For practical purposes, we often say that a vector is

something with a

direction and a

magnitude. This

idea actually comes from the more commonly used

Euclidean vector space, in which the concepts of

direction and

magnitude are defined (via the associated

inner product). Thus, in a Euclidean vector space (which is what we most often work with), a vector can indeed be considered a direction along with a magnitude.

However, a vector need not satisfy these properties. A vector is in

general an element of the

set associated with a given

vector space. This vector space need not be a Euclidean vector space, which would be required to define the concepts of "direction" and "magnitude". For example, take the set of

continuous functions from (0,1) to R. This set, called C(0,1), together with the operations +:V*V->V (

valuewise addition) and *:R*V->V (

valuewise scalar multiplication), forms a vector space. In this

context, the function f defined by f(x):=x is a "vector", but has no clearly identifiable direction or magnitude. (We can in fact define an

inner product, thus making this a Euclidean vector space and giving the function some sort of direction and magnitude, but we need not do so).