A parametric vector equation of the line through A and B is
L(t) = A + t*V
where t is a scalar called the parameter.
This is shown in the following figure.
Note that for this form of the equation of the line when t=0 then L(t) = A, and when t=1 then L(t) = B. So in this case t is just the fractional distance from point A to point B. The range of t is not restricted so the line extends infinitely in both directions.
An alternate form of the line is to use the unit vector,
U = V/|V|, along
the line. Then L(t) = A + t*U.
In this case t is the actual distance a point on line
L(t) is from point A toward point B, in the same
units as the coordinate system.
If C is a point not on line AB then a line through point C and
parallel to line AB is simply:
LC(s) = C + s*V,
where s is a scalar parameter.