Cho hình vuông ABCD cạnh a. Gọi điểm N thuộc cạnh DC sao cho \(2DN=NC\). Tính \(\overrightarrow{AN}.\overrightarrow{BN}\).
\(\dfrac{7}{9}a^2\).\(-\dfrac{2}{9}a^2\).\(\dfrac{2}{3}a^2\).\(\dfrac{3}{5}a^2\).Hướng dẫn giải:
\(\overrightarrow{AN}=\overrightarrow{AD}+\overrightarrow{DN}=\overrightarrow{AD}+\dfrac{1}{3}\overrightarrow{DN}\).
\(\overrightarrow{BN}=\overrightarrow{BC}+\overrightarrow{CN}=\overrightarrow{BC}+\dfrac{2}{3}\overrightarrow{CD}\).
Do đó:
\(\overrightarrow{AN}.\overrightarrow{BN}=\left(\overrightarrow{AD}+\dfrac{1}{3}\overrightarrow{DN}\right)\left(\overrightarrow{BC}+\dfrac{2}{3}\overrightarrow{CD}\right)\) \(=\overrightarrow{AD}.\overrightarrow{BC}+\dfrac{2}{3}\overrightarrow{AD}.\overrightarrow{CD}+\dfrac{1}{3}\overrightarrow{DN}.\overrightarrow{BC}+\dfrac{1}{3}\overrightarrow{DN}.\dfrac{2}{3}\overrightarrow{CD}\)
\(=a^2+\dfrac{1}{3}.\dfrac{2}{3}.a.a.\left(-1\right)\) \(=\dfrac{7}{9}a^2\).
