Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by \(2 \theta\) when the mirror is rotated by an angle \(\theta\).

We are given a light ray that is reflected off a mirror. We need to find the change in direction of the light ray as the mirror rotates by an angle θ. First, let's draw a diagram of this situation. Draw a light ray incident on a mirror and label the angle between the light ray and the normal to the mirror as α. Draw the reflected light ray and label the angle between it and the normal as α as well, since the angle of incidence is equal to the angle of reflection. Now, rotate the mirror counterclockwise by an angle θ and draw the normal to the rotated mirror. Draw the final reflected light ray and label the angle between the final reflected light ray and the normal of the rotated mirror as β.

Now, we need to find a relationship between the angles α, β, and θ. From our diagram, notice that the vertical angles formed by the normals to the mirrors are equal, so we have: \( \angle 1 = \angle 4 \) Also, since the mirror rotates counterclockwise by an angle θ, we can express: \( \angle 2 = \alpha + \theta \) Now consider the angles within the triangle formed by the normals and the final reflected light ray. The angles within a triangle add up to 180°, therefore: \( \angle 3 + \angle 4 + \beta = 180^\circ \)

We now substitute the relationships we found for the angles in the triangle equation: \( (\alpha + \theta) + \angle 1 + \beta = 180^\circ \) Since \( \angle 1 = \angle 4\), we can substitute \( \angle 4\) for \( \angle 1 \): \( (\alpha + \theta) + \angle 4 + \beta = 180^\circ \) Now, we know that the angles α and β are equal to the angles of incidence and reflection, respectively, so the sum of these angles and their respective normals is also equal to 180°, as per our initial diagram: \( \alpha + \angle 4 = 180^\circ \) Now, we can substitute this equation into the previous equation: \( (\alpha + \theta) + (180 - \alpha) + \beta = 180^\circ \) Simplifying this equation: \( \theta + \beta = 180^\circ - \alpha \) Since we are given that the mirror rotates by an angle θ, we need to find the change in direction of the reflected light ray, which is the difference between its final angle and its initial angle: Change in direction = Final angle - Initial angle = β - α Now, we can substitute our equation for β - α: Change in direction = (180° - θ) - α We can then substitute the relationship α = β - θ: Change in direction = (180° - θ) - (β - θ) Simplifying this equation, we have: Change in direction = 2θ

We have now proven that the change in direction of a light ray reflected from a mirror is equal to 2θ when the mirror is rotated by an angle θ.