Let subscript r indicate in red coordinate, subscript b indicate in blue coordinate.
as the image above, the $Z_r$ axis is parallel to $Y_b$ axis, $X_r$ is parallel to $X_b$, and $Y_r$ is parallel to $Z_b$ but direction is opposite.
The origin of the red coordinate shifts t on the $Z_b$ axis relative to the blue one.
For a same point (x, y, z), what is the relation between ($x_b$, $y_b$, $z_b$) and ($x_r$, $y_r$, $z_r$)?
Actually, I realize that the the red coordinate system rotates 90 degrees count clockwise around the X axis of the blue coordinate system.
So in my mind, the $R$ matrix is $\begin{Bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{Bmatrix} $ and the T matrix is $\begin{Bmatrix}0 & 0 & t \end{Bmatrix}$
For a same point (x, y, z) in space
$\begin{Bmatrix} x_r \\ y_r \\ z_r \\ 1 \\ \end{Bmatrix}$ = $\begin{Bmatrix} R & T \\ 0^T & 1 \\ \end{Bmatrix} $ $\begin{Bmatrix} x_b \\ y_b \\ z_b \\ 1 \\ \end{Bmatrix}$
But, the result is not what I want. So what is wrong?