$\overrightarrow{\mathit{a}}$ = $3\hat{\mathit{i}}+2\hat{\mathit{j}}+2\hat{\mathit{k}}$ , $\overrightarrow{\mathit{b}}$ = $\hat{\mathit{i}}+2\hat{\mathit{j}}-2\hat{\mathit{k}}$
∴ $\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}$ = $4\hat{\mathit{i}}+4\hat{\mathit{j}}$ and $\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}$ = $2\hat{\mathit{i}}+4\hat{\mathit{k}}$
$\left(\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}\right)\times \left(\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}\right)$ = $\left|\begin{array}{ccc}\hat{\mathit{i}}& \hat{\mathit{j}}& \hat{\mathit{k}}\\ 4& 4& 0\\ 2& 0& 4\end{array}\right|$
= $\hat{\mathit{i}}$ (16) - $\hat{\mathit{j}}$ (16) + $\hat{\mathit{k}}$ (-8) = $16\hat{\mathit{i}}-16\hat{\mathit{j}}-8\hat{\mathit{k}}$
∴ $\left|\left(\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}\right)\times \left(\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}\right)\right|$ = $\sqrt{{16}^2+{\left(-16\right)}^2+{\left(-8\right)}^2}$
= $\sqrt{256+256+64}$
= $\sqrt{576}$ = 24
So the unit vector, perpendicular to each of the vectors $\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}$ and $\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}$ is given by ± $\frac{\left(\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}\right)\times \left(\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}\right)}{\left|\left(\overrightarrow{\mathit{a}}+\overrightarrow{\mathit{b}}\right)\times \left(\overrightarrow{\mathit{a}}-\overrightarrow{\mathit{b}}\right)\right|}$ = ± $\frac{16\hat{\mathit{i}}-16\hat{\mathit{j}}-8\hat{\mathit{k}}}{24}$ = ± $\frac{2\hat{\mathit{i}}-2\hat{\mathit{j}}-\hat{\mathit{k}}}{3}$ = ± $\frac{2}{3}\hat{\mathit{i}}\mp \frac{2}{3}\hat{\mathit{j}}\mp \frac{1}{3}\hat{\mathit{k}}$