Determine vectors similarity weighing the vectors lengths

Determine vectors similarity weighing the vectors lengths

I have many vector couples, for which I would like to assign a single
number (for each couple) that signify their similarity. I am running a
minimization algorithm on the whole group of vector couples so the
relative 'similarity grade' between couples is important.
If they were all at the same length, cosine similarity would be my obvious
choice. However, the vectors are not of same length and I would like to
take their size differences into account.
My intuition of vectors similarity are :
Vectors with the same direction (d) but of different sizes (s1,s2) are
more similar than vectors with same sizes (s1,s2) but of different
directions (d1,d2).
Vectors of the same size (Sshort) with different directions (d1,d2) are
more similar than the vectors of sam direction (d1,d2) with a larger size
((Slong).
In my context, this makes sense as (somewhat abstractly) if I wanted to
align the 2 vectors I would have needed less "energy" to rotate the
shorter ones.
Is there a standard method that determines vectors similarity along these
lines?
Thanks!