I know this isn't exactly "advanced" or "applied" maths, but I can't think where to put it...
How would I write down the vector through the two points and ?
Because then, for instance, the vector through the points (1, 1, 2) and (2, 3, 4) is (1, 2, 2), and neither of the points is a scalar multiple of the vector coordinates...
The vector calculates the variations between the two points.
Suppose you're moving along the vector. If you increase the abscissa by , then the ordinate will be increased by and the third coordinate by
If you want to describe the points that are moving along the line passing through the two points, then they're in the form (note the difference with what you noted)
I hope this is clear... Maybe a sketch would help ya