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Home » \( n \geq 5 \), so \( n = 5 \) - Abu Waleed Tea

\( n \geq 5 \), so \( n = 5 \) - Abu Waleed Tea

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: For two vectors to be orthogonal, their dot product must be zero. Compute the dot product:

📌 \vec{v} \cdot \vec{w} = 3x + (-2)(4) + x(-1) = 3x - 8 - x = 2x - 8

📌 Set the dot product equal to zero:

📌 2x - 8 = 0 \quad \Rightarrow \quad 2x = 8 \quad \Rightarrow \quad x = 4

📌 Thus, the value of $x$ that makes the vectors orthogonal is $\boxed{4}$.

📌 Question: An educator is using a STEM project to teach vector geometry. In a 3D coordinate system, a student plots three vertices of a regular tetrahedron: $A(1, 0, 0)$, $B(0, 1, 0)$, and $C(0, 0, 1)$. Find the integer coordinates of the fourth vertex $D$ such that all edges of the tetrahedron are of equal length.

📌 Solution: Let the fourth vertex be $D = (x, y, z)$. In a regular tetrahedron, all six edges must be equal. First, compute the distance between known points:

📌 AB = \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{1 + 1} = \sqrt{2}

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