9.
a, \(BC=\sqrt{7^2+10^2}=\sqrt{149}\)
\(cos\left(\overrightarrow{AB};\overrightarrow{AC}\right)=cosA=cos90^o=0\)
\(cos\left(\overrightarrow{AB};\overrightarrow{BC}\right)=-cos\left(\overrightarrow{BA};\overrightarrow{BC}\right)=-cosB=-\dfrac{7\sqrt{149}}{149}\)
\(cos\left(\overrightarrow{AB};\overrightarrow{CB}\right)=cosB=\dfrac{7\sqrt{149}}{149}\)
b, \(\overrightarrow{HB}.\overrightarrow{HC}=HB.HC.cos\widehat{BHC}\)
\(=HB.HC.cos180^o\)
\(=-HB.HC=-AH^2\)
\(=-\dfrac{AB^2.AC^2}{AB^2+AC^2}=-\dfrac{4900}{149}\)
10.
\(BC^2=AB^2+AC^2-2AB.AC.cosA=109\Rightarrow BC=\sqrt{109}\)
a, \(\overrightarrow{AB}.\overrightarrow{AC}=AB.AC.cosA=-\dfrac{35}{2}\)
\(\overrightarrow{AB}.\overrightarrow{BC}=\overrightarrow{AB}\left(\overrightarrow{AC}-\overrightarrow{AB}\right)=\overrightarrow{AB}.\overrightarrow{AC}-\overrightarrow{AB}^2=-\dfrac{133}{2}\)
b, \(AM^2=\dfrac{AB^2+AC^2}{2}-\dfrac{BC^2}{4}=\dfrac{39}{4}\)
