Let FF be a number field, and let F⊂KF⊂K be a field extension of degree nn. Suppose that we are given 2r2r sufficiently general linear polynomials in rr variables over FF. Let XX be the variety over FF such that the FF-points of XX bijectively correspond to the representations of the product of these polynomials by a norm from KK to FF. Combining the circle method with descent we prove that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of XX.