User:Egm6322.s09.bit.sahin/r3-edit

$$\overrightarrow{\overline{O}P}=\overline{x}\left (m\widehat{i}+n\widehat{j} \right )+\overline{y}\left (p\widehat{i} +q\widehat{j}\right )$$

$$x\widehat{i}+y\widehat{j}=x_{0}\widehat{i}+y_{0}\widehat{j}+\overline{x}\left (m\widehat{i}+n\widehat{j} \right )+\overline{y}\left (p\widehat{i}+q\widehat{j} \right )$$

then,

$$x=x_{0}+\overline{x}m+\overline{y}p$$

$$y=y_{0}+\overline{x}n+\overline{y}q$$

from this equation system $$\overline{x}$$ and $$\overline{y}$$ are obtained as

$$\overline{x}=\frac{q}{\left (mq-np \right )}x-\frac{p}{\left (mq-np \right )}y-\frac{q}{\left (mq-np \right )}x_{0}+\frac{p}{\left (mq-np \right )}y_{0}$$

$$\overline{y}=\frac{m}{\left (mq-np \right )}y-\frac{n}{\left (mq-np \right )}x+\frac{n}{\left (mq-np \right )}x_{0}-\frac{m}{\left (mq-np \right )}y_{0}$$

Recall:

$$\begin{Bmatrix} \overline{x}\\ \overline{y} \end{Bmatrix} = \underline{E} \begin{Bmatrix} x\\ y \end{Bmatrix} + \underline{F}$$

$$\begin{Bmatrix} \overline{x}\\ \overline{y}

\end{Bmatrix} =\begin{bmatrix} e_{11} &e_{12} \\ e_{21} &e_{22} \end{bmatrix}\begin{Bmatrix} x\\ y

\end{Bmatrix}+\begin{Bmatrix} f_{11}\\ f_{21}

\end{Bmatrix}$$

therefore,

$$\overline{x}=e_{11}x+e_{12}y+f_{11}$$

$$\overline{y}=e_{21}x+e_{22}y+f_{21}$$

Hence the coefficient are,

$$e_{11}=\frac{q}{\left (mq-np \right )}$$

$$e_{12}=\frac{-p}{\left (mq-np \right )}$$

$$f_{11}=\frac{py_{0}-qx_{0}}{\left (mq-np \right )}$$

$$e_{21}=\frac{m}{\left (mq-np \right )}$$

$$e_{22}=\frac{-n}{\left (mq-np \right )}$$

$$f_{21}=\frac{nx_{0}-my_{0}}{\left (mq-np \right )}$$

Eventually we obtain $$\underline{E}$$ and $$\underline{F}$$ as following

$$\underline{E}=\begin{bmatrix} \frac{q}{\left (mq-np \right )}&\frac{-p}{\left (mq-np \right )} \\ \frac{m}{\left (mq-np \right )}&\frac{-n}{\left (mq-np \right )} \end{bmatrix}$$

$$\underline{F}=\begin{Bmatrix} \frac{py_{0}-qx_{0}}{\left (mq-np \right )}\\ \frac{nx_{0}-my_{0}}{\left (mq-np \right )} \end{Bmatrix}$$