  We characterize the sets of norm one vectors
$\mathbf{x}_1,\ldots,\mathbf{x}_k$ in a Hilbert space $\mathcal H$ such that
there exists a $k$-linear symmetric form attaining its norm at
$(\textbf{x}_1,\ldots,\mathbf{x}_k)$. We prove that in the bilinear case, any
two vectors satisfy this property. However, for $k\ge 3$ only collinear vectors
satisfy this property in the complex case, while in the real case this is
equivalent to $\mathbf{x}_1,\ldots,\mathbf{x}_k$ spanning a subspace of
dimension at most 2. We use these results to obtain some applications to
symmetric multilinear forms, symmetric tensor products and the exposed points
of the unit ball of $\mathcal L_s(^k\mathcal{H})$.
