Đặt hệ trục Oxyz vào chóp với \(O\equiv A\); \(Ox\equiv AD\); \(Oy\equiv AB\); \(Oz\equiv AS\)
Quy ước \(a\) bằng 1 đơn vị độ dài
Ta có các tọa độ: \(A\left(0;0;0\right);B\left(0;1;0\right);C\left(1;1;0\right);D\left(2;0;0\right);S\left(0;0;1\right)\)
\(\Rightarrow M\left(0;\frac{1}{2};\frac{1}{2}\right);N\left(\frac{3}{2};\frac{1}{2};0\right)\) \(\Rightarrow\overrightarrow{MN}=\left(\frac{3}{2};0;-\frac{1}{2}\right)=\frac{1}{2}\left(3;0;-1\right)\)
\(\left\{{}\begin{matrix}\overrightarrow{SA}=\left(0;0;1\right)\\\overrightarrow{CA}=\left(1;1;0\right)\end{matrix}\right.\) \(\Rightarrow\overrightarrow{n_{\left(SAC\right)}}=\left[\overrightarrow{SA};\overrightarrow{CA}\right]=\left(-1;1;0\right)\)
\(\Rightarrow sin\widehat{\left(MN;\left(SAC\right)\right)}=cos\left(\overrightarrow{n};\overrightarrow{MN}\right)=\frac{\left|-1.3+0.1-1.0\right|}{\sqrt{3^2+0^2+1^2}\sqrt{1^2+1^2+0^2}}=\frac{3\sqrt{5}}{10}\)
