expectations 1375660 📰 See How Myapps Transforms Your Lifestart Using It Now 9818437 📰 The Shocking Difference Between Mri And Ct Scans Youve Never Heard Before 815878 📰 Croc That Feeds In Darkness The Absolute Killer Protesting Every Shadow 8494562 📰 Why Investors Are Obsessively Analyzing Yahoo Finance Historical Share Price Charts 3721038 📰 Audio Id Roblox 5244 📰 Add Bang To Your Brand With These Trendy Coffee Clipart Imagesfree Eye Catching 5485723 📰 Aroma Joes Whisper Trick Youve Never Seentruth Is Astounding 9537498 📰 The Untold Truth About Karima Jacksons Hidden Life You Didnt Know 4329935 📰 Latinico 4620535 📰 Do Fox Eat Domestic Cats 7068082 📰 Gym Franchise 1554300 📰 Rapid Environment Editor 2663014