Multiple dimensional vectors

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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 if we have n dimensions, numbered from 1 to n:

\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k

or when there are a countably infinite number of dimensions

\vec \bold w = \sum_{k=- \infty}^\infty v_k \vec \bold a_k .

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 as discussed here.