Let  $\overline{a}=2\overline{i}-7\overline{j}+5\overline{k};\overline{b}=\overline{i}+\overline{k},\overline{c}=\overline{i}+2\overline{j}-3\overline{k}$  be three vectors. If $\overline{r}$ is a vector such that $\overline{r}\times \overline{a}=\overline{c}\times \overline{a}$ and $\overline{r}.\overline{b}=0$ then  $\left|\overline{r}\right|=$ 

$\overline{r}\times \overline{a}=\overline{c}\times \overline{a},\overline{r}.\overline{b}=0$

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

$\Rightarrow \overline{r}=\overline{c}+t\overline{a}$

$0=\overline{c}.\overline{b}+t(\overline{a}.\overline{b})$

$(1-3)+t(2+5)=0$

$t=\frac{2}{7}$

$\overline{r}=\overline{c}+\frac{2}{7}\overline{a}$

$=\frac{1}{7}[(7,4,-21)+(2,14,10)]$

$=\frac{1}{7}[11,0,-11]$

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