a: ABCD là hình chữ nhật
=>\(AB^2+BC^2=AC^2\)
=>\(AC=\sqrt{3^2+4^2}=5\)
ABCD là hình chữ nhật nên \(\overrightarrow{AB}+\overrightarrow{AD}=\overrightarrow{AC}\)
\(\left|\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}\right|\)
\(=\left|\overrightarrow{AC}+\overrightarrow{AC}\right|=2\left|\overrightarrow{AC}\right|=2\cdot AC=2\cdot5=10\)
b: \(\overrightarrow{AM}+\overrightarrow{AN}\)
\(=\overrightarrow{AB}+\overrightarrow{BM}+\overrightarrow{AD}+\overrightarrow{DN}\)
\(=\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{BC}+\overrightarrow{AD}+\dfrac{1}{2}\overrightarrow{DC}\)
\(=\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{BC}+\overrightarrow{BC}\)
\(=\dfrac{3}{2}\left(\overrightarrow{AB}+\overrightarrow{BC}\right)=\dfrac{3}{2}\overrightarrow{AC}\)
=>\(\left|\overrightarrow{AM}+\overrightarrow{AN}\right|=\dfrac{3}{2}\cdot AC=\dfrac{3}{2}\cdot5=\dfrac{15}{2}\)
