In an amusement park maze with all the walls covered with...

In an amusement park maze with all the walls covered with mirrors, Pilar sees Hernando's reflection from a series of three mirrors. If the reflected angle from mirror 3 is $55^{\circ}$ for the mirror arrangement shown in the figure, what is the angle of incidence on mir. ror $1 ?$

Key Concepts

Law of Reflection

This principle states that when a light ray strikes a smooth surface, the angle of incidence is equal to the angle of reflection. It is fundamental in optics and underpins the behavior of light in mirror systems, ensuring that the path of light can be accurately traced and predicted when reflecting off surfaces.

Multiple Reflection Geometry

When light reflects successively off several surfaces, each reflection follows the law of reflection. The cumulative effect of these reflections can be analyzed using geometric methods, summing or relating the individual angles to determine the overall deviation or direction of the ray. This concept is crucial for solving problems involving chains of reflective surfaces.

Angle of Incidence in Optical Systems

The angle of incidence is the angle between the incoming light ray and the normal (a perpendicular line) to the surface at the point of contact. Understanding this concept is essential in calculating the subsequent path of reflected rays, especially in complex arrangements like multiple mirror systems.

Transcript

00:01 One of the nice things about geometric optics is that, yes, you can use geometry to figure some things out.

00:09 Here we are going to trace back a reflected ray and use the law of reflection, tracing the path of a final ray back to the first mirror, mirror number one, that it struck.

00:28 But the law of reflection simply says that the angle of reflection is equal to the angle of incidence.

00:38 And those angles are typically measured to the normal with respect to the normal, to the mirror, or the perpendicular.

00:55 But here we have flat mirrors, and we can also look angles to the flat surface.

01:03 So we have the final ray, and i'm just going to kind of write down some obvious things that it came in at a 55 -degree angle to the normal.

01:17 And if we can use, there's a right triangle that we can kind of trace where the two normals from mirror 2 and mirror 3 meet at 90 degrees.

01:29 So we can solve for the angle of incidents with respect to mirror 2, and that is 35 degrees.

01:44 Okay, normal, normal 2 intersects normal 3 at 90 degrees.

02:00 And so we have the angle with respect to mirror 2 of incidence equal to the reflection as 35 degrees.

02:10 So the harder picture is mirror number one.

02:18 Not exactly the prettiest picture, but that mirror one is tilted at 15 degrees to the vertical, which means that it's normal is tilted at 15 degrees relative to the horizontal.

02:38 Okay, there's a kind of complementary thing going on there.

02:41 If you're having a hard time seeing that, but you can imagine the mirror straight up and down, and of course, its normal will be 0 degrees to the x -axis.

02:54 So mirror 1 makes theta.

03:03 I don't want to call that theta...

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