Parellel Lines in 3D:

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

L2:

are parallel.

If the lines are parallel, their vectors must be 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 = -2i + j + 3k

For L2, the components of the vector are the respective denominators:

n₂ = 4i - 2j - 6k.

We see that n₂ = -2*n₁. Hence the lines are parallel.