#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 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 check the distances from each vertex to another.
#Distance of P and Q = sqrt((0-0)^2 + (1-0)^2 + (0-1)^2) = sqrt(2)
#Distance of P and R = sqrt((1-0)^2 + (0-0)^2 + (0-1)^2) = sqrt(2)
#Distance of Q and R = sqrt((1-0)^2 + (0-1)^2 + (0-0)^2) = sqrt(2)

#Since all of these distances are sqrt(2) and therefore all equal, this means that this is an equilateral triangle since all sides are equal.




#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)

#Step 1 is to expand the equations and simplify.
#r1(t) = [1, 0, 0] + [t, 2t, 3t] = [t+1, 2t, 3t]
#r2(t) = [0, 1, 0] + [2t, t, 3t] = [2t, t+1, 3t]

#Step 2 is to set the z, y, and x equal to each other.
#[t+1=2t, 2t=t+1, 3t=3t]

#Step 3 is to see if there is a consistent solution amongst all 3 equations.
#[t=1, t=1, INF SOLUTIONS]

#The two lines, r1(t) and r2(t), meet because they intersect at x = 1, y = 1, and z = ANYTHING.