a)
n = [1, 4, 1] ⨯ [1, 2, 2] = [6, -1, -2]
Ebene in Normalenform
E: (x - [1, 1, 5])·[6, -1, -2] = 0
b)
E: 6·x - y - 2·z = -5
AB = B - A = [2, 1, 4] - [1, 2, 3] = [1, -1, 1]
g: x = A + r·AB = [1, 2, 3] + r·[1, -1, 1] = [r + 1, 2 - r, r + 3]
g in E einsetzen
6·(r + 1) - (2 - r) - 2·(r + 3) = -5 --> r = -3/5 = -0.6
S = [1, 2, 3] - 3/5·[1, -1, 1] = [2/5, 13/5, 12/5] = [0.4, 2.6, 2.4]
c)
α = ASIN([1, -1, 1]·[6, -1, -2] / (|[1, -1, 1]|·|[6, -1, -2]|)) = 0.4677 = 26.80°