Final answer:
The direction vectors of the two lines are (2, 3, 4) and (1, 6, -2). They are neither proportional nor is their dot product zero, and they are not coplanar, indicating that the lines are skew.
Step-by-step explanation:
To determine whether two lines given by symmetric equations are parallel, perpendicular, or skew, we need to look at their direction vectors.
The first line is given by (x-1)/2 = (y-2)/3 = (z-3)/4, which gives us a direction vector of (2, 3, 4).
The second line's equation can be written as x+1 = 6(y-3) = -2(z+y), which simplifies to x+1 = 6y-18 = -2z-2y. Rearranging, we get x = 6y-19 = -2z-2y-1. This gives us a direction vector of (1, 6, -2).
To determine if the lines are parallel, their direction vectors must be proportional. To determine if they are perpendicular, their dot product must be zero. Since neither of these conditions is satisfied, and the lines are not coplanar, they are skew lines.