Áp dụng công thức $\sin$ ta được:
$\begin{array}{l}\quad S_{ABC}=\dfrac12AB\cdot AC\cdot\sin A\\\to S_{ABC}=\dfrac12\sqrt{AB^2\cdot AC^2\cdot \sin^2A}\\\to S_{ABC}=\dfrac12\sqrt{AB^2\cdot AC^2\cdot(1-\cos^2A)}\\\to S_{ABC}=\dfrac12\sqrt{AB^2\cdot AC^2-AB^2\cdot AC^2\cdot \cos^2A}\\\to S_{ABC}=\dfrac12\sqrt{AB^2\cdot AC^2-\left(\left|\overrightarrow{AB}\right|\cdot \left|\overrightarrow{AC}\right|\cdot \cos A\right)^2}\\\to S_{ABC}=\dfrac12\sqrt{AB^2\cdot AC^2 - \left(\overrightarrow{AB}\cdot \overrightarrow{AC}\right)^2}\end{array}$
