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Home » The Untold Story of Zhongli: How This Icon Shocked the World! - DNSFLEX

The Untold Story of Zhongli: How This Icon Shocked the World! - DNSFLEX

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 = \mathbf{i}((-2)(-1) - (3)(0)) - \mathbf{j}((1)(-1) - (3)(4)) + \mathbf{k}((1)(0) - (-2)(4))

📌 = \mathbf{i}(2) - \mathbf{j}(-1 - 12) + \mathbf{k}(0 + 8) = 2\mathbf{i} + 13\mathbf{j} + 8\mathbf{k}

📌 Thus, the vector is $\boxed{\begin{pmatrix} 2 \\ 13 \\ 8 \end{pmatrix}}$.

📌 Question:** A linguist modeling word embeddings in 3D space defines two unit vectors $\mathbf{u}$ and $\mathbf{v}$ with an angle of $60^\circ$ between them. What is the maximum possible value of $|\mathbf{u} \cdot \mathbf{v}|$?

📌 Solution:** The dot product of two unit vectors is given by $\mathbf{u} \cdot \mathbf{v} = \cos\theta$, where $\theta$ is the angle between them. Since $\theta = 60^\circ$, we compute:

📌 |\mathbf{u} \cdot \mathbf{v}| = |\cos 60^\circ| = \left|\frac{1}{2}\right| = \boxed{\frac{1}{2}}

📌 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:

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