Final answer:
If no vectors are multiples of each other, they are considered linearly independent, suggesting that each vector provides unique information in the vector space. A coordinate system with perpendicular axes like the x-axis and y-axis can help illustrate their independence.
Step-by-step explanation:
If none of the vectors are multiples of each other, then we can consider these vectors to be linearly independent. Linear independence refers to the concept where no vector in a set can be constructed as a linear combination of the others. When there are two-dimensional vector problems, choosing a convenient coordinate system such as one with an x-axis (horizontal) and y-axis (vertical) allows for easy projection and analysis of the vectors. If no two vectors are parallel in this system, which means they are not scalar multiples of each other, they maintain their independence and allow for a diverse span in two-dimensional space.
Furthermore, vector addition in this coordinate system is both commutative and associative, which implies that the order of adding vectors does not affect the final resultant vector. The horizontal and vertical components of a vector are independent of one another, meaning that motion or direction in one does not influence the other.