All you have to do to identify two points is to choose two arbitrary values for t. The easiest ones are usually 0 and 1.
(2-7x0, 3+4x0, -0) = (2,3,0) and that point is on line l
(2-7x1, 3+4x1, -1) = (-5,7,-1) and that point will also be on line l
I have a slight problem in the way the next part of the question is worded.
Vectors have magnitude and direction, they can be anywhere; it is usually best to think of the originating at the origin, because it simplifies the math a lot.
if you mean that a (-7,4,-1) vector that you have extending from the origin is a different vector than one extending from the point (1,0,0) then, yes, they will be co-planar and no linear multiple of either vector will ever yield a common value with any ljnear multiple of the other, except when both are multiplied by zero. This exception seems trivial, but is actually vitally important when you start considering the Nullspace of a system.
If you think of all vectors as dimension and magnitude only and therefore analyze them as if they all originate from the origin, you no longer have any quandary how the linear multiples.
Thinking of parametric equations of parallel lines: all parallel lines will have a reference point and a vector component. If the vector component of line c is a linear multiple of the vector component of line d, then the lines are either parallel or colinear. If they are colinear, then the reference point for c will be found on line d, and vice-versa. you only have to satisfy Your curiosity once, of course).
(2-7x1, 3+4x1, -1) = (-5,7,-1) and that point will also be on line l
I have a slight problem in the way the next part of the question is worded.
Vectors have magnitude and direction, they can be anywhere; it is usually best to think of the originating at the origin, because it simplifies the math a lot.
if you mean that a (-7,4,-1) vector that you have extending from the origin is a different vector than one extending from the point (1,0,0) then, yes, they will be co-planar and no linear multiple of either vector will ever yield a common value with any ljnear multiple of the other, except when both are multiplied by zero. This exception seems trivial, but is actually vitally important when you start considering the Nullspace of a system.
If you think of all vectors as dimension and magnitude only and therefore analyze them as if they all originate from the origin, you no longer have any quandary how the linear multiples.
Thinking of parametric equations of parallel lines: all parallel lines will have a reference point and a vector component. If the vector component of line c is a linear multiple of the vector component of line d, then the lines are either parallel or colinear. If they are colinear, then the reference point for c will be found on line d, and vice-versa. you only have to satisfy Your curiosity once, of course).
all that aside, (-7,4,-1) as a vector has the exact same direction as (-14,8,-2). Any lines using a linear multiple of (-7,4,-1) will be parallel or colinear to any line using another linear multiple of (-7,4,-1).
