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v = rac4012 = rac103. - Portal da AcÃºstica

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Question:** In a 3D vector space modeling neural network parameters, suppose $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are unit vectors such that $\mathbf{x} + \mathbf{y} + \mathbf{z} = \mathbf{0}$. Find the maximum possible value of $\mathbf{x} \cdot \mathbf{y}$.

📌 Solution:** Since $\mathbf{x} + \mathbf{y} + \mathbf{z} = \mathbf{0}$, then $\mathbf{z} = -(\mathbf{x} + \mathbf{y})$.

📌 Since $\mathbf{z}$ is a unit vector:

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

📌 \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:

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