Given data as below:

$\overrightarrow{a}=2\hat{i}-7\hat{\mathit{j}}+5\hat{\mathit{k}}$

$\overrightarrow{b}=\hat{i}+\hat{k}$

$\overrightarrow{c}=\hat{\mathit{i}}+2\hat{\mathit{j}}-3\hat{\mathit{k}}$

Given, $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{a}$

$\Rightarrow (\overrightarrow{r}-\overrightarrow{c})\times \overrightarrow{a}=0$

It means $(\overrightarrow{r}-\overrightarrow{c})$ is parallel to $\overrightarrow{a}$,

$\therefore \overrightarrow{r}=\overrightarrow{c}+\lambda \overrightarrow{a}.......\left(1\right)$

Given, $\overrightarrow{r}·\overrightarrow{b}=0$

From equation (1),

$\left(\overrightarrow{c}+\lambda \overrightarrow{a}\right)·\overrightarrow{b}=0$

$\Rightarrow \left(\overrightarrow{c}·\overrightarrow{b}\right)+\lambda \left(\overrightarrow{b}·\overrightarrow{a}\right)=0$

$\Rightarrow -2+\lambda \left(7\right)=0\Rightarrow \lambda =\frac{2}{7}$

On putting values in equation (1) we get,

$\therefore \overrightarrow{r}=\overrightarrow{c}+\frac{2}{7}\overrightarrow{a}=\frac{1}{7}(11\hat{i}-11\hat{k})$

$\left|\overrightarrow{r}\right|=\frac{11\sqrt{2}}{7}$