Metrics help define the concept of distance in Euclidean space (denoted by 
). Metric spaces, however, needn't always be vector spaces. We use them because they allow us to define limits for objects besides real numbers.
So far, we have been dealing with vectors, but what we don't yet know is how to calculate the length of a vector or the distance between two or more vectors, as well as the angle between two vectors, and thus the concept of orthogonality (perpendicularity). This is where Euclidean spaces come in handy. In fact, they are the fundamental space of geometry. This may seem rather trivial at the moment, ...
