Final answer:
By examining the direction vectors, it is evident that Line 2's direction vector is -2 times Line 1's direction vector, which means the two lines are parallel.
Step-by-step explanation:
To determine if the two lines given by the equations Line 1: r = (1,2,3) + t(1,-2,3) and Line 2: r = (3,2,1) + s(-2,4,-6) are parallel, we need to compare the direction vectors of both lines. The direction vector for Line 1 is given by (1,-2,3), and for Line 2, it is (-2,4,-6). To be parallel, the direction vectors must be scalar multiples of each other.
By comparing the components of the direction vectors, we can see that each component of Line 2's direction vector is -2 times the corresponding component of Line 1's direction vector. This means the lines have direction vectors that are scalar multiples of each other and, therefore, the lines are parallel to each other.