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Home » Mario’s Height Shock: The Average Height (Spoiler: It’s Not What You Think!) - Abbey Badges

Mario’s Height Shock: The Average Height (Spoiler: It’s Not What You Think!) - Abbey Badges

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 Question:** A cloud computing system routes data along vectors in $\mathbb{R}^3$. Given $\mathbf{v} \times \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix}$, find $\mathbf{v}$.

📌 Solution:** Let $\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$. Then:

📌 \mathbf{v} \times \mathbf{a} = \begin{pmatrix} y(4) - z(-1) \\ z(2) - x(4) \\ x(-1) - y(2) \end{pmatrix} = \begin{pmatrix} 4y + z \\ 2z - 4x \\ -x - 2y \end{pmatrix} = \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix}

📌 4y + z = 5 \\

📌 2z - 4x = -6 \\

📌 -x - 2y = -1

📌 From the third equation: $x = 1 - 2y$.

📌 2z - 4(1 - 2y) = -6 \Rightarrow 2z - 4 + 8y = -6 \Rightarrow 2z + 8y = -2 \Rightarrow z + 4y = -1

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