#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: Archimedes, Newton, Einstein



# B. WHAT BRANCH OF AI USES Multivariable calculs?:Deep Learning and Neural Nets are based on Multivariable calculus

# C. WHAT IS THE "DISTANCE" IN SPECIAL RELATIVITY?: The "distance" in Relativity between two events (x1,y1,z1,t1) and
(x2,y2,z2,t2) is sqrt ((x2-x1)^2+ (y2-y1)^2+ (z2-z1)^2- c^2*(t2-t1)^2)  


##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)

# Step 1 is to use the distance formula to determine the length
# Distance of PQ = sqrt[((0-1)^2) + ((1-0)^2) + ((0-0)^2)] = sqrt(2)
# Distance of PR = sqrt[((0-1)^2) + ((0-0)^2) + ((1-0)^2)] = sqrt(2)
# Distance of QR = sqrt[((0-0)^2) + ((0-1)^2) + ((1-0)^2)] = sqrt(2)

#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)

#The first step is to expand 
#r1(t) = [1, 0, 0] + [t, 2t, 3t] = [t+1, 2t, 3t]
#r2(t) = [0, 1, 0] + [2t, t,b3t] = [2t, t+1, 3t]

#The next step is to set the i, j, and k components of each vector equal to eachother
#[t+1 = 2t, 2t = t+1, 3t = 3t]

#The final step is to see if there are any solutions to the equation that are consistent.
#[t = 1, t = 1, t = infinity]

#The lines intersect at t = 1 because it works for all 3 equations.