Vector Subtraction (Component Form)

Vectors subtract exactly as they add: one axis at a time, with no interaction between components. Apply this formula once per axis and you have the whole difference vector. The result a − b is the arrow that runs from the tip of b to the tip of a, which is why it answers every "relative to" question: an aircraft flying at v_plane through wind v_air has a ground velocity found by subtraction, and a police radar reports the difference between two velocity vectors rather than either one alone.

Order is everything. a − b and b − a have the same length but point opposite ways, and swapping them is the standard way to get a relative-velocity answer that is right in size and backwards in direction. A quick check: if a_x = 7 and b_x = −3, then c_x = 7 − (−3) = 10. Subtracting a negative component moves the difference further out, not closer in.