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buffy vampire slayer xander - NBX Soluciones

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \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 $\|\vec{d}(t)\|^2$:

📌 \|\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 $\vec{r}(t)$:

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