Three similarities agree on the unit sphere

ProofFoundation⏱ 15 minRAGcosinedot producteuclideannormalisation

Theory: Embedding Geometry and Similarity →

Problem

Prove $\|u-v\|_2^2 = 2-2\langle u,v\rangle$ for unit vectors, deduce that ranking by cosine, dot product and negative Euclidean distance coincide on the sphere, and give an explicit example off the sphere where dot-product and Euclidean rankings disagree.

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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