Three mirrors and a ray !!

 You may sense this to have some physical flavour but I am talking something of mathematical interest ( and perhaps of physical too! )

Consider a system of three mutually perpendicular plane mirrors P, Q, R as shown :

Theorem :
A ray of light incident on the above system of mirrors gets reflected at most 3 times before escaping !!

Proof:
We employ the fact that an incident ray (in above system) if reflected exactly 3 times then the emergent ray is parallel and in a direction opposite to that of the incident ray. The proof employs an easy use of vectors and is left to the reader as an exercise. ( Use Ê = Î - 2 (Î . ñ)ñ  where Î is the unit vector along the incident ray, Ê is the unit vector along the reflected ray and ñ is the unit vector along the normal to a plane mirror, during reflection.) [I discovered the above theorem while solving this.]

Suppose there is an incident ray a₁ that gets reflected more than 3 times....we will obtain a contradiction.
Without loss of generality, let incident ray a₁ strikes P and gets reflected (call this ray a₂ ) to strike the mirror Q. The ray obtained after reflection at Q (call it a₃ ) will again get reflected (since more than 3 reflections take place) at R. Call the now emerging ray a₄. Since 3 reflections have taken place until now,  a₄ has to be parallel and opposite to a₁. And since more than three reflections take place in all a₄ will strike some mirror again and hence a₁ being parallel to a₄, when extended backwards (opposite to incident direction) will also intersect some mirrror...... both a₁ and a₄ being parallel will intersect the same pair of mirrors and they are thus P (which a₁ strikes), R (which a₄ emerges from).

The situation looks like this: a₁ strikes P ( and has a direction from R to P), gets reflected, then strikes Q, then R and emerges as a₄ which moves to P to strike it again, thus a₁ and a₄ both are parallel and have same direction ( from R to P)....a contradiction to the fact that they have opposite direction.

 Q.E.D.

Ok! Now lets get a little serious......

There was a blonde, a brunette, and a redhead. They came across this mirror... Written on the mirror it said "Tell the truth and get a million bucks, tell a lie and disappear forever."
So the brunette's like "I think I'm the prettiest girl in the school"
*POOF* she disappeared.
Then the redhead steps up and says "I think I'm the smartest girl in the whole school"
*POOF* she dissappeared.
Then the blonde steps up and says"I think -"
*POOF* she dissappeared.

Allright....... that must have really made u serious!!!!

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