3 dimensional matching shortest solution NP-hard?
Your problem is NP-complete, even in two dimensions. 
There is a straightforward reduction from MINIMUM MAXIMAL MATCHING in bipartite graphs:


  MINIMUM MAXIMAL MATCHING in bipartite graphs
  INSTANCE: a bipartite graph $G=(V_1\cup V_2,E)$ with $E\subseteq V_1\times V_2$; an integer $k$
  QUESTION: Does $G$ possess a maximal matching $E'$ of cardinality at most $k$?


(A maximal matching is a subset $E'\subseteq E$ such that no two edges in $E'$ share a common endpoint, and such that every edge in $E-E'$ shares a common endpoint with some edge in $E'$.)

NP-hardness of MINIMUM MAXIMAL MATCHING in bipartite graphs has been shown in 


  M. Yannakakis, F. Gavril
  Edge dominating sets in graphs.
  SIAM J. Appl. Math. 38, 364–372 (1980)
  https://epubs.siam.org/doi/10.1137/0138030


For the reduction, take the vertex set $V_1\cup V_2$ as ground set.
For every edge $\{u,v\}\in E$ with $u\in V_1$ and $v\in V_2$, create a corresponding vector $[u,v]$.
The equivalence to your problem is immediate.

(And if you insist on 3-dimensional vectors, then you should add a third component to these vectors and fill it with a lot of different values.)