Question

What is the cosine of the angle between vectors u and v? Show how you set up the solution, and your actual calculations.
u = [ 4 2 0 -1 ] v = [ 3 -3 2 0 ]
II/ [ 2 points each] For each example below, state whether the feature is

i)    Numerical: discrete or continuous
ii)    Categorical or nominal: ordinal
If you’re doubtful about any part of an answer, add a few words of explanation.

a) Brightness outdoors as measured by a light meter.
b) Brightness outdoors as indicated by people’s subjective judgments (e.g. “it’s pretty bright”, “hard to see”).
c) Angles as measured in degrees between 0 and 360.
d) Telephone area code.
e) Bronze, silver, and gold medals as awarded at the Olympics.
f) Height above sea level in meters.
g) Number of patients in a hospital.
h) Ability to pass light in terms of the following values: opaque, translucent, transparent.
i) Military rank.
j) Density of a substance in grams per cubic centimeter.
k) Coat check number. (When you attend an event, you often can give your coat to someone who, in turn, gives you a number to use to claim your coat when leaving.)

Answer 1

cosine of the angle between vectors u and v is,

$cos \theta = \frac{u.v}{|u||v|}$

u.v = 4*3 + 2 * -3 + 0 * 2 -1 * 0 = 6

$|u| = \sqrt{4^2 + 2^2 + 0^2 + (-1)^2} = 4.582576$

$|v| = \sqrt{3^2 + (-3)^2 + 2^2 + 0^2} = 4.690416$

$cos \theta = \frac{6}{4.582576 * 4.690416} = 0.2791452$

a) Brightness outdoors as measured by a light meter -   Numerical and continuous
b) Brightness outdoors as indicated by people’s subjective judgments (e.g. “it’s pretty bright”, “hard to see”). - Categorical and nominal
c) Angles as measured in degrees between 0 and 360. - Numerical and continuous
d) Telephone area code. - Numerical and discrete
e) Bronze, silver, and gold medals as awarded at the Olympics. - Categorical and Ordinal
f) Height above sea level in meters. - Numerical and continuous
g) Number of patients in a hospital. - Numerical and discrete
h) Ability to pass light in terms of the following values: opaque, translucent, transparent. - Categorical and Ordinal
i) Military rank. - Categorical and Ordinal
j) Density of a substance in grams per cubic centimeter. - Numerical and continuous
k) Coat check number. - Numerical and discrete