\(OB=\sqrt{SB^2-SO^2}=\frac{a\sqrt{3}}{3}\) \(\Rightarrow OA=\sqrt{AB^2-OB^2}=\frac{a\sqrt{6}}{3}\)
Đặt hệ trục Oxyz vào hình chóp với \(Oz\) trùng tia \(OS\); \(Ox\) trùng tia OB, Oy trùng tia OA, \(\frac{a}{3}\) bằng 1 đơn vị độ dài
\(\Rightarrow S\left(0;0;\sqrt{6}\right)\); \(A\left(0;\sqrt{6};0\right);B\left(\sqrt{3};0;0\right);D\left(-\sqrt{3};0;0\right)\)
\(\Rightarrow\overrightarrow{SA}=\left(0;\sqrt{6};-\sqrt{6}\right)=\sqrt{6}\left(0;1;-1\right)\)
\(\overrightarrow{SB}=\left(\sqrt{3};0;-\sqrt{6}\right)=\sqrt{3}\left(1;0;-\sqrt{2}\right)\)
\(\overrightarrow{SD}=\left(-\sqrt{3};0;-\sqrt{6}\right)=-\sqrt{3}\left(1;0;\sqrt{2}\right)\)
\(\Rightarrow\overrightarrow{n_{\left(SAB\right)}}=\left[\overrightarrow{SA};\overrightarrow{SB}\right]=\left(\sqrt{2};1;1\right)\)
\(\overrightarrow{n_{\left(SAD\right)}}=\left[\overrightarrow{SA};\overrightarrow{SD}\right]=\left(\sqrt{2};-1;-1\right)\)
\(\Rightarrow cos\alpha=\frac{\left|2-1-1\right|}{\sqrt{2+1+1}\sqrt{2+1+1}}=0\Rightarrow\alpha=90^0\)
