Question

Cho tam giác ABC, gọi M là trung điểm AB, D là trung điểm BC, N thuộc cạnh AC sao cho \(\overrightarrow{CN}=2\overrightarrow{NA}\), K là trung điểm MN. Biểu diễn \(\overrightarrow{KD}=m\overrightarrow{AB}+n\overrightarrow{AC}\), giá trị m - n = ...

Answer 1

\(\overrightarrow{CN}=2\overrightarrow{NA}\Leftrightarrow\overrightarrow{CA}+\overrightarrow{AN}=-2\overrightarrow{AN}\Leftrightarrow\overrightarrow{AN}=\frac{1}{3}\overrightarrow{AC}\)

\(\overrightarrow{AK}=\frac{1}{2}\overrightarrow{AM}+\frac{1}{2}\overrightarrow{AN}=\frac{1}{4}\overrightarrow{AB}+\frac{1}{6}\overrightarrow{AC}\Rightarrow\overrightarrow{KA}=-\frac{1}{4}\overrightarrow{AB}-\frac{1}{6}\overrightarrow{AC}\)

\(\overrightarrow{KD}=\overrightarrow{KA}+\overrightarrow{AD}=\left(-\frac{1}{4}\overrightarrow{AB}-\frac{1}{6}\overrightarrow{AC}\right)+\left(\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AC}\right)\)

\(=\frac{1}{4}\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}\Rightarrow\left\{{}\begin{matrix}m=\frac{1}{4}\\n=\frac{1}{3}\end{matrix}\right.\) \(\Rightarrow m-n=-\frac{1}{12}\)