$\overrightarrow{\mathit{a}}$ = $\hat{\mathit{i}}-\hat{\mathit{j}}+\hat{\mathit{k}}$
$\overrightarrow{\mathit{b}}$ = $\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}$
Now , $\overrightarrow{\mathit{a}}.\overrightarrow{\mathit{b}}$ = $\left|\overrightarrow{\mathit{a}}\right||\overrightarrow{\mathit{b}}|\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$
Where θ is the angle between $\overrightarrow{\mathit{a}}$ and $\overrightarrow{\mathit{b}}$ ; also 0 ≤ θ ≤ π
$\hat{\mathit{i}}-\hat{\mathit{j}}+\hat{\mathit{k}}$ , $\hat{\mathit{i}}+\hat{\mathit{j}}-\hat{\mathit{k}}$ = $\left(\sqrt{1^2+{\left(-1\right)}^2+1^2}\right)$ $\left(\sqrt{1^2+1^2+{\left(-1\right)}^2}\right)$ cos θ
⇒ 1 - 1 - 1 = 3 cos θ
- 1 = 3 cos θ
cos θ = $-\frac{1}{3}$ ⇒ θ = $\mathit{c}\mathit{o}{\mathit{s}}^{-1}\left(-\frac{1}{3}\right)$