Act as a Machine Learning Researcher. Provide a mathematical derivation and explanation of the Scaled Dot-Product Attention mechanism used in Transformer models. Specifically, explain: 1) The role of the scaling factor 1/sqrt(d_k) in preventing vanishing gradients in the softmax, 2) The geometric interpretation of Queries, Keys, and Values, and 3) How multi-head attention differs from simply increasing the dimensionality of a single head. Use LaTeX formatting for equations and provide a Python implementation from scratch using only NumPy and PyTorch tensor operations.