What would the angular separation Δθ be between the rays reflected from the two surfaces?

Suppose that the spotlight shines so that different parts of the beam reflect off of different two surfaces, one inclined at an angle α (from the horizontal) and one inclined at an angle β. What would the angular separation Δθ be between the rays reflected from the two surfaces? Assume that the light comes at an angle θa to the vertical. Express your answer in terms of some or all of the angles θa, α, and β.

Answer 1

For some surface angle α and spotlight angle θa, the angle between the normal to the surface and the incoming ray is θa – α.

Any ray reflects at the same angle it came in at.
Hence think of the reflected ray being rotated from the incoming one, twice, for an angle θa – α toward the vertical in picture.
Angle of ray reflected off red surface is then θa – 2(θa – α) = 2α – θa, relative to vertical.

Now imagine there’s another surface inclined at angle β. Angle of ray reflected off of that surface is 2β – θa, relative to vertical.

We now have angles of the two reflected rays relative to the same direction (vertical). The angle BETWEEN those rays is then the difference of those angles.
2α – θa – (2β – θa) = 2(α + β)

Answer 2

Part A: θa
Part B: α – (θa – α)
Part C: 2(α – β)

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