回復 1# arend 的帖子
\(BC\mbox{向量}=AC\mbox{向量}-AB\mbox{向量}\)
\(\displaystyle AD\mbox{向量}=\frac{2}{3}AC\mbox{向量} +\frac{1}{3}AB\mbox{向量}\)
\(\displaystyle\Rightarrow AD\mbox{向量}\cdot BC\mbox{向量}=\frac{2}{3}\left|AC\mbox{向量}\right|^2-\frac{1}{3}AC\mbox{向量}\cdot AB\mbox{向量}-\frac{1}{3}\left|AB\mbox{向量}\right|^2\)
\(\displaystyle=\frac{2}{3}\cdot2^2-\frac{1}{3}\cdot3\cdot2\cos150^\circ-\frac{1}{3}\cdot3^2\)
