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 $E$ 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)