A laser beam in air is directed such that its initial (unit) direction vector in spherical polar form is I(θ=60∘,ϕ=45∘), where θ is the angle from the positive z–axis and ϕ is the azimuthal angle from the positive x–axis measured in the xy–plane. The beam falls on a plane mirror whose equation in Cartesian coordinates is 2x−y+2z=5.
(i) Convert the incident direction vector to Cartesian unit‐vector form.
(ii) Obtain a unit normal vector to the mirror directed towards the region of incidence.
(iii) Compute the Cartesian unit vector that gives the direction of the reflected beam.
(iv) Express the reflected direction obtained in part (iii) back in spherical polar form, specifying the new (θ,ϕ) values to the nearest degree.