Match each similarity metric to its formal definition

Four similarity metrics, four shapes. - **Cosine similarity** divides the dot product by the product of the norms, leaving you with the cosine of the angle between the vectors. Range: -1 to 1. - **Dot product** is the sum of element-wise products. On unit-length vectors it equals cosine. - Euclidean distance is the straight-line distance between the vector tips. On unit-length vectors it is a monotone function of cosine, so ranking by either gives the same order. - Hamming distance counts bit positions that differ. It is the metric for binary-quantized embeddings, not for normal float vectors. In practice most 2026 embedding APIs return unit-norm vectors. Cosine, dot product, and Euclidean then produce identical rankings and you pick by computational cost. Binary quantization swaps to Hamming.

**Cosine** measures the angle between vectors. **Dot product** is the raw inner product. Euclidean measures straight-line distance. Hamming counts differing bit positions in binary vectors. The relationships matter: $$\cos(u, v) = \frac{u \cdot v}{\|u\| \, \|v\|}$$ $$\|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2 u \cdot v$$ On unit-norm vectors all three (cosine, dot, Euclidean) rank items identically. Production code stores normalized vectors and queries with dot product because it is the fastest of the three. Hamming distance only makes sense on binary vectors. After binary quantization (sign of each dim), Hamming replaces cosine and runs as XOR + popcount on packed 64-bit words. On modern CPUs this is 5-10x faster than float cosine at acceptable recall cost. The summary table: dense float vectors → cosine or dot on normalized data. Binary-quantized vectors → Hamming.