The point $T ( 7 | 10 | 0 )$ lies on the edge $\left[ \mathrm { A } _ { 3 } \mathrm {~A} _ { 4 } \right]$. Investigate computationally whether there are points on the edge $\left[ \mathrm { B } _ { 3 } \mathrm {~B} _ { 4 } \right]$ for which the following holds: The line segments connecting the point to the points $B _ { 1 }$ and $T$ are perpendicular to each other. If applicable, give the coordinates of these points.

Given is the line $g : \vec { X } = \left( \begin{array} { l } 1 \\ 7 \\ 2 \end{array} \right) + \lambda \cdot \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$, as well as another line $h$, which is parallel to $g$ and passes through the point $A ( 2 | 0 | 0 )$. The point $B$ lies on $g$ such that the lines AB and $h$ are perpendicular to each other.
Determine the coordinates of $B$.

Given is the line $g : \vec { X } = \left( \begin{array} { l } 1 \\ 7 \\ 2 \end{array} \right) + \lambda \cdot \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$, as well as another line $h$, which is parallel to $g$ and passes through the point $A ( 2 | 0 | 0 )$. The point $B$ lies on $g$ such that the lines AB and $h$ are perpendicular to each other.
Calculate the distance between $g$ and $h$.

The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$.
Show by calculation that the triangle ABS is isosceles. Give the coordinates of point $D$ and describe the special position of plane $E$ in the coordinate system.

The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$.
Determine an equation of plane $F$ in coordinate form.

The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$.
Calculate the volume $V$ of the pyramid ABCDS.

A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality.
Show that the highest point of the fountain is approximately 64 cm above the ground.

Water fountains emerge at four points on the surface of the marble sphere. One of these exit points is described in the model by the point $L _ { 0 } ( 1 | 1 | 6 )$. The corresponding fountain is modeled by points $L _ { t } \left( t + 1 | t + 1 | 6,2 - 5 \cdot ( t - 0,2 ) ^ { 2 } \right)$ with suitable values $t \in \mathbb { R } _ { 0 } ^ { + }$.
The point $P$ lies inside the triangle ABS and describes in the model the location where the fountain hits the bronze bowl (see figure). Determine the coordinates of $P$.

Water fountains emerge at four points on the surface of the marble sphere. One of these exit points is described in the model by the point $L _ { 0 } ( 1 | 1 | 6 )$. The corresponding fountain is modeled by points $L _ { t } \left( t + 1 | t + 1 | 6,2 - 5 \cdot ( t - 0,2 ) ^ { 2 } \right)$ with suitable values $t \in \mathbb { R } _ { 0 } ^ { + }$.
Investigate whether the highest point of the water fountain is higher than the highest point of the fountain.

Water fountains emerge at four points on the surface of the marble sphere. One of these exit points is described in the model by the point $L _ { 0 } ( 1 | 1 | 6 )$. The corresponding fountain is modeled by points $L _ { t } \left( t + 1 | t + 1 | 6,2 - 5 \cdot ( t - 0,2 ) ^ { 2 } \right)$ with suitable values $t \in \mathbb { R } _ { 0 } ^ { + }$.
A total of 80 ml of water flows per second from the four exit points into the bronze bowl. Determine the time in seconds that passes until the initially empty fountain is completely filled with water.

Given are the points $P ( 4 | 5 | - 19 ) , Q ( 5 | 9 | - 18 )$ and $R ( 3 | 7 | - 17 )$, which lie in the plane $E$, as well as the line $g : \vec { X } = \left( \begin{array} { c } - 12 \\ 11 \\ 0 \end{array} \right) + \lambda \cdot \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$.
Determine the length of the line segment $[ \mathrm { PQ } ]$. Show that the triangle PQR is right-angled at $R$, and use this to justify that the line segment $[ \mathrm { PQ } ]$ is the diameter of the circumcircle of triangle PQR.
(for verification: $\overline { \mathrm { PQ } } = 3 \sqrt { 2 }$ )

Determine an equation of $E$ in coordinate form and show that the line $g$ lies in $E$.
(for verification: $E : 2 x _ { 1 } - x _ { 2 } + 2 x _ { 3 } + 35 = 0$ )

Justify without calculation that $g$ lies in the $x _ { 1 } x _ { 2 }$-plane.

In a model for a coastal section by the sea, the $x _ { 1 } x _ { 2 }$-plane describes the horizontal water surface and the line $g$ describes the shoreline. The plane $E$ represents the sea floor in the considered section. A buoy floats on the water surface at the location corresponding to the coordinate origin $O$. One unit of length corresponds to one meter in reality.
Determine the angle at which the sea floor slopes down relative to the water surface.

A photographer is to take underwater photos for a travel magazine.
The photographer swims along the shortest possible path from the shoreline to the buoy. Determine the length of this path.

From the buoy, the photographer dives vertically with respect to the water surface downward to a location whose distance to the sea floor is exactly three meters and is represented in the model by the point $K$.
Determine by calculation what depth below the water surface the photographer reaches during this dive.

Three small colorful starfish are located on the sea floor and are represented in the model by the points $P , Q$ and $R$. The photographer moves for his shots from the location described in the model by the point $K$, parallel to the sea floor. The camera lens points perpendicular to the sea floor and has a cone-shaped field of view with an opening angle of $90 ^ { \circ }$.
Assess whether the photographer can reach a location in this way where he can see all three starfish simultaneously in the camera's field of view.

Show that $g$ lies in the plane with the equation $x _ { 1 } + x _ { 2 } + x _ { 3 } = 2$.

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: $k = 19$ )

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.

Show that the edge length of the cube is 12.

Determine the coordinates of one of the two vertices of the octahedron that do not lie in $H$.

Calculate the length of the line segment [AB] and state the special position of this line segment in the coordinate system. (for verification: $\overline { \mathrm { AB } } = \sqrt { 2 }$ )