playa mill valley - Belip

🔗 Related Articles You Might Like:

📰 Solution: The matrix $\mathbf{M}$ is constructed by placing the images of the standard basis vectors as its columns. Thus, $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}$. Verifying, $\mathbf{M} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, confirming correctness. $\boxed{\begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}}$ 📰 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. $\boxed{(4, 3)}$ 📰 Guys Big Breakthrough The Best Cd Case That Carries More Than Just Discs 9706018 📰 Jin Jia 8488376 📰 Best Running Apps 7856367 📰 Employee Centrals Unhinged Policy Will Shock Every Worker Inside 7209819 📰 Jjk Wallpaper 8485540 📰 Lineleap Breakthrough The Simple Fix That Transformed Fitness Click To Learn How 6364810 📰 From Fresh Producer To World Class Rap Genius His Story Will Shock You 355566 📰 Crossy Road 2206918 📰 Darato Sport Shocks Every Athleteyou Wont Believe What This Gear Changes 658666 📰 You Wont Guess Whos Bringing The Andy Capp Comic To Lifethis Comic Changed Everything 2154984 📰 God Vishnu Sahasranamam 8083154 📰 When Does Chicago Fire Return 355537 📰 How A Bush Plane Changed My Life Shocking Stories From The Wild 8447188 📰 Indiana Fever Caitlin Clark Injury 1082084 📰 X Files Star Tom Selleck Young Shocking Secrets Revealed About His Early Days 7165117