#Solutions to ATTENDANCE QUIZ for Lecture 1 of Math251(Dr. Z.)

#ANSWERS TO RANDOM FACTS IN THE LECTURE

#[Note: I forgot to ask them. In the future I will try to remember. Sometimes I will forget so you should answer
#None were asked.

#Here are the answers anyway:

# 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(x^2+y^2+z^2- c^2*t^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)

#Distance(P,Q)= sqrt((1-0)^2+(0-1)^2+(0-0)^2)= sqrt(2)
#Distance(P,R)= sqrt((1-0)^2+(0-0)^2+(0-1)^2)= sqrt(2)
#Distance(Q,R)= sqrt((0-0)^2+(1-0)^2+(0-1)^2)= sqrt(2)

#They are ALL THE SAME! Hence the 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)

#IMPORTANT NOTE: In THE LECTURE I MESSED UP! YOU NEET DIFFERENT SYMBOLS FOR r1 and r2 , s and t RESPECTIVELY

#THE CRUCIAL STEP IS TO REPLACE t by s in r2(t)

#r1(t)=[1,0,0]+ t*[1,2,3]=[1,0,0]+ [t,2*t,3*t]= [1+t,2*t,3*t]


#r2(s)=[0,1,0]+ s*[2,1,3]=[0,1,0]+ [2*s,s,3*s]= [2*s,1+s,3*s]


#We need r1(t)=r2(s)  this means that each componet must be equal to it corresponding one
#We get THREE equations in ONE unkown (t)

# 1+t=2*s , 2*t=1+s, 3*t=3*s

#From the last equation we have s=t, so  1+t=2 and t=1 so also s=1. Plugging into the second equation we  get that it agrees

#r1(1)= [1+1,2*1,3*1]=[2,2,3]
#r2(1)= [2*1,1+1,3*1]=[2,2,3] (yea! They agree)

#Ans. to 2: The two lines meet at the point (2,2,3)