Given two vectors are $\hat{i} - \hat{2j} + \hat{k}$ and $\hat{3i} + \hat{2j} + \hat{k}$ ⇒ sin θ = $\frac{\left|\vec{a} \times \vec{b}\right|}{\left|\vec{a}\right| \left|\vec{b}\right|}$ ... (i) To find $\vec{a} \times \vec{b}$ $\vec{a} \times \vec{b}$ = $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 3 \\ 3 & -2 & 1 \end{vmatrix}$ ⇒ $\vec{a} \times \vec{b}$ = [- 2 - (- 6)] $\hat{i}$ - (1 - 9) $\hat{j}$ + [- 2 - (- 6)] $\hat{k}$ ⇒ $\vec{a} \times \vec{b}$ = $\hat{4i} + \hat{8j} + \hat{4k}$ ⇒ $\left|\vec{a} \times \vec{b}\right|$ = $\sqrt{4^2+8^2+4^2}$ = $\sqrt{96}$ = 4 $\sqrt{6}$ ... (i) $\left|\vec{a}\right|$ = $\sqrt{1^2+(-2)^2+3^2}$ = $\sqrt{14}$ ... (ii) and $\left|\vec{b}\right|$ = $\sqrt{3^2+(-2)^2+3^2}$ = $\sqrt{14}$ ... (iii) Since sin θ = $\frac{\left|\vec{a} \times \vec{b}\right|}{\left|\vec{a}\right| \left|\vec{b}\right|}$ ⇒ sin θ = $\frac{4\sqrt{6}}{\sqrt{14}\sqrt{14}}$ ... From (i), (ii) and (iii) ⇒ sin θ = $\frac{2\sqrt{6}}{7}$