If you have a vector v=(1,0) that points to the right, you can scale this vector infinitely in that direction by multiplying it by a positive scalar.
5v = (5,0)
62.1v = (62.1,0)
Similarly, you can scale that vector infinitely in the opposite direction (i.e. left) by multiplying it by a negative scalar:
-987v = (-987,0)
If we call this scalar c, the expression cv allows us to represent any point along the X axis simply by varying c, meaning that cv defines a line along that axis.
Similarly, we can do the same for a vector w=(0,1) along the Y axis, scaling it by d.
Now we have a method for moving to any point on the XY plane simply by varying c and d in the linear combination: cv + dw, meaning that we've defined a plane using two vectors.
Two caveats:
- this won't work if v and w are parallel; for example, if v = -w (and neither are zero) then we can only move along a line instead of a plane
- it also won't work if either of the vectors are zero, because no matter what you multiply by, a zero vector can only represent a single point
If you do the same for another 1x3 vector, and it is not parallel to the first, you get all the points on a different line.
These two lines define a plane (and the cross product of the two vectors defines its normal vector)
Also, if you have 3 dimensional vectors you were always in 3D.