Set every component of the position vector to zero at the same value of time or parameter. In two or three dimensions, satisfying only one coordinate condition identifies an intersection with an axis or plane, not the coordinate origin. Solving the simultaneous equations therefore tests whether the complete trajectory reaches the reference point and determines the associated event.
The coordinate origin is tied to the selected reference frame, so changing that frame changes the coordinate functions used to describe the motion. A trajectory that reaches the origin in one description may not reach it in another. Stating the frame makes the calculated time or parameter meaningful and prevents confusion when comparing position equations or graphs.
Each zero of a coordinate function marks a candidate value at which that coordinate is at the reference location. The complete result requires checking all relevant coordinates together. If the equations yield several valid parameter values, the motion can be examined at each one to distinguish repeated passages from a single event and to relate the results to the trajectory.
Write the position vector or coordinate functions in the chosen frame, then impose zero on every position component. Solve the resulting equation or simultaneous system for time, or for the parameter that labels the trajectory. Substitute each candidate back into the original coordinate functions, because only values satisfying the full set of conditions represent the required passage.
On a position-time graph, a zero of the relevant coordinate indicates a candidate crossing of the reference coordinate value, while a spatial trajectory shows where the path meets the origin. Reading graphs alone can be ambiguous when several coordinates are involved, so the plotted intersection should be checked against the coordinate equations before reporting a passage time.
It provides a concrete test for motion models: predicted coordinate functions can be checked for whether and when they reach the reference point. In kinematics, this supports crossing-time calculations and trajectory analysis; in coordinate-based physics, it links algebraic solutions with geometric paths. The same check can also help interpret field trajectories when their coordinates are explicitly defined.