The following Geogebra worksheet is designed to help students visualize what happens when we represent lines as vectors.
The worksheet's sliders are designed to be used with problems presented as "given a point, find the representation of a line in the direction of a vector."

If the problem gives you two points and asks you to find the vector representation, how can you use this worksheet?
On the left, you'll see that the point C and the line f are hidden.
For point C: "unhide" it. What happens when you adjust t?
Now "unhide" the line f. What do you notice?
Do you understand that as the value of t changes, the vector r points to a new ordered pair that falls on the line?