Final answer:
To verify if a set is a vector space, it must follow the rules of vector addition and scalar multiplication. Properties include commutative, associative, and distributive laws related to vector and scalar quantities. This concept applies to analyzing one-dimensional and two-dimensional relative motion problems.
Step-by-step explanation:
To determine whether a set is a vector space, it must satisfy several properties, including but not limited to the commutative property of addition which states that the order of vector addition does not affect the result, so for any vectors A and B, the equation A + B = B + A must hold true. Additionally, a vector space requires that vector addition is associative, there exists an additive identity (zero vector), every vector has an additive inverse (negative vector), and scalar multiplication is distributive over vector addition and scalar addition.
For one-dimensional and two-dimensional problems, the concepts of position and velocity can be represented by vectors. When analyzing relative motion problems using these vectors, one must consider both the magnitude and the direction of the vectors to properly understand their combined effect. These principles apply in both one-dimensional and two-dimensional vector spaces.