In below image I’d like to move the position of the green anchor point along the green dotted line (I freehanded in there, image it’s a straight extension of the anchor point handle’s line). I’d like to do so while keeping the cyan anchor point in place as well as the direction of the green line locked. Note that this is not a snapping angle (90, 45, etc) nor do I have any lines I could snap to.
Is this possible somehow?
The only way I can think of is to draw a line the direction of the green line and snap to it, but this seems very backward.
You can use the scale tool with the tangent. Once the tangent is selected with the direct selection tool (a) click on the vertex tangent to move scale pivot to vertex. By holding Shift you can now constrain the scaling to be equal in both direction, which achieves what you want. You can also scale both tangents in unison this way.
Screencast 1: Moving tangent along existing direction by scaling uniformly.
Making a support line is also not as stupid as it sounds. But there is also a third alternative and that is to rotate the document grid temporarily to the tangent angle.