Question

Cho hình vuông ABCD có tâm O

Tính tích vô hướng của\(\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\cdot\left(\overrightarrow{AO}+\overrightarrow{AD}\right)\)

Answer 1

\(=\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\left(\frac{1}{2}\overrightarrow{AC}+\overrightarrow{AD}\right)=\left(\overrightarrow{AB}+\overrightarrow{AB}+\overrightarrow{AD}\right)\left(\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}+\overrightarrow{AD}\right)\)

\(=\left(2\overrightarrow{AB}+\overrightarrow{AD}\right)\left(\frac{1}{2}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AD}\right)\)

\(=\overrightarrow{AB}^2+3\overrightarrow{AB}.\overrightarrow{AD}+\frac{1}{2}\overrightarrow{AB}.\overrightarrow{AD}+\frac{3}{2}\overrightarrow{AD}^2\)

\(=AB^2+\frac{3}{2}AD^2=\frac{5}{2}AB^2\)