3.3.3 Line and plane in space

Problem: By the coordinates of the vertices of the pyramid \( A_{1} A_{2} A_{3} A_{4} \) find: 1) the length of the edge \( A_{1} A_{2} ; 2 \) ) the angle between edges \( A_{1} A_{3} \) and \( \left.A_{1} A_{4} ; 3\right) \) the angle between faces \( A_{1} A_{2} A_{3} \) and \( \left.A_{1} A_{2} A_{4} ; 4\right) \) the equation of the straight line passing through the vertices \( A_{4} \) and the center of gravity of the face \( A_{1} A_{2} A_{3} ; 5 \) ) the length and the equation of the height from the vertex \( A_{4} \) to the face \( \left.A_{1} A_{2} A_{3} ; 6\right) \) the distance between crossing edges \( A_{1} A_{2} \) and \( A_{3} A_{4} \). \[ A_{1}(1 ; 1 ; 2), \quad A_{2}(-1 ; 1 ; 3), \quad A_{3}(2 ;-2 ; 4), \quad A_{4}(-1 ; 0 ;-2) \text {. } \]