There is a Cartesian coordinate {XYZ}, and three points A, B and C such that they specify coordinate {xyz}. The coordinates of points A, B and C is known in {XYZ}. How the transformation matrix $T=[R \hspace{0.2cm}t;0\hspace{0.2cm}0\hspace{0.2cm}0\hspace{0.2cm}1]$ (where $R$ is the rotation matrix and $t$ is the translation vector) can be computed?
Please see the Figure.
Note 1: It is obvious that the translation vector $t$ is equal to the coordinate of point A.
I greatly appreciate if it is possible to give me some hints.
Let $A$ be described in the ${XYZ}$ system by $(a_X,a_Y,a_Z)$, and similarly for $B$ and $C$. The translation vector is obviously given by $A$ so $t=(a_X,a_Y,a_Z)$. Now let's translate ${xyz}$ to the origin of ${XYZ}$. For simplicity I would assume $AB=AC=1$, but if it's not the case, you can just normalize these vectors. Then ${\bf x'}=(b_X-a_X,b_Y-a_Y,b_Z-a_Z)=(x'_X,x'_Y,x'_Z)$, ${\bf y'}=(c_X-a_X,c_Y-a_Y,c_Z-a_Z)=(y'_X,y'_Y,y'_Z)$. For ${\bf z'}$ we use the cross product. The rotation matrix $R$ when applied to column vector ${\bf X}=\left(\begin{align}1\\0\\0\end{align}\right)$ will yield the column vector ${\bf x'}$, and similarly for ${\bf Y}$ and ${\bf Z}$. We can write this in a matrix form: $$R\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)=\left(\begin{array}{ccc}x'_X&y'_X&z'_X\\x'_Y&y'_Y&z'_Y\\x'_Z&y'_Z&z'_Z\end{array}\right)$$