In a certain online dating service, participants are given a 4-statement survey to determine their compatibility with other participants. Based on the questionnaire, each participant is notified if they are compatible with another participant. Each question is multiple choice with the possible responses of "Agree" or "Disagree," and these are assigned the numbers 1 or −1, respectively. Participant’s responses to the survey are encoded as a vector in R4, where coordinates correspond to their answers to each question. Here are the questions: Question #1. I prefer outdoor activities, rather than indoor activities.
Question #2. I prefer going out to eat at restaurants, rather than cooking at home.
Question #3. I prefer texting, rather than talking on the phone.
Question #4. I prefer living in a small town, rather than a big city.
Here are the results from the questionnaire, with a group of 5 participants:
Question#1 Question# 2 Question#3 Question#4
Participant A 1 1 -1 -1
Participant B -1 1 1 1
Participant C -1 -1 1 1
Participant D 1 -1 -1 -1
Participant E 1 -1 1 1
Two participants are considered to be "compatible" with each other if the angle between their compatibility vectors is 60 degree or less. Participants are considered to be "incompatible" if the angle between their compatibility vectors is 120 degree or larger. For angles between 60 degree and 120 degree, pairs of participants are warned that they "may or may not be compatible.
a) Which pairs of participants are compatible?
b) Which pairs of participants are incompatible?
c) How would this method of testing compatibility change if the questionnaire also allowed the answer "Neutral, " which would correspond to the number zero in a participant's vector? Would this be better than only allowing "Agree" or "Disagree"? Could anything go wrong if we allowed "Neutral" as an answer?