Multiple dimensional vectors

If there are more than three dimensions then we just sum from over more indices. That is the beauty of the sum notation for vectors. For example:

${\vec {\mathbf {v}}}=\sum _{k=1}^{n}v_{k}{\vec {\mathbf {a}}}_{k}$

or even

${\vec {\mathbf {w}}}=\sum _{k=-\infty }^{\infty }v_{k}{\vec {\mathbf {a}}}_{k}$

when there are a countably infinite number of dimensions. If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral. See Functions and Vectors, an Analogy.