# If $\overrightarrow r\times\overrightarrow b=\overrightarrow c\times\overrightarrow b$ and $\overrightarrow r.\overrightarrow a=0$, where $\overrightarrow a=2\hat i+3\hat j-\hat k,\:\overrightarrow b=3\hat i-\hat i+\hat k\:and\:\:\overrightarrow c=\hat i+\hat j+3\hat k$, then $\overrightarrow r=?$

Let $\overrightarrow r=x\hat i+y\hat j+z\hat k$
Given: $\overrightarrow r.\overrightarrow a=0$
$\Rightarrow\: 2x+3y-z=0..............(i)$
$\overrightarrow c\times\overrightarrow b=4\hat i+8\hat j-4\hat k$
$\overrightarrow r\times\overrightarrow b=(y-z)\hat i+(3z-x)\hat j+(-x-3y)\hat k$
Given: $\overrightarrow r\times\overrightarrow b=\overrightarrow c\times\overrightarrow b$
$\Rightarrow\:(y-z)\hat i+(3z-x)\hat j+(-x-3y)\hat k=4\hat i+8\hat j-4\hat k$
$\Rightarrow\:y-z=4,\:3z-x=8,\:x+3y=4 .............(ii)$
Solving $(i)\:\:and\:\:(ii)$
$x=-2,\:\:y=2\:\:and\:\:z=2$
$\therefore\:\overrightarrow r=-2\hat i+2\hat j+2\hat k$.