An n-dimensional index where the search key specifies an exact match on certain dimensions?
If you only want to be able to search on a single dimension at a time (as in your example), this is readily solved by precomputing an index.  Basically, you scan the entire database and build up an index for each dimension.  For instance, an index for the first dimension is a hashtable that maps from the value of $x_1$ to the tuple $(x_1,\dots,x_n)$, for each tuple.  You then build an index for the second dimension, for the third dimension, etc.  You do this once, in advance.  Once you have the index, it's easy to answer queries.  Also, it's easy to update the index when you insert or delete vectors.

A simple way to program this would be to insert your data into a database.  Most databases will automatically build indices and update them as you change the data in the database.  This will spare you having to implement that logic yourself.