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Home » Fall in Love with Italian Goodbye Words: The Top 5 Most Memorable Phrases! - Veritas Home Health

Fall in Love with Italian Goodbye Words: The Top 5 Most Memorable Phrases! - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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

📌 \|\vec{d}(t)\|^2 = (2 - 2t)^2 + (-1 - t)^2 + (5 + t)^2

📌 Add them up:

📌 \|\vec{d}(t)\|^2 = (4 - 8t + 4t^2) + (1 + 2t + t^2) + (25 + 10t + t^2) = 6t^2 + 4t + 30

📌 Take derivative and set to 0:

📌 \frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0 \Rightarrow t = -\frac{1}{3}

📌 Substitute $t = -\frac{1}{3}$ into $\vec{r}(t)$:

📌 \vec{r}\left(-\frac{1}{3}\right) = \begin{pmatrix} 1 + 2(-\frac{1}{3}) \\ -\frac{1}{3} \\ -3 - (-\frac{1}{3}) \end{pmatrix} = \begin{pmatrix} 1 - \frac{2}{3} \\ -\frac{1}{3} \\ -3 + \frac{1}{3} \end{pmatrix} = \begin{pmatrix} \frac{1}{3} \\ -\frac{1}{3} \\ -\frac{8}{3} \end{pmatrix}

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