Given the lines
$$r \equiv \frac { x - 2 } { 1 } = \frac { y + 1 } { 1 } = \frac { z + 4 } { - 3 } , \quad s \equiv \left\{ \begin{array} { l } x + z = 2 \\ - 2 x + y - 2 z = 1 \end{array} . \right.$$
a) (1.5 points) Write an equation of the common perpendicular line to $r$ and $s$.
b) (1 point) Calculate the distance between $r$ and $s$.

Let the line $r \equiv \left\{ \begin{array} { l } - x - y + z = 0 \\ 2 x + 3 y - z + 1 = 0 \end{array} \right.$ and the plane $\pi \equiv 2 x + y - z + 3 = 0$. It is requested:\ a) ( 0.75 points) Calculate the angle formed by $r$ and $\pi$.\ b) (1 point) Find the symmetric point of the intersection of line $r$ and plane $\pi$ with respect to the plane $z - y = 0$.\ c) ( 0.75 points) Determine the orthogonal projection of line $r$ onto plane $\pi$.

Given the line $r \equiv \frac { x - 1 } { 2 } = \frac { y } { 1 } = \frac { z + 1 } { - 2 }$, the plane $\pi : x - z = 2$ and the point A(1,1,1), find:\ a) ( 0.75 points) Study the relative position of $r$ and $\pi$ and calculate their intersection, if it exists.\ b) ( 0.75 points) Calculate the orthogonal projection of point A onto the plane $\pi$.\ c) (1 point) Calculate the point symmetric to point A with respect to the line $r$.

Let the line $r \equiv \left\{ \begin{array} { l } x = \lambda \\ y = 0 \\ z = 0 \end{array} \right.$ and the plane $\pi : z = 0$.
a) (1 point) Find an equation of the line parallel to the plane $\pi$ whose direction is perpendicular to $r$ and passes through the point $( 1,1,1 )$.
b) (1.5 points) Find an equation of a line that forms an angle of $\frac { \pi } { 4 }$ radians with the line $r$, is contained in the plane $\pi$ and passes through the point ( $0,0,0$ ).

In coordinate space, let $E$ be the plane passing through the three points $A ( 0 , - 1 , - 1 )$ , $B ( 1 , - 1 , - 2 )$ , $C ( 0,1,0 )$ . Assume $H$ is a point in space satisfying $\overrightarrow { AH } = \frac { 2 } { 3 } \overrightarrow { AB } - \frac { 1 } { 3 } \overrightarrow { AC } + 3 ( \overrightarrow { AB } \times \overrightarrow { AC } )$ . Based on the above, answer the following questions.
(1) Find the volume of tetrahedron $ABCH$ . (4 points) (Note: The volume of a tetrahedron is one-third of the base area times the height)
(2) Let $H ^ { \prime }$ be the symmetric point of point $H$ with respect to plane $E$ . Find the coordinates of $H ^ { \prime }$ . (4 points)
(3) Determine whether the projection of point $H ^ { \prime }$ onto plane $E$ lies inside $\triangle ABC$ . Explain your reasoning. (4 points) (Note: The interior of a triangle does not include the three sides of the triangle)

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
Continuing from question 19, it is known that point $P$ is on plane $E$ and $b = 0$. Find the maximum possible range of $c$ and the minimum possible length of line segment $\overline{OP}$. (Non-multiple choice question, 8 points)