The original note, as written at the time
You may have straight linear directions but for any motion you can never find a straight linear path. All paths of motion are curvilinear. Consequently, you can never have a single fundamental space vector defining a path; you will always have at least two fundamental vectors, which implies a planar formation. I suspect this is even volumnar, so all motion is ultimately defined three dimensionally.
Ongoing development
You may have straight linear directions, but for any motion you can never find a straight linear path. All paths of motion are curvilinear. Consequently you can never have a single fundamental space vector defining a path — there will always be at least two fundamental vectors, which implies a planar formation. And it is probably volumnar, so all motion is ultimately defined three dimensionally.
I think this is right, and the reason is that a straight line is an idealisation that requires an isolated body. Nothing is isolated. Every moving thing is in a field, and a field curves the path, so the straight line is the limit case of a body with no forces on it, which does not exist.
The practical consequence for engineering is that we draw straight lines and then correct for reality with tolerances. A road is designed as a series of tangents and then eased with transition curves, because a vehicle cannot change direction instantly. The mathematics of the design is linear; the mathematics of the motion is not.
Your vector argument is the interesting half. If every path needs at least two vectors, then a path is not an object but a relation — you cannot specify where a thing goes without also specifying what it is going around. That is a different way of thinking about motion, and it is closer to how a navigator works than how a draughtsman works.
Three-dimensionality follows, and it is why the best paths in a city are not the shortest ones. They are the ones that follow something.
Nothing travels in a straight line. The straight line is what we draw when we have stopped looking.