#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, Isaac Newton, and Albert Einstein
# 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)
#An equilateral triangle is a triangle whose sides are equal to each other. So, if we can show that all the distances between vertices are the same, the triangle is equilateral.
#Length of side PQ = sqrt((0-1)^2 + (1-0)^2) = sqrt(2)
#Length of side QR = sqrt((0-1)^2 + (1-0)^2) = sqrt(2)
#Length of side PR = sqrt((1-0)^2 + (0-1)^2) = sqrt(2)
The lengths are equal to each other, so, the PQR triangle is equilateral.


#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 enter all the parametric equations into Maple
R1:= [1,0,0] + t*[1,2,3];
R2:= [0,1,0] + t*[2,1,3];
#We expand both equations
expand(R1);
#Result: [t + 1, 2*t, 3*t]
expand(R2);
#Result: [2*t, t + 1, 3*t]
#We come up with an equation in which these conditions are true:
eq:={R1[1]=R2[1], R1[2]=R2[2], R1[3]=R3[3]};
solve(eq,t);
#The result is empty - there is no solution, so, the lines never meet