   Chapter 12.2, Problem 52E

Chapter
Section
Textbook Problem

# Suppose the three coordinate planes are all mirrored and a light ray given by the vector a = ⟨a1, a2, a3⟩ first strikes the xz-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by b = ⟨a1, −a2, a3⟩. Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate precisely the distance from the earth to the moon.) To determine

To show: The initial ray that is reflected on three mirrors placed at coordinate planes is parallel to the final resultant reflected output ray.

Explanation

Given data:

Light ray given by vector a=a1,a2,a3 and reflected ray is given by vector b=b1,b2,b3 .

The ray of light strikes all three mirrors that are placed at coordinate planes. Hence, the ray is not parallel to any of the mirrors as the planes are always perpendicular to each other. Therefore,

a10a20a30

Draw the diagram of reflected light ray on mirrors as shown in Figure 1.

The angle of reflection is same as angle of reflection, hence the angle is θ on both edges of point B in Figure 1.

Consider the magnitude of vector b is equal to magnitude of vector a.

|b|=|a|

From Figure 1, write the expression for sinθ .

sinθ=|b2||b| (1)

sinθ=|a2||a| (2)

Equate the equations (1) and (2).

|b2||b|=|a2||a||b2|=|a2|(|b||a|)

Substitute |a| for |b| ,

|b2|=|a2|(|a||a|)=|a2|(1)=|a2|

From Figure 1, it is observed that |AB|=|BC| .

Write the expression for sinϕ .

sinϕ=|b3||BC| (3)

sinϕ=|a3||AB| (4)

Equate the equations (3) and (4).

|b3||BC|=|a3||AB||b3|=|a3||BC||AB|

Substitute |BC| for |AB| ,

|b3|=|a3||BC||BC||b3|=|a3|(1)|b3|=|a3|

Write the expression for cosϕ

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