Skew Lines in 3D:

L1: x = -2 + t, y = 4 - 3t, z = -5 - t and

L2: x = 3 - s, y = 5 + 2s, z = 4 + s

are skew.

First, check that the lines are not parallel, or that their vectors are not proportional, so get the vectors for each line.

For L1, the components of the vector are the coefficients of t in the parametric equations:

n₁ = i - 3j - k

For L2, the components of the vector are the coefficients of s in the parametric equations:

n₂ = -i + 2j + k.

We see that n₂ and n₁ are not proportional. Hence the lines are not parallel.

To check for intersection, we'll set the x and y coordinates equal and solve for the values of s and t required. If the lines intersect, these values of s and t can be substituted into the equations for z and we should get equality of the z coordinates also.

For x: -2 + t = 3 - s

For y: 4 - 3t = 5 + 2s

Upon solving, we find s = 16, t = -11.

Now substitute these values into the equations for z:

For L1: z = 5 - t = 5 - (-11) = 6

For L2: z = 4 + s = 4 + 16 = 20

Since we do not have equality for the z coordinates also, the lines must not intersect and are skew.