I did this problem for you recently, Nick.  Here it is again.
Is there something about it that you didn't understand about
lines in space?  Would you understand it better if I used
the i,j,k notation instead of the ‹a,b,c› notation? 

If you need further help, you may email me at AnlytcPhil@aol.com

A line parallel to the vector v = ‹a,b,c› and passing through the
point P(,,) is represented by the
parametric equations

, , 

or as the symmetric equations:



if none of a,b, or c are 0.

Begin by using the points P(2,1,-3) and Q(-1,4,1) 
to find a direction vector for the line passing through 
P and Q, given by
    __
v = PQ = ‹-1-(2),4-1,1-(-3)› = ‹-3,3,4›

So we substitute in 

, , 

with ‹a,b,c› = ‹-3,3,4›  and the point P(,,) = P(2,1,-3)

, , 

That's the parametric equations for the line.

If you want the symmetric equation of the line, we
substitute in







Edwin