#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: Albert Einstein, Isaac #Newton, Archimedes

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

# 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)

#First: we must find the distance from P and Q and using the distance formula we find that
#sqrt((1 - 0)^2 + (0 - 1)^2 + (0 - 0)^2);
#                              (1/2)
#                             2     
#Distance between P and R: sqrt((1-0)^2+(0-0)^2+(0-1)^2);
#				 (1/2)
#				2
#Distance between Q and R: sqrt((0-0)^2+(1-0)^2+(0-1)^2;
#				 (1/2)
#				2
#As we can see the distance from all vertices to one another is 2^(1/2) or the sqrt(2) so
#the vertices form 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)
#First: we look at the direction vectors of both lines
#the first line being <1,2,3> and the second line being <2,1,3>
#The direction vectors are not scalars of one another so they are not parallel and 
#neither are the lines
#Second: the lines would intersect if there is a pair say x,y where r1 of x = r2 of y
#First component: 1+x = 0+2y
#Second component: 0+2x = 1+1y
#Third Component: 0+3x = 0+3y
#using the 2nd and 3rd component in a system of equations:
#Solving for x in the 2nd component, x = (1+1y)/2
#Plug this x into the 3rd component, (3+3y)/2 = 3y
#Multiply 2 on both sides, (3+3y) = 6y
#Solve for y so y=1
#Plug this y into the x we solved for from the second component so x = 1
#Finally plug both values into the first component to verify the x component so we get
#1+1 = 0+2(1) = 2=2
#Thus we have determined that these lines intersect at (2,2,3) found by plugging in one 
#x and y values