Astrologers claim to be able to recognize trends in personality and fortune that depend on an individual's birthday by somehow incorporating where the stars were 2000 years ago. Suppose that instead of star-gazers coming up with stuff, math teachers who like linear algebra (we'll call them vectologers) had come up with a similar system as follows: Consider your birthday as a row vector (month day). For instance, I was born on July 12 so my vector would be (7 12). Vectologers have made the rule that how well individuals get along with each other depends on the angle between vectors. The smaller the angle, the more harmonious the relationship.
(a) Find the angle between your vector and mine, in radians.
(b) Would you get along better with me, or with a professor born on September 19?
(c) For maximum harmony in a relationship, when should the other person be born?
(d) Is there a person with whom you have a "worst case" relationship, i.e., your vector and theirs are orthogonal? If so, what are the birthdate(s) for such people? If not, explain why not.