#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 calculus?:
# Deep Learning and Neural Nets

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

#In order to prove that the triangle PQR is an equilateral triangle,
#the distance between the vertices must be equal or |P Q|=|P R|=|Q R|.

#The distance formula: |P1P2| = sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)
#Substitute the values of each pair of vertices into the formula:
#|P Q|= sqrt((0-1)^2+(1-0)^2+(0-0)^2)=sqrt(2)
#|P R|= sqrt((0-1)^2+(0-0)^2+(1-0)^2)=sqrt(2)
#|Q R|= sqrt((0-0)^2+(0-1)^2+(1-0)^2)=sqrt(2)
#|P Q|=|P R|=|Q R|= sqrt(2); therefore, the distance between the vertices are equal and #triangle PQR is 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)

#Expand both line equations:
#r1(t)=[t+1,2t,3t]
#r2(t)=[2t,t+1,3t]
#Then set each component of the equations equal to each other...
#eq := {2t = t+1, 3t = 3t, t+1 = 2t}
#...in order to solve for the value of t to determine if the two will meet or not.
#t=1; r1(1)=r2(1)=[2,2,3]
# Since t=1, the lines will meet at the point P(2,2,3).