A smooth, planar mirror in three-dimensional space has a unit normal vector n=⟨0,0.60,0.80⟩. A light ray is observed after reflection to travel in the direction given by the vector r=⟨0.40,0.40,−0.82⟩ (coordinates expressed in an orthonormal basis).
(a) Normalize the reflected direction vector r to obtain a unit vector r^.
(b) Using the law of reflection expressed in vector form, determine the unit vector i^ that represents the direction of the incident ray before striking the mirror. Show all intermediate steps of your vector calculations.
(c) Calculate the angle of incidence—defined as the angle between i^ and the mirror’s normal n—to the nearest tenth of a degree.
(d) Verify explicitly that the angle obtained in part (c) equals the angle of reflection between r^ and n, thereby confirming consistency with the mirror law.