Answer
Since the two lines are not parallel and they do not intersect, the two lines are skew.
Work Step by Step
$l_1 : (x,y,z) = (-2,-2,-2)+n(1,2,3)$
$l_2 : (x,y,z) = (1,1,1)+r(1,-3,5)$
We can verify the ratio of each coordinate of the direction vectors of each line:
$x-coordinates: \frac{1}{1} = 1$
$y-coordinates: \frac{2}{-3} = -\frac{2}{3}$
$z-coordinates: \frac{3}{5} = \frac{3}{5}$
Since the ratios are not the same for all three coordinates, the two lines are not parallel.
Let's assume that the two lines intersect at the point $(a,b,c)$.
Then:
$-2+n = 1+r = a$
$n = r+3$
Then:
$-2+2n = 1-3r = b$
$-2+2(r+3) = 1-3r$
$-2+2r+6 = 1-3r$
$5r = -3$
$r = -\frac{3}{5}$
We can find $n$:
$n = r+3$
$n = -\frac{3}{5}+3$
$n = \frac{12}{5}$
However:
$c = -2+3n = -2+3(\frac{12}{5}) = \frac{26}{5}$
$c = 1+5r = 1+5(-\frac{3}{5}) = -2$
Clearly, this is a contradiction. Therefore, the assumption that the two lines intersect is false.
Since the two lines are not parallel and they do not intersect, the two lines are skew.
