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I am looking to visually represent some properties of vectors when plotted on the plane versus when the are plotted on the torus. Plotting them on the plan is relatively straight forward, however I was hoping to find a way to plot on the torus as my next step and found this to be quite the challenge. I saw a previously answered question that was connecting two points of the torus:

torus = ParametricPlot3D[{Cos[θ] (1 + .3 Cos[φ]), 
    Sin[θ] (1 + .3 Cos[φ]), .3 Sin[φ]}, {\
θ, 0, 2 π}, {φ, 0, 2 π}, 
   PlotStyle -> Opacity[0.6]];

line = ParametricPlot3D[{Cos[.2 + 
       t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), 
    Sin[.2 + t (.8 - .2)] (1 + .3 Cos[.1 + t (2.5 - .1)]), .3 Sin[.1 +
        t (2.5 - .1)]}, {t, 0, 1}, PlotStyle -> Directive[Thick, Red]];

Show[torus, line, Boxed -> False, Axes -> None]

If this is the best way to answer my question I may need suggestions on how to convert a vector from the plane into a path on the torus, if not my ask stands.

This plotted the path from one point to another on the torus which is close to what I am looking for, but not exactly what I was hoping to achieve. The idea for me is I'd like to plot a pair of vectors in stages as they extend around the torus, making note of a property that occurs in the intersection of the vectors.

So for example, if we were to take a vector starting at the origin and ending at [1,1] how could you plot that on the torus accordingly?

I have poked around the site a little but am a bit new to the process of asking so I apologize if this is not the correct channel!

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