Final answer:
Two vectors of different magnitudes can add up to zero if they are equal in magnitude and opposite in direction, known as antiparallel. Furthermore, three or more vectors can add up to zero if they are arranged to form a closed shape. Scalars cannot be added to vectors because they lack the directional component necessary for vector addition.
Step-by-step explanation:
When considering the question of whether you can end up at your starting point after taking two steps of different sizes, it's helpful to think in terms of vectors and vector addition. In two or more dimensions, two vectors with different magnitudes can add up to zero if they are in exactly opposite directions. For instance, if one vector represents a step of 5 meters east, and the other represents a step of 5 meters west, their resultant, or sum, would be zero, bringing you back to your starting point.
More generally, two vectors can only add to zero if they are equal in magnitude and opposite in direction, forming a pair of antiparallel vectors. However, when you have three or more vectors, these vectors can have different magnitudes and still add up to zero if they are arranged such that they form a closed shape, like a triangle, where the vectors represent the sides of the shape.
It is important to note that scalar quantities cannot be added directly to vectors because they lack direction, and vector addition requires both magnitude and direction for computations. The analytical methods of vector addition, such as the parallelogram rule or component method, provide precise ways to calculate the results of vector addition and subtraction.