How embedding arithmetic work?

Probably I do not understand fully the emeddings - can you answer this question?

  • Using Open AI API I have got embedding of the words emb(man), emb(woman), emb(king), emb(queen)
  • I have programmed the embeddings operations for:
    • addition - adds elements of vectors
    • subtraction - subtracts elements of vectors
    • scalar multiply - provides sum of vector elements product
    • normalization - multiplies the vector by 1/SQRT(vector scalar multiply)
  • where possible I have checked, that my operations are working correctly
  • emb(x) is embedding of word x
  • I have calculated: embQueen = (emb(king)-emb(man))+emb(woman), always after subtraction or addition normalized the result
  • I have compared scalar multiply of embQueen ↔ emb(queen) being very low in comparison with emb(queen) ↔ emb(king)
  • I do not understand this result… What I am doing wrong? Can your provide working example of Open AI embedding arithmetic?