# If $\overrightarrow a=\hat i+\hat j+\hat k,\:\:\overrightarrow a.\overrightarrow b=1$ and $\overrightarrow a\times\overrightarrow b=\hat j-\hat k,$ then $\overrightarrow b=?$

Let $\overrightarrow b=x\hat i+y\hat j+z\hat k$
Given : $\overrightarrow a.\overrightarrow b=1$
$\Rightarrow\:x+y+z=1.........(i)$
and
$\overrightarrow a \times\overrightarrow b=\hat j-\hat k$
$\Rightarrow\:(z-y)\hat i-(z-x)\hat j+(y-x)\hat k=\hat j-\hat k$
$\Rightarrow\:z=y,\:x-z=1\:and\:x-y=1$
Substituting in (i) we get
$x+2y=1$
$\Rightarrow\:3y=0\:\:or\:\:y=0=z$ and $x=1$
$\therefore\:\overrightarrow b=\hat i$