I have two different coordinate systems, $A$ and $B$, with the same origin but rotated. I know that the axes of system $B$ have the following coordinates in system $A$. $$ \begin{align} x_B &= (-0.021, -0.723, 0.689)\\ y_B &= (-0.070, 0.689, 0.721)\\ z_B &= (-0.997, -0.033, -0.066) \end{align} $$
I also have a vector $C$ of length $3$ in the $y_A$- direction. Now, I'm wondering what are the coordinates of $C$ in the $B$ coordinate system. At first, I was thinking that I could simply use the transformation
x$_C$ = 3*0.723
y$_C$ = 3*0.689
z$_C$ = 3*0.021
But I don't think that's correct. How do I do this?
This approach may be helpful. Define $B$ to be the matrix formed from the vectors of the system-B axes relative to system-A:
$$ B= \begin{bmatrix} −0.021 & −0.070 & −0.997 \\ −0.723 & 0.689 & −0.033 \\ 0.689 & 0.721 & −0.066 \\ \end{bmatrix} $$
This matrix will transform coordinates from system-B to system-A via:
$$ \mathbf{x}_{A}=B\mathbf{x}_{B} $$
where $\mathbf{x}_{A}$ are the coordinates in system-A and $\mathbf{x}_{B}$ are the coordinates in system-B.
To determine a coordinate tranformation from system-A to system-B, one then needs the inverse of $B$:
$$ B^{-1}\mathbf{x}_{A}=\mathbf{x}_{B} $$
$$ B^{-1}= \begin{bmatrix} -0.0217157 & -0.724614 & 0.690345 \\ -0.0705677 & 0.68942 & 0.72129 \\ -0.997597 & -0.0331419 & -0.0651831 \\ \end{bmatrix} $$
Now, this can be multiplied by the vector in the system-A:
$$ \mathbf{x}_{B}=B^{-1}\mathbf{x}_{A}= \begin{bmatrix} -0.0217157 & -0.724614 & 0.690345 \\ -0.0705677 & 0.68942 & 0.72129 \\ -0.997597 & -0.0331419 & -0.0651831 \\ \end{bmatrix} \begin{bmatrix} 0 \\ 3 \\ 0 \\ \end{bmatrix} $$
$$ \mathbf{x}_{B}=B^{-1}\mathbf{x}_{A}= 3 \begin{bmatrix} -0.724614 \\ 0.68942 \\ -0.0331419 \\ \end{bmatrix} $$
I hope this helps.