For 𝐿1:𝑟= −3,2,8 +𝑠(7,−1,−6),𝑠𝜖𝑅 and 𝐿2:𝑟= 3,7,2 +𝑡(1,−6,0),𝑡𝜖𝑅, determine the points of intersection, if any exist.

Answer 1

well, if they lines intersect, then you will need
(x+3)/7 = (x-3)/1 ... x = 4
(y-2)/-2 = (y-7)/-6 ... y = 1
z = 2
So, the lines intersect at (4,1,20)
You can verify that s=1, t=1 produces this point.