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Home » This $10,000 / 6 Breakthrough Shocked Millions—See How It Changed Everything! - Sterling Industries

This $10,000 / 6 Breakthrough Shocked Millions—See How It Changed Everything! - Sterling Industries

📅 April 16, 2026 👤 scraface

Apr 16, 2026

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📌 \vec{w} = 2\vec{u} - 3\vec{v} = 2\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} - 3\begin{pmatrix} -2 \\ 4 \\ 1 \end{pmatrix}

📌 Distribute the scalars:

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

📌 Subtract the vectors component-wise:

📌 \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 $P = (3, -1, 2)$.

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