Does arm pit hair grow at the same rate as other hair?

No. Different parts of body have different % and duration of hair follicles in the growth phase (anagen). In general the rate of anagen is proportional to length of the hair. In other words, people who have difficulty growing long hair on their scalp have a shorter anagen. Normal anagen duration: scalp 2-6 yrs, eyebrows 4-8 wks, beard/chin 1 yr, armpits 4 months etc..
Hair growth. Hair follows a specific growth cycle with three distinct and concurrent phases: anagen, catagen, and telogen phases. Each phase has specific characteristics that determine the length of the hair. All three phases occur simultaneously; one strand of hair may be in the anagen phase, while another is in the telogen phase.