Deriving principal component out of cosine similarity

Question

Cosine similarity is defined as

$\frac{A\cdot B}{|A||B|}$

Now, if you multiply it by $|B|$, so that you have

$\frac{A \cdot B}{|A|}$,

what is the name for this?

Could this be considered a principal component?

and what kind of properties / interpretation would it have,

particularly in comparison to cosine distance or Euclidean distance?

Answer 1

It's a length of the projection (multiplied by sign of $\cos$) of the vector $B$ on the vector $A$.