Quite easily, thankfully (assuming you want to move in the direction of your LookVector). A LookVector is just a normalized vector, which means it has a magnitude (think “distance”) of 1. Because of this, you can just multiply that LookVector by the desired distance (e.g. 20) to get the position offset. You then add this position to your starting point (e.g. 0, 0, 0).
local partPosition = somePart.Position
local lookVector = someLookVector
local distance = 20
local offset = lookVector * distance
local newPosition = partPosition + offset
-- Or the two above lines into one:
local newPosition = partPosition + (lookVector * distance)