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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 $\theta$ when the mirror is rotated by an angle $2\theta$.
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OpenStax College Physics Solution, Chapter 25, Problem 3 (Problems & Exercises) (2:08)

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This is College Physics Answers with Shaun Dychko. We're going to show that when a ray hits a flat mirror and that mirror is rotated by an angle of <i>theta</i> degrees the direction of the ray will change by two times that angle. So, we have this picture here initially and the ray is going to an angle of <i>alpha</i> with respect to this dotted line here. And then later on, we're going to show that the ray is going at an angle of <i>theta</i> plus <i>alpha</i> plus <i>theta</i> or <i>alpha</i> plus two <i>theta</i> with respect to the same horizontal dotted line. So, after the mirror is rotated by an angle <i>theta</i>, the ray is still incident at the same angle as it was before. But now, it has a new angle with respect to the perpendicular of the mirror. So, this is the new perpendicular of the mirror in blue here and here's the old perpendicular to the mirror. And, this perpendicular or normal, you might say, has moved by an angle <i>theta</i> because it's attached to the mirror, of course. And now, the new angle of incidence is this total <i>alpha</i>, the angle with respect to the original perpendicular plus this new <i>theta</i>. So, the angle of incidence is <i>alpha</i> plus <i>theta</i>. And so, the angle of reflection has to be the same, <i>alpha</i> plus <i>theta</i>. And so, the question is what is the new direction of the ray with respect to the original perpendicular? So, that's going to be this <i>alpha</i> plus <i>theta</i> that we found here plus this additional <i>theta</i>. And so, that's <i>alpha</i> plus two <i>theta</i>. And so, the change in direction is the new angle, which is <i>alpha</i> plus two <i>theta</i> minus the original angle of <i>theta</i> and then that makes the difference of two <i>theta</i>.