12. Given in $\triangle A B C$, $\boldsymbol{A} \boldsymbol{B}=\boldsymbol{a}, \boldsymbol{A} \boldsymbol{C}=\boldsymbol{b}$. Try to express $S_{\triangle A B C}=$ $\qquad$ using vector operations of $\boldsymbol{a}$ and $\boldsymbol{b}$.
$\begin{array}{l}\text { 12. } \frac{1}{2} \sqrt{(|\boldsymbol{a}| \cdot|\boldsymbol{b}|)^{2}-(\boldsymbol{a} \cdot \boldsymbol{b})^{2}} . \\ \text { Given } S_{\triangle A B C}=\frac{1}{2}|A \boldsymbol{B}| \cdot|A C| \sin A \\ =\frac{1}{2}|\boldsymbol{a}| \cdot|\boldsymbol{b}| \sin A, \\ \text { we have } S_{\triangle A B C}^{2}=\frac{1}{4}|\boldsymbol{a}|^{2} \cdot|\boldsymbol{b}|^{2}\left(1-\cos ^{2} A\right) \\ =\frac{1}{4}|\boldsymbol{a}|^{2} \cdot|\boldsymbol{b}|^{2}\left[1-\left(\frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| \cdot|\boldsymbol{b}|}\right)^{2}\right] \\ =\frac{1}{4}\left[(|\boldsymbol{a}| \cdot|\boldsymbol{b}|)^{2}-(\boldsymbol{a} \cdot \boldsymbol{b})^{2}\right] .\end{array}$