Vectors cannot be solely defined as quantities with magnitude and direction because their mathematical definition is independent of coordinate systems. They are elements of a vector space, emphasizing their nature as geometrical objects rather than mere numerical arrays. The behavior of vector components under coordinate transformations, such as rotations, highlights their intrinsic properties. This perspective reinforces that vectors exist beyond the choice of any specific coordinate system. Understanding vectors in this way allows for a more robust and universal application in mathematics and physics.
