Re-express as a linear system Ac = b where c=[c_1, c_2] is a vector of the unknown coefficients, A is a 1000 \times 2 matrix whose columns are v_{app} u_s and \sin(\psi) \cos(\beta) /v_{app}, and b is the vector psi_dot.
Then solve with c = A\b. That’ll get you a least-squares solution to the problem, the one that minimizes sum res^2, using QR decomposition.