Three large plane mirrors form the three sides of an equilateral triangular tunnel of side length 1.0 m. The mirrors are vertical and meet at 60° angles along the edges of the triangle. A narrow laser beam enters the tunnel through a small aperture centered on one side, 25 cm above the base and 30 cm from the adjacent corner. The beam lies in a horizontal plane and initially travels perpendicular to the side through which it enters.
After a sequence of reflections from the interior mirror surfaces, the beam eventually leaves the triangular tunnel through the same side on which it entered, but at a different location.
Assume ideal specular reflection and answer the following.
(a) Using coordinate geometry with the origin at one corner of the triangle, assign coordinates to the entry point and write the initial direction vector of the beam.
(b) Determine algebraically the coordinates of the first three reflection points in the order in which they are encountered.
(c) Prove that after exactly three reflections the beam emerges parallel to its original direction, regardless of its initial position along the entry side, provided the incident direction is perpendicular to that side.
(d) For the given numerical entry location, calculate the coordinates of the exit point and the total distance traveled by the beam inside the tunnel.