Let $\mathbb { R } ^ { 3 }$ be the set of the three-dimensional real column vectors and $\mathbb { R } ^ { 3 \times 3 }$ be the set of the three-by-three real matrices. Let $n _ { 1 } , n _ { 2 }$, and $n _ { 3 } \in \mathbb { R } ^ { 3 }$ be linearly independent unit-length vectors and $n _ { 4 } \in \mathbb { R } ^ { 3 }$ be a unit-length vector not parallel to $n _ { 1 } , n _ { 2 }$, or $n _ { 3 }$. Let A and B be square matrices defined as
$$\mathbf { A } = \left( \begin{array} { l } n _ { 1 } ^ { \mathrm { T } } - n _ { 2 } ^ { \mathrm { T } } \\ n _ { 2 } ^ { \mathrm { T } } - n _ { 3 } ^ { \mathrm { T } } \\ n _ { 3 } ^ { \mathrm { T } } - n _ { 4 } ^ { \mathrm { T } } \end{array} \right) , \quad \mathbf { B } = \sum _ { i = 1 } ^ { 4 } n _ { i } n _ { i } ^ { \mathrm { T } }$$
Here, $\mathrm { X } ^ { \mathrm { T } }$ and $\boldsymbol { x } ^ { \mathrm { T } }$ denote the transpose of a matrix X and a vector $\boldsymbol { x }$, respectively. Answer the following questions.
(1) Find the condition for $n _ { 4 }$ such that the rank of $\mathbf { A }$ is three.
(2) In the three-dimensional Euclidean space $\mathbb { R } ^ { 3 }$, consider four planes $\Pi _ { i } = \{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } \boldsymbol { x } - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) that satisfy the following three conditions: (i) the rank of A is three, (ii) $\Omega = \left\{ x \in \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } \geq 0 , i = 1,2,3,4 \right\}$ is not the empty set, and (iii) there exists a sphere $\mathrm { C } ( \subset \Omega )$ to which $\Pi _ { i } ( i = 1,2,3,4 )$ are tangent. The position vector of the center of C is represented by $\mathbf { A } ^ { -1 } \boldsymbol { u }$ using a vector $\boldsymbol { u } \in \mathbb { R } ^ { 3 }$. Express $\boldsymbol { u }$ using $d _ { i } ( i = 1,2,3,4 )$.
(3) Show that B is a positive definite symmetric matrix.
(4) Consider the point P from which the sum of squared distances to four planes $\{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) is minimized. The position vector of P is represented by $\mathrm { B } ^ { -1 } v$ using a vector $v \in \mathbb { R } ^ { 3 }$. Express $v$ using $n _ { i }$ and $d _ { i } ( i = 1,2,3,4 )$.
(5) Let $l _ { i }$ be a straight line through a point $Q _ { i }$, the position vector of which is $x _ { i } \in \mathbb { R } ^ { 3 }$, parallel to $n _ { i } ( i = 1,2,3 )$ in $\mathbb { R } ^ { 3 }$. Let $\mathrm { R } _ { i }$ be the orthogonal projection of an arbitrary point $R$, the position vector of which is $y \in \mathbb { R } ^ { 3 }$, onto $l _ { i }$. The position vector of $R _ { i }$ is represented by $y - \mathrm { W } _ { i } \left( y - x _ { i } \right)$ using a matrix $\mathrm { W } _ { i } \in \mathbb { R } ^ { 3 \times 3 }$. The identity matrix is denoted by $I \in \mathbb { R } ^ { 3 \times 3 }$.
(a) Express $\mathrm { W } _ { i }$ using $n _ { i }$ and I.
(b) Show that $\mathrm { W } _ { i } ^ { \mathrm { T } } \mathrm { W } _ { i } = \mathrm { W } _ { i }$.
(c) Consider a plane $\Sigma = \left\{ \boldsymbol { x } \in \mathbb { R } ^ { 3 } \mid \boldsymbol { a } ^ { \mathrm { T } } \boldsymbol { x } = b \right\} \left( \boldsymbol { a } \in \mathbb { R } ^ { 3 } \right.$ is a non-zero vector, and $b$ is a real number). Let $\mathrm { S } \in \Sigma$ be the point from which the sum of squared distances to $l _ { 1 } , l _ { 2 }$, and $l _ { 3 }$ is minimized. When $n _ { 1 } , n _ { 2 }$, and $n _ { 3 }$ are orthogonal to each other, the position vector of $S$ is represented by
$$\left( \mathrm { I } - \frac { a a ^ { \mathrm { T } } } { a ^ { \mathrm { T } } a } \right) w + \frac { a b } { a ^ { \mathrm { T } } a }$$
using a vector $\boldsymbol { w } \in \mathbb { R } ^ { 3 }$ which is independent of $a$ and $b$. Express $\boldsymbol { w }$ using $\mathbf { W } _ { i }$ and $x _ { i } ( i = 1,2,3 )$.