  Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's
theorem on linear forms needs to be modified when one requires that the vectors
of coefficients of the linear forms make a bounded acute angle with respect to
a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point
of $\mathbb{R}^n$ that we are approximating has linearly independent
coordinates over $\mathbb{Q}$, we obtain best possible exponents of
approximation which surprisingly depend only on the dimension of $V$. Our
estimates are derived by reduction to a result of Thurnheer, while their
optimality follows from a new general construction in parametric geometry of
numbers involving angular constraints.
