Baltic Way 1991 · Problem 17

Geometry

Let the coordinate planes have the reflection property. A beam falls onto one of them. How does the final direction of the beam after reflecting from all three coordinate planes depend on its initial direction?

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Topics

Coordinates and vectors · Solid geometry

Solutions

Solution

Solution: Let the velocity vector of the beam be $\vec{v} = (\alpha, \beta, \gamma)$ . Reflection from each of the coordinate planes changes the sign of exactly one of the coordinates $\alpha$ , $\beta$ and $\gamma$ , and thus the final direction will be opposite to the initial one.

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AoPS · Baltic Way 1991