Six handcrafted 2D vectors use equal coordinate weights. Query components are bounded to −4..4 and snapped to half units; Top k is 1..6. The fixed −5..5 square plot shows actual toy coordinates with equal unit scales, not a projection. Exact coincidence groups glyphs and labels only; table identities remain separate. Scenarios preserve query, scoring rule and k while changing stored vectors.

Exact brute-force scoring evaluates all six items. Euclidean sorts by squared distance (exact on this half-unit grid), then fixed ID A..F; the table displays its square root. Cosine is dot divided by both nonzero lengths, sorted highest first. Its comparison key is rounded to 12 decimals for deterministic numerical ties; fixed ID resolves equal keys. Display rounding to six decimals is separate. Tie order is reproducible, not semantic superiority.

A zero query or zero item has no direction, so geometric cosine is Undefined and this demo excludes it. A zero query yields no eligible cosine results. Some libraries assign numerical zero to these cases; that convention does not create a direction. Euclidean works at zero. Nonzero cosine 0 means perpendicular, 1 means same direction rather than identical length, and −1 means opposite. No score is a relevance probability.

The demo does not normalize coordinates when you choose Cosine. If both vectors were normalized, squared Euclidean distance would equal 2−2×cosine and rankings would agree; raw vectors need not agree. Query/scenario/metric/k edits clear stale explanations. Prediction or Reset restores the current baseline; free-exploration Reset starts Experiment 1. There is no training, semantic query text, approximate index, model inference, task ground truth or retrieval-quality metric here.