#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

# 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)

  Dist:=sqrt ((x2-x1)^2+ (y2-y1)^2+ (z2-z1)^2- c^2*(t2-t1)^2);      (the equation that calculate the distance between two points)
  P:=[1,0,0] ; Q:=[0,1,0] ; R:=[0,0,1]
  Dist (P,Q) :=sqrt(2)    (distance between point P and Q)
  Dist (P,R) :=sqrt(2)     (distance between point P and R)
  Dist (Q,R):=sqrt(2)     (distance between point Q and R)
  Dist (P,Q) = Dist (P,R) = Dist (R,Q) = sqrt(2)    (the distance between each two points are equal to sqrt(2), which satisfied the poverty of 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)

  r1(t)=[1,0,0]+ t*[1,2,3]=[1+t,2,3]
  r2(t)=[0,1,0]+ t*[2,1,3]=[2,1+t,3]
  when two lines meet, r1(t) should equal to r2(t)
  [x(r1(t)),t(r1(t)),z(r1(t))]=[x(r2(t)),t(r2(t)),z(r2(t))]
  therefore we get 1+t=2 and 2=1+t which is the same. Then solve for 1+t=2, we get t=1
  So when t=1,  r1(t) and r2(t) will meet at [2,2,3]