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Home » è¡¨è¾¾å¼åç®: \(rac2(x - 2)(x + 2)4x = rac(x - 2)(x + 2)2x\) - Abu Waleed Tea

è¡¨è¾¾å¼åç®: \(rac2(x - 2)(x + 2)4x = rac(x - 2)(x + 2)2x\) - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \vec{w} = \begin{pmatrix} 6 - (-6) \\ -2 - 12 \\ 4 - 3 \end{pmatrix} = \begin{pmatrix} 12 \\ -14 \\ 1 \end{pmatrix}

📌 Thus, the vector $\vec{w}$ is:

📌 \boxed{\begin{pmatrix} 12 \\ -14 \\ 1 \end{pmatrix}}

📌 Question: A hydrologist models the flow of groundwater in a 3D aquifer using vectors. If the flow vector $\vec{f} = \begin{pmatrix} 4 \\ -2 \\ 5 \end{pmatrix}$ represents the direction and rate of flow, find the point on the line $\vec{r}(t) = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix} + t\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ that is closest to the point $P = (3, -1, 2)$.

📌 Let the point on the line be $\vec{r}(t) = \begin{pmatrix} 1 + 2t \\ t \\ -3 - t \end{pmatrix}$.

📌 Let $P = (3, -1, 2)$. The vector from the point on the line to $P$ is:

📌 \vec{d}(t) = \vec{P} - \vec{r}(t) = \begin{pmatrix} 3 - (1 + 2t) \\ -1 - t \\ 2 - (-3 - t) \end{pmatrix} = \begin{pmatrix} 2 - 2t \\ -1 - t \\ 5 + t \end{pmatrix}

📌 To find the closest point, minimize the magnitude of $\vec{d}(t)$, or equivalently, minimize $\|\vec{d}(t)\|^2$:

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