How many vectors are needed to define a plane/hyperplane in n-dimensional space? In 3 dimensions, if there are 2 vectors with tails at the origin and the heads in differing locations (and the vectors aren't parallel), that information is sufficient to define a plane. In higher dimensions, how many vectors with their tails at the origin are needed to define a plane/hyperplane?
If by plane you mean a 2-dimensional subspace, then the answer is $2$. Since you also asked for a hyperplane, which is a subspace of codimension $1$, meaning an $(n-1)$-dimensional subspace of a vector space of dimension $n$, you need $n-1$ linearly independent vectors to span a hyperplane.
If you happen to have an inner product on your $n$-dimensional space, then you can specify a $(n-1)$-dimensional hyperplane using a single normal vector for it (plus a point in the hyperplane if you don't want it to go through the origin).