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: $\overline { \mathrm { PQ } } = 3 \sqrt { 2 }$ )
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: $\overline { \mathrm { PQ } } = 3 \sqrt { 2 }$ )