Let the transformation rule between two coordinate systems $ (x_1, x_2, x_3) $, and $ (u_1, u_2, u_3) $ be $ x_1 = a_{11} u_1 + a_{12} u_2 + a_{13} u_3 \\ x_2 = a_{21} u_1 + a_{22} u_2 + a_{23} u_3 \\ x_3 = a_{31} u_1 + a_{32} u_2 + a_{33} u_3 \\ $
If $ \hat e_{x1}, \hat e_{x2}, \hat e_{x3}$ be the unit vectors in $ (x_1, x_2, x_3) $ and $ \hat e_{u1}, \hat e_{u2}, \hat e_{u3}$ be the unit vectors in $ (u_1, u_2, u_3) $, What would be the transformation relation between unit vectors (both from $ \hat e_{x1}, \hat e_{x2}, \hat e_{x3}$ to $ \hat e_{u1}, \hat e_{u2}, \hat e_{u3}$ and vice versa)?
New specific question posted: Transformation of unit vectors from cartesian coordinate to cylindrical coordinate .