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Understanding the Context
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📰 Question: A volcanic fissures edge is modeled by the line $x = 5 - 3t$, $y = 2 + t$, $z = -1 + 4t$. Determine the integer coordinates of the point on this line closest to a seismic sensor at $(1, 5, -3)$.
📰 $(-4 + 3t)(-3) + (3 - t)(1) + (-2 - 4t)(4) = 0$
📰 Question: Given three vertices of a regular tetrahedron at $(0, 0, 0)$, $(1, 1, 1)$, and $(1, 0, 0)$, find the coordinates of the fourth vertex with integer components.
