What I know
Original two points that establish a box (points are opposite corners):
P1 = (-3,3,1)
P2 = (3,3,15)
Starting axis (at origin):
X vector = (1,0,0)
Y vector = (0,1,0)
Z vector = (0,0,1)
What I wish to do
I wish to move my box to my new coordinate system which is shown below:
X vector = (0.9616,-0.0006,-0.2745)
Y vector = (-0.0008,-1,-0.0003)
Z vector = (-0.2745,0.0005,-0.9616)
What is a simple way to translate my box coordinates systematically? The given numbers above are just one example I have, mostly to ensure that the new axis uses unit vectors.
What I have tried so far
I have tried to use the translational matrix formula:
to solve for A, B and C which coorespond to my new vector, resulting in Rz, Rx, and Ry with respect to my image. From there I have tried to then use my angles to translate my two points. However, the resulting points I get are either correct, mirrored across an axis, or flipped. Is there a simpler way to calculate this or am I going about it the right method at least?
Visual example
Coordinate system in red is the coordinate relative to the box. The Z axis needs to align with the new Z axis, Y with the new Y, and Z with the new Z, moving the box with it
A handy fact to remember when trying to solve such problems is that the columns of a transformation matrix are the images of the basis vectors. Thus, the matrix that maps that standard basis to the new $X$, $Y$ and $Z$ vectors is simply $$M = \begin{bmatrix} 0.9616 & -0.0008 & -0.2745 \\ -0.0006 & -1.0000 & 0.0005 \\ -0.2745 & -0.0003 & -0.9616 \end{bmatrix}.$$ Apply this matrix to your points to find where they end up as they’re dragged along with the coordinate axes by this transformation. For your two sample points this gives $$M\begin{bmatrix}-3\\3\\1\end{bmatrix}=\begin{bmatrix}-3.1617\\ -2.9977\\ -0.1390\end{bmatrix} \\ M\begin{bmatrix}3\\3\\15\end{bmatrix} = \begin{bmatrix}-1.2351\\ -2.9943\\ -15.2484\end{bmatrix}.$$