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. Determine the diameter of the marble sphere to the nearest centimeter.
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.