A line on a two-dimensional plane can be expressed as $\alpha x + \beta y + \gamma = 0$, where $( x , y )$ is a point on the line in the Cartesian coordinate system. We call the column vector $( \alpha , \beta , \gamma ) ^ { \mathrm { T } }$ a coefficient vector of the line. Answer the following questions. Note that the coefficient vector in your answer must satisfy $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
(1) Find a coefficient vector of a line that passes through a point $\vec { a }$ and is perpendicular to a unit vector $\vec { v }$ on a two-dimensional plane.
(2) Let a line B pass through a point $\vec { b }$ and be perpendicular to a unit vector $\vec { n }$. Given a line $A$, let the line $A ^ { \prime }$ be the mirror transformation of the line $A$ over the line $B$. Using $\vec { b }$ and $\vec { n }$, write a three-dimensional square matrix that transforms a coefficient vector of the line A to a coefficient vector of the line $\mathrm { A } ^ { \prime }$.
(3) Find the determinant of the matrix derived in Question (2).
(4) Consider the movement of the line $\mathrm { D } _ { t }$ whose coefficient vector changes with the real variable $t$ as $\left( 4 t , 4 t ^ { 2 } - 1 , t \right) ^ { \mathrm { T } }$. This line passes through a point regardless of $t$. Find the coordinate of that point.
(5) Suppose that, with the mirror transformation over a line $M _ { t }$, which also changes with $t$, the line $\mathrm { D } _ { t }$ in Question (4) is transformed to the line with a coefficient vector $( 0,1 , - t ) ^ { \mathrm { T } }$. Find the coefficient vector $\left( \alpha _ { t } , \beta _ { t } , \gamma _ { t } \right) ^ { \mathrm { T } }$ of the line $\mathrm { M } _ { t }$, where $\alpha _ { t } > 0$ and $\beta _ { t } > 0$ for $t > 0$.
(6) When $t$ changes from 0 to $+ \infty$, consider the region where the line $\mathrm { M } _ { t }$ in Question (5) can exist. Describe the region using a simple mathematical expression and draw a diagram of the region.