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We determine the first non‐stableA   1   ‐homotopy sheaf of    SL   n   . Using techniques of obstruction theory involving theA   1   ‐Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank at least    d  ‐  1    on split smooth affine quadrics of dimension    2  d  ‐  1   . These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations ofA   1   ‐homotopy sheaves with real and complex realization.

Algebraic vector bundles on spheres