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Suppose the three coordinate planes are all mirrored and a light ray given by the vector $ a = \langle a_1, a_2, a_3 \rangle $ 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 = \langle a_1, -a_2, a_3 \rangle $. 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 very precisely the distance from the earth to the moon.)

$$

\mathbf{d}=\left\langle- a_{1},-a_{2},-a_{3}>=(-1)<a_{1}, a_{2}, a_{3}>=(-1) \mathbf{a}\right.

$$

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Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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