Furthermore, given is the family of lines $h _ { a } : \vec { X } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right) + \mu \cdot \left( \begin{array} { l } 1 \\ a \\ 0 \end{array} \right)$ with $\mu \in \mathbb { R }$ and $a \in \mathbb { R }$. Prove that $g$ and $h _ { a }$ are skew lines for every value of $a$.
Determine an equation of $L$ in coordinate form and the angle $\varphi$ that $L$ makes with the $x _ { 1 } x _ { 2 }$-plane. (for verification: $x _ { 1 } + x _ { 2 } + x _ { 3 } - 19 = 0 ; \varphi \approx 55 ^ { \circ }$ )
Determine computationally the value of $k$ for which the pyramid $\mathrm { EFGHS } _ { k }$ completes the solid ABCDEFGH to form a large pyramid $\mathrm { ABCDS } _ { k }$. (for verification: $k = 19$ )
Draw the pyramid EFGHS${}_{15}$ in Figure 1. The lateral face $\mathrm { EFS } _ { 15 }$ and the base EFGH of this pyramid form an angle. Justify without further calculation that the measure of this angle is less than $45 ^ { \circ }$; use the following information for this purpose: For the midpoint $M$ of the square EFGH and the point $N$ with $\vec { N } = \frac { 1 } { 2 } \cdot ( \vec { E } + \vec { F } )$, we have $\overline { M S _ { 15 } } < \overline { M N }$.
Determine the change in height of the structure caused by the change in construction plan, in meters. Justify that in the lower part of the structure, the angle of inclination of the lateral faces with respect to the ground is more than $9 ^ { \circ }$ greater than in the upper part of the structure.
The shadow region of the entire pyramid on the ground consists in the model of two congruent quadrilaterals. Draw this shadow region in Figure 3 and specify the special form of the mentioned quadrilaterals.