The angle between the lines x-3 y -4=0,4 y - z +5=0 and x+3 y -11=0,2 y - z +6=0 is

Step-by-step solution

Angle between lines defined by plane intersections The direction vector of a line formed by two planes' intersection is the cross product of their normal vectors. For the first line, with planes x - 3y - 4 = 0 and 4y - z + 5 = 0 , the normals are n₁ = 1, -3, 0 and n₂ = 0, 4, -1 . The cross product yields the direction vector: d₁ = n₁ n₂ = 3, 1, 4 For the second line, given by x + 3y - 11 = 0 and 2y - z + 6 = 0 , the normals are n₃ = 1, 3, 0 and n₄ = 0, 2, -1 . The direction vector is: d₂ = n₃ n₄ = -3, 1, 2 The angl