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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 .

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Let's take the unit (column) vectors to be $\bf e_1,e_2,e_3$ and $\bf \hat{e}_1,\hat{e}_2,\hat{e}_3$ for the $x_1x_2x_3$ and $u_1u_2u_3$ coordinate systems respectively. You have a linear system describing their relationship:

$$\begin{pmatrix}\bf e_1 & \bf e_2 & \bf e_3\end{pmatrix}=\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{pmatrix} \begin{pmatrix}\bf \hat{e}_1 & \bf \hat{e}_2 & \bf \hat{e}_3\end{pmatrix}. \tag{$*$}$$

Note that since the unit vector are column vectors, above we have three $3\times 3$ matrices. This also helps transform arbitrary vectors between coordinate systems, because we can write

$$\begin{pmatrix}\bf e_1 & \bf e_2 & \bf e_3\end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=x_1\mathbf{e_1}+x_2\mathbf{e_2}+x_3\mathbf{e_3}$$

and similarly for the $u$ coordinate system. The system $(*)$ can be inverted by matrix inversion; that is, writing $A$ for the matrix of scalars, left-multiply both sides of $(*)$ by $A^{-1}$. Note that $A$ will be invertible, or else it won't be a valid coordinate transformation.

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