Name:
• I will be checking for organization, conceptual understanding, and proper mathematical communication, as
well as completion of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are
doing.
• Use correct mathematical notation.
• You may work with your classmates. However, please submit your own work!
1
. (10 points) Consider the triangle with sides v = 〈v1,v
2
〉,w = 〈w1,w2〉 (in R2), and v − w, where θ is
the angle between v and w.
x
y
v−
w
w
v
(w1,w2)
(v1,v2)
θ
Show that
‖v‖‖w‖cos(θ) = v1w1 + v2w2 (1)
and hence conclude that the dot product of v and w can also be defined as
v · w = ‖v‖‖w‖cos(θ) (2)
(i.e., the geometric definition of the dot product).
2. In a certain online dating service, participants are given a 4-statement survey to determine their com-
patibility 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.
(OVER)
1
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 compati-
bility vectors is 60◦ or less. Participants are considered to be “incompatible” if the angle between their
compatibility vectors is 120◦ or larger. For angles between 60◦ and 120◦, pairs of participants are warned
that they “may or may not be compatible.”
(a) (5 points) Which pairs of participants are compatible and which pairs of participants are incompat-
ible?
(b) (2 points) 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? (Note: This is more of an open-ended question. I want you guys to think
about the positives or negatives to allowing a third response and explain if it would be better or
worse and give reasons.)
3. (3 points) If v×w = 2i− 3j + 5k, and v · w = 3, find tan(θ) where θ is the angle between v and w.
2