Question #41750

Let $P_1$ and $P_2$ be two planes whose normal vectors $\vec{n}_1$ and $\vec{n}_2$ are given by the first two column vectors of the matrix $A = \begin{pmatrix} 8 & x & 1 \\ 6 & 20 & 1 \\ 0 & z & 1 \end{pmatrix}$ where $x$ is an integer. The third column vector is denoted by $\vec{n}_3$. The volume of the parallelepiped formed by the vectors $\vec{n}_1, \vec{n}_2,$ and $\vec{n}_3$ is $70$. If the plane $P_2$ is perpendicular to the $xy$-plane, find the value of $\left[ 26 \cos \theta \right]$, where $\theta$ is the acute angle between the planes $P_1$ and $P_2$, and $[ \cdot ]$ denotes the greatest integer function.