As I was trained in engineering and physics I sometimes get confused when trying to abstract my thoughts mathematically and this maybe a terminology problem, temporary misconception, for me as much as much as anything.
If two connected, but arbitrarily directed straight line slop vectors, are drawn in a Euclidean cartesian space, i.e. point $A$ leading to point $B$ along the first straight line path and point $B$ leading to point $C$ along the second straight lint path, then we can obviously compute a path leading from point $A$ to point $C$ directly. As there are only three points defined at this stage, they can all be considered to lie in the same plane $P_1$.
If we then add a new arbitrary straight path from point $C$ to point $D$, this point will not necessarily lie in the plane $P_1$, and most likely not. We could choose to define a new plane $P_2$ in which the last 3 points on the path $B$, $C$ and $D$ lie.
If I then draw a straight line path between $A$ and $D$, the first and last point, then can they be assigned a single defined plane (using only the two points specified), that is mathematically meaningful in this context?
For example, for each of the two planes $P_1$ and $P_2$ already defined, a plane normal vector can be defined, and those two normal vectors can be vector summed, to define a third "resultant" plane normal vector for $P_3$.
Does this third plane $P_3$ have any meaning (useful application or connection) to the resultant path between $A$ and $D$, above and beyond the unconstrained number of other planes that $A$ and $D$ could be considered to lie within? I suppose the resultant plane encodes partial historical information about the random path already taken, but I am otherwise confused about the relationship of $P_3$ to the resultant path $A$-$D$.
No, there is no "meaningful" plane with A and D in it, since you can reach D from A in many different ways. so only the "history" could vote for a plane but even than it could be ABD or ACD