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-v_2 = -2 \implies v_2 = 2 - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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}

📌 So all edges must be $\sqrt{2}$. Set up equations for $AD = BD = CD = \sqrt{2}$:

📌 $AD^2 = (x - 1)^2 + y^2 + z^2 = 2$

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