Question #41657

Let $A$ be a $3 \times 3$ matrix with real entries such that: $A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \\ 2 \end{pmatrix}$ Let $P_1$ and $P_2$ be two planes passing through the origin. The normal vector to $P_1$ is parallel to the first row vector of $A$, and the normal vector to $P_2$ is parallel to the first column vector of $\text{adj}(A)$. Let $\vec{n}_1$ and $\vec{n}_2$ be the normal vectors of $P_1$ and $P_2$ respectively, such that their $\hat{k}$ components are strictly positive and they both have a magnitude of $\sqrt{2}$. If $\theta$ is the angle between the vectors $\vec{n}_1$ and $\vec{n}_2$, what is the value of $\cos \theta$?