\textbf{Problem 4}

In a three-dimensional Cartesian coordinate system $x y z$, consider the positional relationship among three planes defined by Equations (1)-(3), and the positional relationship among the three planes and a sphere defined by Equation (4).

$$\begin{aligned}
& a _ { 11 } x + a _ { 12 } y + a _ { 13 } z = b _ { 1 } , \\
& a _ { 21 } x + a _ { 22 } y + a _ { 23 } z = b _ { 2 } , \\
& a _ { 31 } x + a _ { 32 } y + a _ { 33 } z = b _ { 3 } , \\
& x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3 ,
\end{aligned}$$

where $a _ { i j }$ and $b _ { i } ( i , j = 1,2,3 )$ are constants.

For the three planes, let $\mathrm { A } = \left( \begin{array} { l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } \end{array} \right)$ be the coefficient matrix and $\mathbf { B } = \left( \begin{array} { l l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } & b _ { 1 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } & b _ { 2 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } & b _ { 3 } \end{array} \right)$ be the augmented coefficient matrix.

I. Let $\mathbf { A } = \left( \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c c c } 1 & 1 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & - c \end{array} \right)$ where $c$ is a positive constant.

\begin{enumerate}
  \item Find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
  \item Among the three planes, the plane that is tangential to the sphere defined by Equation (4) at a point $\mathrm { P } ( 1,1,1 )$ is called Plane 1. Between the other two planes, the plane with the shorter distance to P is called Plane 2. Find the distance between P and Plane 2. Then, find the volume of the part of the sphere existing between Planes 1 and 2.
\end{enumerate}

II. When the three planes intersect in a line, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.

III. Suppose that the three planes are tangential to the sphere at three different points. Illustrate all possible positional relationships among the three planes and the sphere. In addition, for each relationship, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.