Question

Cho tam giác ABC, gọi M là điểm thuộc cạnh BC sao cho MB = 2 MC, biểu diễn \(\overrightarrow{AM}=m\overrightarrow{AB}+n\overrightarrow{AC}\). Giá trị m.n bằng...

Answer 1

\(\overrightarrow{MB}=-2\overrightarrow{MC}\Leftrightarrow\overrightarrow{MB}=-2\left(\overrightarrow{MB}+\overrightarrow{BC}\right)\)

\(\Rightarrow3\overrightarrow{MB}=-2\overrightarrow{BC}\Rightarrow\overrightarrow{BM}=\frac{2}{3}\overrightarrow{BC}=\frac{2}{3}\left(\overrightarrow{BA}+\overrightarrow{AC}\right)=-\frac{2}{3}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AC}\)

\(\overrightarrow{AM}=\overrightarrow{AB}+\overrightarrow{BM}=\overrightarrow{AB}-\frac{2}{3}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AC}=\frac{1}{3}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AC}\)

\(\Rightarrow\left\{{}\begin{matrix}m=\frac{1}{3}\\n=\frac{2}{3}\end{matrix}\right.\) \(\Rightarrow mn=\frac{2}{9}\)