a)
U1: x = [4, 14, -4] + r·([6, 11, -4] - [4, 14, -4]) = [4, 14, -4] + r·[2, -3, 0]
U2: x = [11, 9, -14] + r·([9, 6, -12] - [11, 9, -14]) = [11, 9, -14] + r·[-2, -3, 2]
U1: |[2, -3, 0]| = √13 LE = 100·√13 m/min = 360.6 m/min = 21.63 km/h
U1: |[-2, -3, 2]| = √17 LE = 100·√17 m/min = 412.3 m/min = 24.74 km/h
Die Richtungsvektoren sind linear unabhängig, damit sind die Geraden nicht parallel.
Abstand zweier Windschiefer Geraden
d = |[2, -3, 0] ⨯ ([11, 9, -14] - [4, 14, -4])| / |[2, -3, 0]| = 10.46 LE = 1046 m
b)
[11, 9, -14] + r·[-2, -3, 2] = [x, y, 0] --> x = -3 ∧ y = -12 ∧ r = 7
|[45, 2,0] - [-3, -12, 0]| = 50 LE = 5000 m
ARCSIN([-2, -3, 2]·[0, 0, 1]/ABS([-2, -3, 2])) = 29.02°
ARCTAN(2/√(2^2 + 3^2)) = 29.02°
U1: x =[4, 14, -4] + 7·[2, -3, 0] = [18, -7, -4]
|[18, -7, -4] - [-3, -12, 0]| = √482 = 21.95 LE = 2195 m