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 $\displaystyle (x_1, y_1, z_1)$ and $\displaystyle (x_2, y_2, z_2)$?
Okay...my next question is then "What does this mean?" Are the points on the vector just scalar products of that vector? I.e. $\displaystyle (x_3, y_3, z_3) \in v$, the set of points on the vector, if there exists an $\displaystyle a$ such that $\displaystyle a(x_1-x_2) = x_3$, $\displaystyle a(y_1-y_2) = y_3$ and $\displaystyle a(z_1-z_2) = z_3$?
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 $\displaystyle x_2-x_1$, then the ordinate will be increased by $\displaystyle y_2-y_1$ and the third coordinate by $\displaystyle z_2-z_1$
If you want to describe the points that are moving along the line passing through the two points, then they're in the form $\displaystyle (x_1+a(x_2-x_1),y_1+a(y_2-y_1),z_1+a(z_2-z_1))$ (note the difference with what you noted)
I hope this is clear... Maybe a sketch would help ya