#ATTENDANCE QUIZ for Lecture 1 of Math251(Dr. Z.)
#EMAIL RIGHT AFTER CLASS (OR RIGHT AFTER YOU WATCHED THE VIDEO) THE EDITED VERSION OF
#THIS .txt FILE (WITH YOUR ANSWERS)
#TO:
#DrZcalc3@gmail.com
#Subject: q1
#with an ATTACHMENT CALLED:
#q1FirstLast.txt
#(e.g. q1DoronZeilberger.txt)



#ANSWERS TO RANDOM FACTS IN THE LECTURE

# A. ACCORDING TO Dr. Z. THE TOP THREE SCIENTISTS OF ALL TIME ARE:

# B. WHAT BRANCH OF AI USES Multivariable calculs?:

# C. WHAT IS THE "DISTANCE" IN SPECIAL RELATIVITY?:   


##THE ACTUAL QUIZ:

#1. Show that the triangle with vertices 
#P=[1,0,0], Q=[0,1,0], R=[0,0,1]  is an equilateral triangle.


#YOUR SOLUTION HERE (EXPLAIN ALL THE STEPS)

#In order to show that the triangle is equilateral,
#We need to show that Dist(P,Q), Dist(P,R), and Dist(Q,R) are equal

#Dist(P,Q) = sqrt((0-1)^2 + (1-0)^2 + (0-0)^2) = sqrt(2)
#Dist(P,R) = sqrt((0-1)^2 + (0-0)^2 + (1-0)^2) = sqrt(2)
#Dist(Q,R) = sqrt((0-0)^2 + (0-1)^2 + (1-0)^2) = sqrt(2)

#Since all of them equal sqrt(2), the triangle is indeed an equilateral triangle


#2.  Determine whether the following two lines ever meet.
#If they do meet, where?

#r1(t)=[1,0,0]+ t*[1,2,3]

#r2(t)=[0,1,0]+ t*[2,1,3]

#YOUR SOLUTION HERE(EXPLAIN ALL THE STEPS)

#Expanding r1(t) gives us: [1,0,0] + [t,2t,3t]
#=[1+t,2t,3t]
#Expanding r2(s) gives us: [0,1,0] + [2s,s,3s]
#=[2s,1+s,3s]

#Now we need to set up a system of equations and solve for t,
#If there is a solution for t, we know that the lines meet as well as where they meet

#1+t=2s
#2t=1+s
#3t=3s

#We now that t=s, plugging into other equations we find t=1 and using that we know that the lines meet

#r1(1) = [1+1, 2*1, 3*1] = [2,2,3]
#Therefore, the lines meet at (2,2,3)