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Home » Unlock the Ultimate Power: 10 Must-Have Minecraft Sword Enchantments You Can’t Miss! - DNSFLEX

Unlock the Ultimate Power: 10 Must-Have Minecraft Sword Enchantments You Can’t Miss! - DNSFLEX

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 Question: Find the value of $x$ such that the vectors $\begin{pmatrix} 2 \\ x \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}$ are orthogonal.

📌 Two vectors are orthogonal if their dot product is zero. Compute the dot product:

📌 \begin{pmatrix} 2 \\ x \\ -3 \end{pmatrix} \cdot \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix} = 2 \cdot 4 + x \cdot (-1) + (-3) \cdot 1 = 8 - x - 3 = 5 - x

📌 - x = 0 \Rightarrow x = 5

📌 Question: A hydrologist is minimizing the energy cost of pumping water through an aquifer modeled by the function $f(x) = 2x^2 - 8x + 11$. Find the minimum value of $f(x)$.

📌 This is a quadratic function in the form $f(x) = ax^2 + bx + c$, with $a = 2 > 0$, so the parabola opens upward and has a minimum at the vertex.

📌 The x-coordinate of the vertex

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