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how do i make a minecraft map - Sourci

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 5Question: Given three vertices of a cube at $(0,0,0)$, $(1,0,0)$, and $(0,1,0)$, find the coordinates of the fourth vertex in the same face, assuming all coordinates are integers.

📌 Solution: The given points lie on the base face of the cube in the $xy$-plane. The fourth vertex of the square face must complete the rectangle. The missing vertex is $(1,1,0)$, as it forms equal sides and right angles with the given points.

📌 \boxed{(1,1,0)}

📌 Question: For a polynomial modeling the trajectory of a seed dispersal algorithm, $z^6 + z^4 + z^2 + 1 = 0$, find the maximum imaginary part of a root expressed as $\cos \theta$, where $\theta$ is an acute angle.

📌 Solution: Factor the polynomial as $\frac{z^8 - 1}{z^2 - 1} = 0$ (excluding roots of $z^2 = 1$). The roots are the 8th roots of unity except $\pm 1$. The roots with maximum imaginary part are $e^{i\pi/4}$ and $e^{i3\pi/4}$, with imaginary part $\frac{\sqrt{2}}{2} = \cos(\pi/4)$.

📌 \boxed{\cos\left(\frac{\pi}{4}\right)}

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