Question

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

Short answer

The end-up at starting point in two steps of different sizes is not possible. It is possible in case of three or more steps.

Step-by-step solution

Explanation for two steps of different sizes

Suppose a person starts from point A and reaches point B in one step of displacement $d_1$ and reaches the initial position by covering displacement $d_2$ in another step of different size.

The vector sum of displacements of the two steps to end up at the starting point can never be zero. The vector sum may be positive or negative, but always other than zero; so it is impossible to end up at the starting point in two steps of different sizes.

Explanation for three or more steps of different sizes

The vector sum of three vectors may be zero if the sum of the first two vectors is equal to the magnitude of the third vector in the exact opposite direction. This condition may also be true for more vectors.

Therefore, the end-up at the starting point in two different-sized steps is not possible, but it is possible for three or more steps.