What defines vectors that are scalar multiples of each other?

Vectors that are scalar multiples of each other are defined by their parallelism. When one vector can be expressed as a scalar (a real number) multiplied by another vector, it means that both vectors have the same or opposite direction. This indicates that they lie along the same line; thus, they are parallel.

For example, if vector A is (2, 3) and vector B is (4, 6), vector B is a scalar multiple of vector A because it can be expressed as 2 times vector A. Both vectors point in the same direction, confirming their parallel nature.

In contrast, vectors that are perpendicular would intersect at a right angle but do not have any scalar relationship such that one is a multiple of the other. Similarly, vectors that intersect can do so without being scalar multiples, simply crossing each other's path at some point. Lastly, vectors having different magnitudes does not determine the relationship between them; they could still be scalar multiples and hence parallel, regardless of their lengths.