So you were given two vectors in space, and (not yet knowing the cross-product trick) you said to yourself, "For a vector to be perpendicular to both of them, it must be perpendicular to *each* of them." So you made the vector be (a,b,c), and then you set the dot product with each of the two given vectors to 0. That gave you two equations with three unknowns, which you solved to eliminate a and b, and that left you with the form you gave, (0, 2c/3, c). Is that a fair summary?

For most pairs of vectors, an analogous thing would happen. If the two vectors, by bad luck, were collinear (not independent), then you would have only been able to eliminate one variable from your equations. That would have left you with a perpendicular *plane*, not a line.

There isn't any failure mode that would leave (0,0,0) as the only solution, and a lucky thing, too. But what it would mean is that there was no perpendicular line, so (0,0,0) would be the only solution. I guess that means that "it's a point" would be the best of the options you gave.