Frame A:
Frame B:
(x1 in frame A is perpendicular to the plane created by x2 and x3 and x3 in frame B is perpendicular to the plane created by x1 and x2)
I am trying to find the rotation matrix from A to B. I know that in order to do this, I need to write the unit vectors of B in terms of the coordinates of A so that: $$R_{AB} = \begin{bmatrix} (R_{AB})_1 & (R_{AB})_2 & (R_{AB})_3\end{bmatrix}$$ How can I write the unit vectors of B in terms of coordinates of A with this little given info?
I still find the description a bit vague. But if we take $e_1=(1,0,0)$, $e_2=(0,1,0)$, and $e_3=(0,0,1)$, then the new coordinate basis should be $e_1' = e_1$, $e_2' = (0,1/\sqrt2,1/\sqrt2)$, and $e_3'=(0,-1/\sqrt2,1/\sqrt2)$. Thus, the rotation matrix should be $$R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1/\sqrt2 & -1/\sqrt2 \\ 0 & 1/\sqrt2 & 1/\sqrt2\end{bmatrix}.$$