If fact, your answers are not lines at all! What you give look like points but I guess you mean them as vectors- and I presume you mean them as "direction" vectors for the lines. The fact that "(1,1,1).(2,3,4) = 7" shows that they are not perpendicular (not really relevant) and the fact that "there does not exist a t such that t(1,1,1) = (2,3,4)" shows that they are not parallel. Neither of those means they do not intersect.
As defennnder said, you need to write them as lines with a parameter such as t. To do that, you will have to specify a point on them- and whether or not they intersect will depend on that point. For example, the lines [itex]\vec{r_1}= t\vec{i}+ t\vec{j}+ t\vec{k}[/itex] and [itex]\vec{r_2}= 2t\vec{i}+ 3t\vec{j}+ 4t\vec{k}[/itex] have direction vectors [itex]\vec{i}+ \vec{j}+ \vec{k}[/itex] and [itex]2\vec{i}+ 3\vec{j}+ 4\vec{k}[/itex] respectively and intersect at (0, 0, 0). You need to find skew lines.