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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 \Rightarrow \|\mathbf{x}\|^2 + \|\mathbf{y}\|^2 + 2\mathbf{x} \cdot \mathbf{y} = 1

📌 Since $\|\mathbf{x}\| = \|\mathbf{y}\| = 1$, this becomes:

📌 + 1 + 2\mathbf{x} \cdot \mathbf{y} = 1 \Rightarrow 2 + 2\mathbf{x} \cdot \mathbf{y} = 1 \Rightarrow \mathbf{x} \cdot \mathbf{y} = -\frac{1}{2}

📌 This is the only possible value under the constraint, so the maximum is:

📌 \boxed{-\frac{1}{2}}

📌 Question:** A digital transformation algorithm rotates a vector in 3D space using $\tan(\theta)$, where $\theta$ is the angle between two planes. If $\cos \theta = \frac{1}{3}$, compute $\tan \theta$.

📌 Solution:** Use identity:

📌 \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}

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