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Download the Secret CAPUT PNG Logo Template—Free & Professional! - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A museum curator is restoring a triangular panel from a historical instrument, where the triangle has vertices at $A = (1, 3, -2)$, $B = (4, 1, 1)$, and $C = (2, -1, 3)$. Find the point $P$ on the line through $A$ and $B$ that is closest to point $C$.

📌 Solution: The line through $A$ and $B$ is parametrized as:

📌 \mathbf{r}(t) = \mathbf{A} + t(\mathbf{B} - \mathbf{A}) = \langle 1, 3, -2

📌 angle + t\langle 3, -2, 3

📌 angle = \langle 1 + 3t, 3 - 2t, -2 + 3t

📌 Let $P = \mathbf{r}(t)$. The vector $\overrightarrow{CP} = \mathbf{r}(t) - \mathbf{C} = \langle 1 + 3t - 2, 3 - 2t + 1, -2 + 3t - 3

📌 angle = \langle -1 + 3t, 4 - 2t, -5 + 3t

📌 To minimize the distance, minimize $\|\overrightarrow{CP}\|^2$:

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