It depends on the definition of your multiplication operator, although, yes, it is true that <0,v>=0 for all v in any vector space. Same goes for scalar multiplication in any vector space.
But nobody can stop me from defining the multiplication operator as addition. (e.g. 1*2 = 3, and 1*0 = 1).
I'm lately studying when I really should be watching anime.
Now I'm behind by one week. Luckily I'm not following that many shows this seasons.
NO. I wish I could make myself study.