This is another seemingly stupid question, but it seems counter-intuitive for me to use a cosine of the angle as a similarity measure (in direction) of two vectors as opposed to the angle itself.
To elaborate, when we take the similarity of vectors v1, v2 and v3
we would expect some linearity in the measure, for example d(v1,v2)+d(v2,v3) = d(v1,v3). A property cosine measures do not have; We are only taking one component in a 2D space.
Do we just use the cosine of the angle for convenience, because we do not want to define and redefine the range of the (negative vs positive) angles? Or is there another, deeper interpretation?
edit: changed "distance" to "similarity in direction"