Gọi mặt phẳng (P) chứa A'C có pt \(ax+by+cz+d=0\)
Do \(A'\left(0;0;1\right);C\left(1;1;0\right)\) \(\Rightarrow\left\{{}\begin{matrix}c+d=0\\a+b+d=0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}c=-d\\b=-a-d\end{matrix}\right.\) (1)
\(\left\{{}\begin{matrix}\overrightarrow{n_{\left(P\right)}}=\left(a;b;c\right)\\\overrightarrow{n_{Oxy}}=\left(0;0;1\right)\end{matrix}\right.\) \(\Rightarrow cos\alpha=\frac{\left|c\right|}{\sqrt{a^2+b^2+c^2}}=\frac{1}{\sqrt{6}}\)
\(\Leftrightarrow6c^2=a^2+b^2+c^2\Leftrightarrow5c^2-a^2-b^2=0\) (2)
Thế (1) vào (2):
\(5d^2-a^2-\left(a+d\right)^2=0\Leftrightarrow2d^2-ad-a^2=0\)
\(\Leftrightarrow\left(2d+a\right)\left(d-a\right)=0\Rightarrow\left[{}\begin{matrix}a=d\Rightarrow b=-2d\\a=-2d\Rightarrow b=d\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\left(a;b;c\right)=\left(d;-2d;-d\right)=d\left(1;-2;-1\right)\\\left(a;b;c\right)=\left(-2d;d;-d\right)=-d\left(2;-1;1\right)\end{matrix}\right.\)
\(\Rightarrow\)Vecto pháp tuyến của hai mặt phẳng lần lượt là \(\left\{{}\begin{matrix}\overrightarrow{n_1}=\left(1;-2;-1\right)\\\overrightarrow{n_2}=\left(2;-1;1\right)\end{matrix}\right.\)
\(\Rightarrow\) Góc giữa 2 mặt phẳng:
\(cos\beta=\frac{\left|\overrightarrow{n_1}.\overrightarrow{n_2}\right|}{\left|\overrightarrow{n_1}\right|.\left|\overrightarrow{n_2}\right|}=\frac{\left|2+2-1\right|}{\sqrt{1+4+1}.\sqrt{4+1+1}}=\frac{1}{2}\) \(\Rightarrow\beta=60^0\)
