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).