  We consider approximation of vectors $\mathbf{z}\in
F\otimes\mathbb{R}\cong\mathbb{R}^r\times\mathbb{C}^s$ by elements of a number
field $F$ and construct examples of badly approximable vectors. These examples
come from compact subspaces of $SL_2(\mathcal{O}_F)\backslash
SL_2(F\otimes\mathbb{R})$ naturally associated to (totally indefinite,
anisotropic) $F$-rational binary quadratic and Hermitian forms, a
generalization of the well-known fact that quadratic irrationals are badly
approximable over $\mathbb{Q}$.
