Let $\overrightarrow{a}=\hat{i}-\hat{j},\overrightarrow{b}=\hat{j}-\hat{k},\overrightarrow{c}=\hat{k}-\hat{i}$ and $\overrightarrow{d}$ is a unit vector such that $\overrightarrow{a}·\overrightarrow{d}=0=\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]$. Find $d$.

$\overrightarrow{a}·\overrightarrow{d}=0\Rightarrow \overrightarrow{d}\perp \overrightarrow{a}$

$\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]=0\Rightarrow \overrightarrow{b}\times \overrightarrow{c}·\overrightarrow{d}=0\Rightarrow \overrightarrow{d}\perp (\overrightarrow{b}\times \overrightarrow{c})$

$\therefore \overrightarrow{d}=\alpha \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\alpha \left[\right(\overrightarrow{a}·\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}·\overrightarrow{b}\left)\overrightarrow{c}\right]$

$=\alpha [-\overrightarrow{b}+\overrightarrow{c}]=\alpha (-\hat{i}-\hat{j}+2\hat{k})$

$d$ is a unit vector $\therefore \overrightarrow{d}=\pm \frac{(\hat{i}+\hat{j}-2\hat{k})}{\sqrt{6}}$