What happens if you add up and down if you are in a intersection on a singular plane

1. What the question likely means: motion on a plane vs. leaving it

If by “add up and down” you mean combining two displacements — one you call “up” and one “down” — those are vectors you can add; their sum is another vector lying in the plane only if both original vectors lie in the plane. If one of the directions points normal to the plane (perpendicular), adding it moves you off the plane; mathematically that is handled by vector addition and line–plane intersection formulations that distinguish movement parallel to the plane from movement along the normal ^{[2] [1]}.

2. When two directions lie in the same plane: they produce a single resultant

Elementary vector geometry says two planar displacements sum to a single resultant displacement — another point on the plane reached by following one then the other. Methods for computing that resultant use parametric equations of lines or vector addition; if you represent movement as r = r0 + v t, substituting into a plane equation lets you check whether a moved point remains on the plane ^{[2] [3]}.

3. When one direction is normal (out of the plane): you leave the plane

If one of your “up/down” components is along the plane’s normal, adding it moves you off the plane. Line–plane intersection theory clarifies that a line (a parametric motion along a direction) either (a) lies entirely in the plane, (b) is parallel to the plane with no intersection, or (c) meets the plane at a single point — so a nonzero normal component produces a line that generally intersects the plane at exactly one point (the original point) and then departs ^{[1] [2]}.

4. If “up” and “down” cancel exactly: you stay put

If “up” and “down” are equal and opposite vectors, their sum is the zero vector and you remain at the same point. That outcome is exact vector arithmetic; sources show solving for a scalar parameter t in parametric equations gives the precise point (or lack of one) of intersection depending on whether the net displacement is zero or nonzero ^{[2] [3]}.

5. If your question intended intersecting planes rather than movement

If instead you meant “what happens when two planes intersect” (two “directions” of orientation), classical results say two nonparallel planes intersect in a line; if they are parallel and distinct there is no intersection, and if they coincide they intersect in the whole plane. One computes the line direction as the cross product of the two normal vectors and finds a point on both planes to parametrize the line ^{[4] [5]}.

6. How to compute concretely (recipes journalists cite for readers)

• To check whether a proposed displacement keeps you on the plane: substitute the resulting point coordinates into the plane’s Cartesian equation and verify equality ^[2].
• To find where a parametric line r = r0 + d t meets a plane given by n·(r − a) = 0, substitute r and solve for t; if a finite t exists you get the intersection point, if no t satisfies the equation the line is parallel and misses the plane, and if every t satisfies it the line lies in the plane ^{[3] [6] [2]}.

7. Conflicting interpretations and missing details

Different sources frame the problem either as line–plane intersection, vector addition on a plane, or plane–plane intersection; your brief phrasing leaves ambiguity. Available sources do not mention any physics or non‑Euclidean interpretations (e.g., gravity, curved surfaces) for “up/down” in your question — they treat it purely geometrically ^{[1] [2]}. If you meant a different context (topology, manifolds, or discrete grid intersections) that is not covered in the provided reporting.

8. Bottom line for a reader who wants to act

Decide whether your “up/down” vectors lie in the plane or include a normal component. If both lie in the plane, add them as vectors to get a new point on the plane; if one includes the normal, you will generally leave the plane and can compute exactly where a parametric line crosses or departs the plane by substituting the line into the plane equation and solving for t ^{[2] [3]}.

If you want, give me the explicit coordinates or equations (point, direction vectors, or plane equation) and I will compute the resultant point or intersection step by step using the methods shown in the cited sources ^{[3] [6]}.