A line has direction, infinite magnitude and is positioned somewhere.

A vector has direction and magnitude. Note that its position is irrelevant (but we usually denote vectors to start at the origin).

So to get the vector form of a line, you need a direction vector, which you then make infinitely long by multiplying by a parameter. But this vector is defined to go through the origin and has no guarantee to actually go through the points you want it to go through. Adding one of the points enables the line to be translated from the origin to that point, and thus is guaranteed to go through the points you require it to. That's why p is added. It would have worked just as well to add q.