# Line through two vectors

• Nov 21st 2013, 11:38 PM
Scurmicurv
Line through two vectors

Find the parametric equation of the line through p = (2, -5) and q = (-3, 1).

I have no problem seeing how the line is parallel to (q - p), but the key then gives p as the constant in the parametric equation without further comment, so if t is the parameter we have

M = p + (q-p)t

My question is, how exactly do we know to use p here? I feel like I'm missing something basic, any help would be greatly appreciated!
• Nov 22nd 2013, 12:28 AM
Prove It
Re: Line through two vectors
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.
• Nov 22nd 2013, 12:44 AM
Scurmicurv
Re: Line through two vectors
M'kay, yea, it was the part about whether it made any difference which of the vectors one choose I was a bit hesitant about. Thanks for clearing that up!