# THE ACTUAL QUIZ: Yash Khangura

#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)
# The following steps find the distance between the two points respective to each side of the triangle
# dist(P,Q) = sqrt((0-1)^2 + (1-0)^2 + (0-0)^2) = sqrt(2)
# dist(P,R) = sqrt((0-1)^2 + (0-0)^2 + (1-0)^2) = sqrt(2)
# dist(Q,R) = sqrt((0-0)^2 + (0-1)^2 + (1-0)^2) = sqrt(2)

# Because the line segments PQ, PR, and QR equal each other, the triangle with vertices
# P=[1,0,0], Q=[0,1,0], R=[0,0,1] is 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)

# I expand each of the vector equations
# r1(t) = <t+1, 2t, 3t>
# r2(t) = <2t, t+1, 3t>

# I set their x, y, z components equal to each other and solve for t. If they equal each other at a certain
# t then the lines intersect at that time. 
# r(t) = <t+1=2t, 2t=t+1, 3t=3t> = <t=1, t=1, t=t>
# The lines r1(t) and r2(t) intersect at t=1 at the point (2,2,3).