no
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Yes, they do exist. In fact, there are infinitely many of them.
It's called a Ray.
This is all a matter of labels ... what certain words mean.-- "Line" means it has no ends, and goes on forever in both directions.-- "Ray" means it has one end. It goes on forever in one direction.-- "Line segment" means it has two ends, and a definite length.
a line a ray goes in only one direction forever No, a line goes on forever in two directions, a ray starts at one point and then goes on forever.
If you were to have 3 points on the same line, then you would actually not be determining a plane, because there are infinitely many planes that can intersect a given line. But if you have 3 points in the form of the points (or vertices) of a triangle, then you determine a plane in the sense that there is only one possible plane upon which that triangle can be drawn (not including a degenerate triangle, which is equivalent to a line).