Final answer:
To determine whether the lines l1 and l2 are parallel, skew, or intersecting, we need to compare their directional vectors. If the cross product of the directional vectors is zero, the lines are parallel. If it is nonzero, the lines are either skew or intersecting.
Step-by-step explanation:
The given equations for lines l1 and l2 are:
l1: x = 5 + 2t, y = 4 - t, z = 1 + 3t
l2: x = 7 + 4s, y = 3 - 2s, z = 4 + 5s
To determine whether these lines are parallel, skew, or intersecting, we need to compare their directional vectors. The directional vector for l1 is (2, -1, 3) and for l2 is (4, -2, 5). If the cross product of these two vectors is zero, the lines are parallel. If it is nonzero, the lines are either skew or intersecting.
Taking the cross product of the directional vectors, we get:
(2, -1, 3) x (4, -2, 5) = (-1, -2, -2)
Since the cross product is nonzero, we can conclude that the lines l1 and l2 are either skew or intersecting.