The reflecting surfaces of two intersecting flat mirrors are...

The reflecting surfaces of two intersecting flat mirrors are at an angle of $\theta\left(0^{\circ}<\theta<90^{\circ}\right)$, as shown in Figure P35.22. If a light ray strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle of $\beta=180^{\circ}-2 \theta$.

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Key Concepts

Geometric Transformation of Reflections

In geometry, the composition of two reflections over lines (or planes in three dimensions) that intersect is equivalent to a rotation whose angle is twice the angle between those lines. This mathematical insight is crucial in analyzing optical paths involving multiple reflections and calculating the angular deviation between the original and final directions of the light ray.

Reflection from Intersecting Mirrors

When light undergoes successive reflections from two mirrors that meet at an angle, the overall effect on the ray’s direction is equivalent to a rotation. Specifically, the rotation angle is twice the angle between the mirrors. This concept allows one to determine the relationship between the incident and emerging rays regardless of the point of reflection.

Law of Reflection

This principle states that when a ray of light strikes a reflective surface, the angle of incidence (the angle between the incident ray and the normal to the surface) is equal to the angle of reflection (the angle between the reflected ray and the normal). It is a fundamental rule in optics that governs the behavior of light when interacting with reflective surfaces.

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