Question

Determine whether the lines intersect(have a common point) and if so, find the coordinates of that point?
r(t) = < 4t+3, 4t + 2, 6t + 6>, for -infinity < t < infinity
R(s) = < s + 2, 3s - 1, -4s + 10>, for -infinity < t < infinity

And if r and R describe the paths of two particles, do the particles collide?

My teacher did an example in class but this question differs from this one so I don't know how to do it.

Answer 1

let's re-write each line as
x = 4t+3
y = 4t+2
z = 6t+6

and

x = s+2
y = 3s-1
z = -4s + 10

so
4t+3 = s+2 ---> 4t - s = -1
4t+2 = 3s-1 --> 4t - 3s= -3

subtract: 2s= 2
s = 1 , then t=0

if 2nd line, when s = 1
x = 3
y=2
z = 6

in 1st line when t=0
x=3
y=2
z=6

Yes, they do intersect at (3,2,6)