Question

How can you show that two vectors (3,0,-4)+t(1,2,3) and (1,5,2)+s(2,1,4) have no points in common?

Answer 1

show that they do not intersect. check if they have any solutions.
equate values of both x y and z for both vectors, and see if there is any solution for any 's' and 't'

OR they can either be parallel, or skew.
for being parallel, they must have same direction ratios (or multiples of one another)... which doesn't seem to be true for above. so now we see if they are skew.