The known data: length of $AO$, $OB$, angles $\alpha$, $\beta$, point value of $B= \left(b_1,b_2\right)$ in the Cartesian coordinate system where $O$ is the origin. It is also known that point value $O= \left(O_1,O_2\right)$ in the Cartesian coordinate system where $A$ is the origin
How can I find the value of a point $B= \left(b_{1new}, b_{2new}\right)$ in Cartesian coordinate system where $A$ is the origin? First I tried to use matrix and map points, but it did not work.
The Cartesian coordinate system with origin $O$ was moved from point $A$ to point $O$ and rotated clockwise through an angle $\alpha$.
We provide you with a diagram depicting the coordinate system transformation described in your problem statement. The coordinate system, which has $O_0 = A=(0,0)$ as its origin, is shown in $\mathrm{\mathbf{\color{red}{red}}}$. The same coordinate system after translation, which is also drawn in $\mathrm{\mathbf{\color{red}{red}}}$, has its origin at $O=(o_\mathrm{x}, o_\mathrm{y})$. This coordinate system is then subjected to a clockwise rotation to obtain the coordinate system drawn in $\mathrm{\mathbf{\color{green}{green}}}$.
We leave it to OP to obtain the sought expressions for the coordinates of point $B=\left(b_\mathrm{xnew}, b_\mathrm{ynew}\right) $ with respect to a coordinate system having its origin at $A=\left(0, 0\right)$. First, you have to find the angle $\measuredangle BON$ in terms of $\alpha$ and $\beta$. Then, express $QB$ and $BN$ in terms of $OA$, $OB$, $\alpha$, and $\beta$.