HCR - Rotation Matrix

Given two reference frames like so:

We can write:

$$\{\begin{array}{l}{x}^{\prime }=x_x^{\prime }\cdot x+x_y^{\prime }\cdot y+x_z^{\prime }\cdot z\\ y^{\prime }=y_x^{\prime }\cdot x+y_y^{\prime }\cdot y+y_z^{\prime }\cdot z\\ z^{\prime }=z_x^{\prime }\cdot x+z_y^{\prime }\cdot y+z_z^{\prime }\cdot z\end{array}\to \left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\overline{\stackrel{ˉ}{R}}\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$$

Where for example $z_x^{\prime },z_y^{\prime },z_z^{\prime }$ is defined as:

So, $R$ is defined as:

$$R=\left[\begin{array}{lcc}{x}_x^{\prime }& x_y^{\prime }& x_z^{\prime }\\ y_x^{\prime }& y_y^{\prime }& y_z^{\prime }\\ z_x^{\prime }& z_y^{\prime }& z_z^{\prime }\end{array}\right]\triangleq R_{O{}^{\prime }}^O$$

$R_{O{}^{\prime }}^O$ $\triangleq$ Rotation Matrix from reference frame $O$ to $O{}^{\prime }$