calculating the location of a point relative to another origin There are two sources of 3-dimensional coordinates, the second source can be traversed from the first origin at any angle and direction, and we also have a point with its coordinates relative to the second origin, the question is the coordinates of this point relative to the first origin.

 A: With respect to rotation part, assuming from your picture that coordinate frames are made by three orthonormal vectors, transforming vectors from frame $\mathbf F$ to frame $\mathbf G$ is about solving a $3 \times 3$ linear system. 
Define two $3 \times 3$ matrices $F = [ f_1, f_2, f_3 ]$ and  $G = [ g_1, g_2, g_3 ]$ where the columns $f_i$ are the vectors of frame $\mathbf F$ and the columns $g_i$ are the vectors of frame $\mathbf G$ 
Then the matrix $F G^{-1}$ send vectors from frame $\mathbf G$ to frame $\mathbf F$:
$F G^{-1} g = f$
And the matrix $G F^{-1}$ send vectors from frame $\mathbf F$ to frame $\mathbf G$:
$G F^{-1} f = g$
Since you are interested also in translation, you need to extend the matrices $F$ and $G$ to be $4 \times 4$ and work with homogeneous coordinates instead of Euclidean vectors i.e., work with 3D vectors embedded in 4D space.
See:
https://en.m.wikipedia.org/wiki/Homogeneous_coordinates
Once you have defined your $4 \times 4$ matrices the mappings defined above still works for mapping vectors between frames $\mathbf F$ and $\mathbf G$.
