We study the question of when a generic stably free module admits a free summand of a given rank, which (as we will show) may be seen as an algebraic version of the vector fields on spheres problem solved by J.F. Adams in the 60s. Following work of M. Raynaud, the question can be reinterpreted in terms of the existence (or nonexistence) of a section of a natural projection map between Stiefel varieties given by "forgetting frames." Using techniques from motivic homotopy theory, we show that a section does not exist in many cases, thereby constructing examples of stably free modules that do not admit free summands of given ranks.