\(CD=\sqrt{AC^2+AD^2}=a\sqrt{2}\)
\(BC=\sqrt{AB^2+AC^2-2AB.AC.cos\widehat{BAC}}=a\sqrt{3}\)
\(\Rightarrow BD^2+CD^2=BC^2\Rightarrow CD\perp BD\)
\(cos\widehat{ADC}=\frac{AD}{CD}=\frac{1}{\sqrt{2}}\)
\(cos\left(\overrightarrow{AB};\overrightarrow{CD}\right)=\frac{\overrightarrow{AB}.\overrightarrow{CD}}{AB.CD}=\frac{\left(\overrightarrow{AD}+\overrightarrow{DB}\right).\overrightarrow{CD}}{a^2\sqrt{2}}=\frac{\overrightarrow{AD}.\overrightarrow{CD}}{a^2\sqrt{2}}=\frac{a.a\sqrt{2}.\frac{1}{\sqrt{2}}}{a^2\sqrt{2}}=\frac{1}{\sqrt{2}}\)
\(\Rightarrow\left(\overrightarrow{AB};\overrightarrow{CD}\right)=45^0\)
