Question

Suppose you take two steps A and B (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point A+B the sum of the lengths of the two steps?

Short answer

Two non-zero displacements can be zero only if they have equal magnitude but are headed in the opposite directions. Yes, the maximum distance we can end up from the starting point is the sum of the lengths of the two steps.

Step-by-step solution

Distance and displacement

A distance is the actual path covered by a moving object. It is a scalar quantity that cannot be zero or negative.

A displacement is the shortest path covered by a moving body from an initial position to reach a final position. It is a vector quantity, which can be zero or negative under certain conditions.

Ending up at the starting point

Two non-zero vectors add up to zero only if their magnitude is equal, but they are headed in the opposite directions.

Hence, if we take one step along the north and another step of the same length along the south, we will end up at the starting point.

Maximum distance

A distance is a scalar quantity, which can be added according to simple algebraic rules.

If you take one step of length A and another step of length B, then the maximum distance traveled is:

$d=A+B$

Hence, the maximum distance one can end up from the starting point is the sum of the lengths of the two steps.